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zXM$WgD2a^9zmsf2BOdYUUdTPcNRl0u#vx;?lzAYH-~vE-(GAv5=d!yQ-@?b(bPn$S z_!OK^v_F!k;NkHDX^G|It0N-e`sZ}^Xm51&+-hNb+$#*J(>@;dsEh-*0)F%l+1K25 zE_M>e-&=m(mBD8ff76DQDzOcnDoWznrpnDB=Sd+LvQ+N3H%x0;U845(7gX7@qnxAZ z%anMa@~B9 zE+cW<6&n`#UwtKU^HGRH&?GZ*5<*lEe)#16_!uP!Z^Kn?d9L-H+oqae7H8STykK;wO(`|_2Yl}?AgEn%F>d4RC+G{d;aY4_lsV7 zgJeq&42#jw&+k9^^8BB?zODwE|f_|2O?SgUpQH6&el{XfqB z{Eek0{rIGLZ_l6Y|FG!l(%{<7&jvrZR_{Sc9e?)UKm6OD`K_fT{ZxDSn}05PcxRBz zs2UAgqM!cRfBMHi{JEti{rqAc%+k{HXaBkUTAtj?1Dz#zN4`?H!fHJ=UHBcHJ%9FB z-!FQazy7, |0\downarrow>, |1\uparrow>, |1\downarrow>`. -# Consequently the :math:`2\times2` and :math:`2\times2\times2\times2` block -# diagonal structures of the matrices will contain the on-site interactions. -# The converse is true for the hopping between the sites. -# From here let us generate a function to build and diagonalize the Hamiltonian. -# We need to generate the Coulomb matrix for the on-site interactions and -# apply it to the block diagonal. The hopping connects off-site indices with -# the same spin. -def diagonalize(U, t, extra_emat=None): - """Diagonalize 2 site Hubbard Hamiltonian""" - umat = np.zeros((norb, norb, norb, norb), dtype=np.complex128) - emat = np.zeros((norb, norb), dtype=np.complex128) - U_mat_1site = edrixs.get_umat_slater('s', U) - umat[:2, :2, :2, :2,] = umat[2:, 2:, 2:, 2:] = U_mat_1site - emat[2, 0] = emat[3, 1] = emat[0, 2] = emat[1, 3] = t - - if extra_emat is not None: - emat = emat + extra_emat - - H = (edrixs.build_opers(2, emat, basis) - + edrixs.build_opers(4, umat, basis)) - - e, v = scipy.linalg.eigh(H) - return e, v - -################################################################################ -# The large :math:`U` limit -# ------------------------------------------------------------------------------ -# Let us see what happens with :math:`U \gg t`. -e, v = diagonalize(1000, 1) -print("Energies are") -print(e) - -################################################################################ -# To analyze what is going on we can determine the spin expectation values -# of the cluster. Building the operators follows the same form as the -# Hamiltonian and the previous example. -spin_mom_one_site = edrixs.get_spin_momentum(ll) -spin_mom = np.zeros((3, norb, norb), dtype=np.complex128) -spin_mom[:, :2, :2] = spin_mom[:, 2:, 2:] = spin_mom_one_site - -opS = edrixs.build_opers(2, spin_mom, basis) -opS_squared = (np.dot(opS[0], opS[0]) + np.dot(opS[1], opS[1]) - + np.dot(opS[2], opS[2])) - -################################################################################ -# This time let us include a tiny magnetic field along the :math:`z`-axis, so -# that we have a well-defined measurement axis and print out the expectation -# values. -zeeman = np.zeros((norb, norb), dtype=np.complex128) -zeeman[:2, :2] = zeeman[2:, 2:] = 1e-8*spin_mom_one_site[2] -e, v = diagonalize(1000, 1, extra_emat=zeeman) - -Ssq_exp = edrixs.cb_op(opS_squared, v).diagonal().real -Sz_exp = edrixs.cb_op(opS[2], v).diagonal().real - -header = "{:<10s}\t{:<6s}\t{:<6s}" -print(header.format("E", "S(S+1)", "")) -for i in range(len(e)): - print("{:<2f}\t{:.1f}\t{:.1f}".format(e[i], Ssq_exp[i], Sz_exp[i])) - -################################################################################ -# For :math:`U \gg t` the two states with double occupancy acquire an energy of -# approximately :math:`U`. The low energy states are a :math:`S=0` singlet and -# and :math:`S=1` triplet, which are split by :math:`4t^2/U`, which is the -# magnetic exchange term. - -################################################################################ -# :math:`U` dependence -# ------------------------------------------------------------------------------ -# Let us plot the changes in energy with :math:`U`. -plt.figure() - -t = 1 -Us = np.linspace(0.01, 10, 50) -Es = np.array([diagonalize(U, t, extra_emat=zeeman)[0] for U in Us]) - -plt.plot(Us/t, Es/t) -plt.xlabel('U/t') -plt.ylabel('Eigenstate energies/t') -plt.show() - -################################################################################ -# To help interpret this, we can represent the eigenvectors in terms of a sum -# of the single particle states. - -def get_single_particle_repesentations(v): - reps = [] - for i in range(6): - rep = sum([vec*weight for weight, vec - in zip(v[:, i], np.array(basis))]) - reps.append(rep) - - return np.array(reps) - -t = 1 -for U in [10000, 0.0001]: - e, v = diagonalize(U, t, extra_emat=zeeman) - repesentations = get_single_particle_repesentations(v) - print("For U={} t={} states are".format(U, t)) - print(repesentations.round(3).real) - print("\n") - -################################################################################ -# For :math:`U \gg t` the ground state maximizes its magnetic exchange -# energy saving. In the :math:`U \ll t` condition the ground state maximizes -# its kinetic energy saving. Since both states share the same parity, the -# cross-over between them is smooth. This type of physics is at play in current -# research on quantum materials [1]_ [2]_. - -############################################################################## -# -# .. rubric:: Footnotes -# -# .. [1] Y. Wang et al., `Phys. Rev. Lett. 122, 106401 (2019) `_. -# .. [2] A. Revelli et al., `Science Advances 5, eaav4020 (2019) `_. diff --git a/edrixs/_downloads/18e0b10c5dbad68c0748b21d78984496/example_6_Hubbard_dimer.py b/edrixs/_downloads/18e0b10c5dbad68c0748b21d78984496/example_6_Hubbard_dimer.py deleted file mode 100644 index 2e1ef44302..0000000000 --- a/edrixs/_downloads/18e0b10c5dbad68c0748b21d78984496/example_6_Hubbard_dimer.py +++ /dev/null @@ -1,176 +0,0 @@ -#!/usr/bin/env python -""" -Hubbard Dimer -===================================== -This exercise will demonstrate how to handle hopping and multi-site problems within -edrixs using the example of a Hubbard dimer. We want to solve the equation - - .. math:: - \\begin{equation} - \\hat{H} = \\sum_{i,j} \\sum_{\\sigma} t_{i,j} \\hat{f}^{\\dagger}_{i,\\sigma} \\hat{f}_{j, \\sigma} - + U \\sum_{i} \\hat{n}_{i,\\uparrow}\\hat{n}_{i,\\downarrow}, - \\end{equation} - -which involves two sites labeled with indices :math:`i` or :math:`j` with two -electrons of spin :math:`\\sigma\\in{\\uparrow,\\downarrow}`. :math:`t_{i,j}` -is the hopping between sites, :math:`\\hat{f}^{\\dagger}_{i,\\sigma}` is the -creation operators, and -:math:`\\hat{n}^{\\dagger}_{i,\\sigma}=\\hat{f}^{\\dagger}_{i,\\sigma}\\hat{f}_{i,\\sigma}` -is the number operator. The main task is to represent this Hamiltonian and -the related spin operator using the EDRIXS two-fermion and four-fermion form -where :math:`\\alpha,\\beta,\\delta,\\gamma` are the indices of the single -particle basis. - - .. math:: - \\begin{equation} - \\hat{H} = \\sum_{\\alpha,\\beta} t_{\\alpha,\\beta} \\hat{f}^{\\dagger}_{\\alpha} \\hat{f}_{\\beta} - + \\sum_{\\alpha,\\beta,\\gamma,\\delta} U_{\\alpha,\\beta,\\gamma,\\delta} - \\hat{f}^{\\dagger}_{\\alpha}\\hat{f}^{\\dagger}_{\\beta}\\hat{f}_{\\gamma}\\hat{f}_{\\delta}. - \\end{equation} - -""" - -################################################################################ -# Initialize matrices -# ------------------------------------------------------------------------------ -# We start by noting that each of the two sites is like an :math:`l=0` -# :math:`s`-orbital with two spin-orbitals each. We will include -# two electron occupation and build the Fock basis. -import numpy as np -import matplotlib.pyplot as plt -import scipy -import edrixs -np.set_printoptions(precision=4) - - -ll = 0 -case = 's' -norb = 4 -noccu = 2 -basis = edrixs.get_fock_bin_by_N(norb, noccu) - -################################################################################ -# Create function to populate and diagonalize matrices -# ------------------------------------------------------------------------------ -# The Coulomb and hopping matrices :code:`umat` and :code:`emat` will be -# represented by :math:`4\times4\times4\times4` and :math:`4\times4` matrices, -# respectively. Note that we needed to specify -# that these are, in general, complex, although -# they happen to contain only real numbers in this case. We follow the convention -# that these are ordered first by site and then by spin: -# :math:`|0\uparrow>, |0\downarrow>, |1\uparrow>, |1\downarrow>`. -# Consequently the :math:`2\times2` and :math:`2\times2\times2\times2` block -# diagonal structures of the matrices will contain the on-site interactions. -# The converse is true for the hopping between the sites. -# From here let us generate a function to build and diagonalize the Hamiltonian. -# We need to generate the Coulomb matrix for the on-site interactions and -# apply it to the block diagonal. The hopping connects off-site indices with -# the same spin. -def diagonalize(U, t, extra_emat=None): - """Diagonalize 2 site Hubbard Hamiltonian""" - umat = np.zeros((norb, norb, norb, norb), dtype=np.complex128) - emat = np.zeros((norb, norb), dtype=np.complex128) - U_mat_1site = edrixs.get_umat_slater('s', U) - umat[:2, :2, :2, :2,] = umat[2:, 2:, 2:, 2:] = U_mat_1site - emat[2, 0] = emat[3, 1] = emat[0, 2] = emat[1, 3] = t - - if extra_emat is not None: - emat = emat + extra_emat - - H = (edrixs.build_opers(2, emat, basis) - + edrixs.build_opers(4, umat, basis)) - - e, v = scipy.linalg.eigh(H) - return e, v - -################################################################################ -# The large :math:`U` limit -# ------------------------------------------------------------------------------ -# Let us see what happens with :math:`U \gg t`. -e, v = diagonalize(1000, 1) -print("Energies are") -print(e) - -################################################################################ -# To analyze what is going on we can determine the spin expectation values -# of the cluster. Building the operators follows the same form as the -# Hamiltonian and the previous example. -spin_mom_one_site = edrixs.get_spin_momentum(ll) -spin_mom = np.zeros((3, norb, norb), dtype=np.complex128) -spin_mom[:, :2, :2] = spin_mom[:, 2:, 2:] = spin_mom_one_site - -opS = edrixs.build_opers(2, spin_mom, basis) -opS_squared = (np.dot(opS[0], opS[0]) + np.dot(opS[1], opS[1]) - + np.dot(opS[2], opS[2])) - -################################################################################ -# This time let us include a tiny magnetic field along the :math:`z`-axis, so -# that we have a well-defined measurement axis and print out the expectation -# values. -zeeman = np.zeros((norb, norb), dtype=np.complex128) -zeeman[:2, :2] = zeeman[2:, 2:] = 1e-8*spin_mom_one_site[2] -e, v = diagonalize(1000, 1, extra_emat=zeeman) - -Ssq_exp = edrixs.cb_op(opS_squared, v).diagonal().real -Sz_exp = edrixs.cb_op(opS[2], v).diagonal().real - -header = "{:<10s}\t{:<6s}\t{:<6s}" -print(header.format("E", "S(S+1)", "")) -for i in range(len(e)): - print("{:<2f}\t{:.1f}\t{:.1f}".format(e[i], Ssq_exp[i], Sz_exp[i])) - -################################################################################ -# For :math:`U \gg t` the two states with double occupancy acquire an energy of -# approximately :math:`U`. The low energy states are a :math:`S=0` singlet and -# and :math:`S=1` triplet, which are split by :math:`4t^2/U`, which is the -# magnetic exchange term. - -################################################################################ -# :math:`U` dependence -# ------------------------------------------------------------------------------ -# Let us plot the changes in energy with :math:`U`. -plt.figure() - -t = 1 -Us = np.linspace(0.01, 10, 50) -Es = np.array([diagonalize(U, t, extra_emat=zeeman)[0] for U in Us]) - -plt.plot(Us/t, Es/t) -plt.xlabel('U/t') -plt.ylabel('Eigenstate energies/t') -plt.show() - -################################################################################ -# To help interpret this, we can represent the eigenvectors in terms of a sum -# of the single particle states. - -def get_single_particle_repesentations(v): - reps = [] - for i in range(6): - rep = sum([vec*weight for weight, vec - in zip(v[:, i], np.array(basis))]) - reps.append(rep) - - return np.array(reps) - -t = 1 -for U in [10000, 0.0001]: - e, v = diagonalize(U, t, extra_emat=zeeman) - repesentations = get_single_particle_repesentations(v) - print("For U={} t={} states are".format(U, t)) - print(repesentations.round(3).real) - print("\n") - -################################################################################ -# For :math:`U \gg t` the ground state maximizes its magnetic exchange -# energy saving. In the :math:`U \ll t` condition the ground state maximizes -# its kinetic energy saving. Since both states share the same parity, the -# cross-over between them is smooth. This type of physics is at play in current -# research on quantum materials [1]_ [2]_. - -############################################################################## -# -# .. rubric:: Footnotes -# -# .. [1] Y. Wang et al., `Phys. Rev. Lett. 122, 106401 (2019) `_. -# .. [2] A. 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This model\nconsiders a set of correlated orbitals, often called the impurity or metal\nstates, that hybridize with a set of uncorrelated orbitals, often called the\nligands or bath states. Everyone's favorite test case for x-ray spectroscopic\ncalculations of the Anderson impurity model is NiO and we won't risk being\noriginal! This means that our correlated states will\nbe the Ni $3d$ orbitals and the uncorrelated states come from O\n$2p$ orbitals. The fact that we include these interactions means that our\nspectrum can include processes where electrons transition from the bath to the\nimpurity, as such the Anderson Impurity Model is often more accurate than atomic\nmodels, especially if the material has strong covalency.\n\nWhen defining the bath states, it is useful to use the so-called\nsymmetry adapted linear combinations of orbitals as the basis. These states\ntake into account the symmetry relationships between the different bath atom\norbitals and the fact that there are bath orbital combinations that do not\ninteract with the impurity by symmetry. By doing this the problem can be\nrepresented with fewer orbitals, which makes the calculation far more efficient.\nThe standard EDRIXS solver that we will use assumes that the bath states are\nrepresented by an integer number of bath sites set by :code:`nbath`, each of\nwhich hosts the same number of spin-orbits as the impurity e.g. 10 for a\n$d$-electron material.\n\nNiO has a rocksalt structure in which all Ni atoms are surrounded by six O\natoms. This NiO cluster used to simulate the\ncrystal would then contain 10 Ni $3d$ spin-orbitals and $6$\nspin-orbitals per O $\\times 6$ O atoms $=36$ oxygen\nspin-orbitals. As explained by, for example, Maurits Haverkort\net al. in [1]_ symmetry allows us to represent the bath with 10 symmetry\nadapted linear combinations of the different O $p_x, p_y, p_z$ states.\nThe crystal field and hopping parameters for\nsuch a calculation can be obtained by post-processing DFT calculations. We will\nuse values for NiO from [1]_. If you use values from a paper the relevant\nreferences should, of course, be cited.\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "import edrixs\nimport numpy as np\nimport matplotlib.pyplot as plt" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Number of electrons\nWhen formulating problems of this type, one usually thinks of a nominal\nvalence for the impurity atom in this case :code:`nd = 8` and assume that the\nbath is full. The solver that we will\nuse can simulate multiple bath sites. In our case we specify\n:code:`nbath = 1` sites. Electrons will be able to transition from O to Ni\nduring our calculation, but the total number of valance electrons\n:code:`v_noccu` will be conserved.\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "nd = 8\nnorb_d = 10\nnorb_bath = 10\nnbath = 1\nv_noccu = nd + nbath*norb_d\nshell_name = ('d', 'p') # valence and core shells for XAS calculation" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Coulomb interactions\nThe atomic Coulomb interactions are usually initialized based on Hartree-Fock\ncalculations from, for example,\n[Cowan's code](https://www.tcd.ie/Physics/people/Cormac.McGuinness/Cowan/).\nedrixs has a database of these.\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "info = edrixs.utils.get_atom_data('Ni', '3d', nd, edge='L3')" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The atomic values are typically scaled to account for screening in the solid.\nHere we use 80% scaling. Let's write these out in full, so that nothing is\nhidden. Values for $U_{dd}$ and $U_{dp}$ are those of Ref. [1]_\nobtained by comparing theory and experiment [2]_ [3]_.\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "scale_dd = 0.8\nF2_dd = info['slater_i'][1][1] * scale_dd\nF4_dd = info['slater_i'][2][1] * scale_dd\nU_dd = 7.3\nF0_dd = U_dd + edrixs.get_F0('d', F2_dd, F4_dd)\n\nscale_dp = 0.8\nF2_dp = info['slater_n'][4][1] * scale_dp\nG1_dp = info['slater_n'][5][1] * scale_dp\nG3_dp = info['slater_n'][6][1] * scale_dp\nU_dp = 8.5\nF0_dp = U_dp + edrixs.get_F0('dp', G1_dp, G3_dp)\n\nslater = ([F0_dd, F2_dd, F4_dd], # initial\n [F0_dd, F2_dd, F4_dd, F0_dp, F2_dp, G1_dp, G3_dp]) # with core hole" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Energy of the bath states\nIn the notation used in EDRIXS, $\\Delta$ sets the energy difference\nbetween the bath and impurity states. $\\Delta$ is defined in the atomic\nlimit without crystal field (i.e. in terms of the centers of the impurity and\nbath states before hybridization is considered) as the energy for a\n$d^{n_d} \\rightarrow d^{n_d + 1} \\underline{L}$ transition.\nNote that as electrons are moved one has to pay energy\ncosts associated with the Coulomb interactions. The\nenergy splitting between the bath and impurity is consequently not simply\n$\\Delta$. One must therefore determine the energies by solving\na set of linear equations. See the `edrixs.utils functions `\nfor details. We can call these functions to get the impurity energy\n$E_d$, bath energy $E_L$, impurity energy with a core hole\n$E_{dc}$, bath energy with a core hole $E_{Lc}$ and the\ncore hole energy $E_p$. The\n:code:`if __name__ == '__main__'` code specifies that this command\nshould only be executed if the file is explicitly run.\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "Delta = 4.7\nE_d, E_L = edrixs.CT_imp_bath(U_dd, Delta, nd)\nE_dc, E_Lc, E_p = edrixs.CT_imp_bath_core_hole(U_dd, U_dp, Delta, nd)\nmessage = (\"E_d = {:.3f} eV\\n\"\n \"E_L = {:.3f} eV\\n\"\n \"E_dc = {:.3f} eV\\n\"\n \"E_Lc = {:.3f} eV\\n\"\n \"E_p = {:.3f} eV\\n\")\nif __name__ == '__main__':\n print(message.format(E_d, E_L, E_dc, E_Lc, E_p))" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The spin-orbit coupling for the valence electrons in the ground state, the\nvalence electrons with the core hole present, and for the core hole itself\nare initialized using the atomic values.\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "zeta_d_i = info['v_soc_i'][0]\nzeta_d_n = info['v_soc_n'][0]\nc_soc = info['c_soc']" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Build matrices describing interactions\nedrixs uses complex spherical harmonics as its default basis set. If we want to\nuse another basis set, we need to pass a matrix to the solver, which transforms\nfrom complex spherical harmonics into the basis we use.\nThe solver will use this matrix when implementing the Coulomb interactions\nusing the :code:`slater` list of Coulomb parameters.\nHere it is easiest to\nuse real harmonics. We make the complex harmonics to real harmonics transformation\nmatrix via\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "trans_c2n = edrixs.tmat_c2r('d',True)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The crystal field and SOC needs to be passed to the solver by constructing\nthe impurity matrix in the real harmonic basis. For cubic symmetry, we need\nto set the energies of the orbitals along the\ndiagonal of the matrix. These need to be in pairs as there are two\nspin-orbitals for each orbital energy. Python\n[list comprehension](https://realpython.com/list-comprehension-python/)\nand\n[numpy indexing](https://numpy.org/doc/stable/reference/arrays.indexing.html)\nare used here. See `sphx_glr_auto_examples_example_1_crystal_field.py`\nfor more details if needed.\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "ten_dq = 0.56\nCF = np.zeros((norb_d, norb_d), dtype=complex)\ndiagonal_indices = np.arange(norb_d)\n\norbital_energies = np.array([e for orbital_energy in\n [+0.6 * ten_dq, # dz2\n -0.4 * ten_dq, # dzx\n -0.4 * ten_dq, # dzy\n +0.6 * ten_dq, # dx2-y2\n -0.4 * ten_dq] # dxy)\n for e in [orbital_energy]*2])\n\n\nCF[diagonal_indices, diagonal_indices] = orbital_energies" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The valence band SOC is constructed in the normal way and transformed into the\nreal harmonic basis.\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "soc = edrixs.cb_op(edrixs.atom_hsoc('d', zeta_d_i), edrixs.tmat_c2r('d', True))" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The total impurity matrices for the ground and core-hole states are then\nthe sum of crystal field and spin-orbit coupling. We further needed to apply\nan energy shift along the matrix diagonal, which we do using the\n:code:`np.eye` function which creates a diagonal matrix of ones.\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "E_d_mat = E_d*np.eye(norb_d)\nE_dc_mat = E_dc*np.eye(norb_d)\nimp_mat = CF + soc + E_d_mat\nimp_mat_n = CF + soc + E_dc_mat" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The energy level of the bath(s) is described by a matrix where the row index\ndenotes which bath and the column index denotes which orbital. Here we have\nonly one bath, with 10 spin-orbitals. We initialize the matrix to\n:code:`norb_d` and then split the energies according to :code:`ten_dq_bath`.\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "ten_dq_bath = 1.44\nbath_level = np.full((nbath, norb_d), E_L, dtype=complex)\nbath_level[0, :2] += ten_dq_bath*.6 # 3z2-r2\nbath_level[0, 2:6] -= ten_dq_bath*.4 # zx/yz\nbath_level[0, 6:8] += ten_dq_bath*.6 # x2-y2\nbath_level[0, 8:] -= ten_dq_bath*.4 # xy\nbath_level_n = np.full((nbath, norb_d), E_Lc, dtype=complex)\nbath_level_n[0, :2] += ten_dq_bath*.6 # 3z2-r2\nbath_level_n[0, 2:6] -= ten_dq_bath*.4 # zx/yz\nbath_level_n[0, 6:8] += ten_dq_bath*.6 # x2-y2\nbath_level_n[0, 8:] -= ten_dq_bath*.4 # xy" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The hybridization matrix describes the hopping between the bath\nand the impurity. This is called either $V$ or $T$ in the\nliterature and matrix sign can either be positive or negative based.\nThis is the same shape as the bath matrix. We take our\nvalues from Maurits Haverkort et al.'s DFT calculations [1]_.\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "Veg = 2.06\nVt2g = 1.21\n\nhyb = np.zeros((nbath, norb_d), dtype=complex)\nhyb[0, :2] = Veg # 3z2-r2\nhyb[0, 2:6] = Vt2g # zx/yz\nhyb[0, 6:8] = Veg # x2-y2\nhyb[0, 8:] = Vt2g # xy" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We now need to define the parameters describing the XAS. X-ray polarization\ncan be linear, circular or isotropic (appropriate for a powder).\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "poltype_xas = [('isotropic', 0)]" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "edrixs uses the temperature in Kelvin to work out the population of the low-lying\nstates via a Boltzmann distribution.\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "temperature = 300" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The x-ray beam is specified by the incident angle and azimuthal angle in radians\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "thin = 0 / 180.0 * np.pi\nphi = 0.0" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "these are with respect to the crystal field $z$ and $x$ axes\nwritten above. (That is, unless you specify the :code:`loc_axis` parameter\ndescribed in the :code:`edrixs.xas_siam_fort` function documentation.)\n\n" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The spectrum in the raw calculation is offset by the energy involved with the\ncore hole state, which is roughly $5 E_p$, so we offset the spectrum by\nthis and use :code:`om_shift` as an adjustable parameters for comparing\ntheory to experiment. We also use this to specify :code:`ominc_xas`\nthe range we want to compute the spectrum over. The core hole lifetime\nbroadening also needs to be set via :code:`gamma_c_stat`.\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "om_shift = 857.6\nc_level = -om_shift - 5*E_p\nominc_xas = om_shift + np.linspace(-15, 25, 1000)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The final state broadening is specified in terms of half-width at half-maximum\nYou can either pass a constant value or an array the same size as\n:code:`om_shift` with varying values to simulate, for example, different state\nlifetimes for higher energy states.\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "gamma_c = np.full(ominc_xas.shape, 0.48/2)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Magnetic field is a three-component vector in eV specified with respect to the\nsame local axis as the x-ray beam. Since we are considering a powder here\nwe create an isotropic normalized vector. :code:`on_which = 'both'` specifies to\napply the operator to the total spin plus orbital angular momentum as is\nappropriate for a physical external magnetic field. You can use\n:code:`on_which = 'spin'` to apply the operator to spin in order to simulate\nmagnetic order in the sample. The value of the Bohr Magneton can\nbe useful for converting here $\\mu_B = 5.7883818012\\times 10^{\u22125}$.\nFor this example, we will account for magnetic order in the sample by\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "ext_B = np.array([0.00, 0.00, 0.12])\non_which = 'spin'" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The number crunching uses\n[mpi4py](https://mpi4py.readthedocs.io/en/stable/). You can safely ignore\nthis for most purposes, but see\n[Y. L. Wang et al., Computer Physics Communications 243, 151-165 (2019)](https://doi.org/10.1016/j.cpc.2019.04.018)\nif you would like more details.\nThe main thing to remember is that you should call this script via::\n\n mpirun -n python example_AIM_XAS.py\n\nwhere :code:`` is the number of processors\nyou'd like to us. Running it as normal will work, it will just be slower.\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "if __name__ == '__main__':\n from mpi4py import MPI\n comm = MPI.COMM_WORLD\n rank = comm.Get_rank()\n size = comm.Get_size()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Calling the :code:`edrixs.ed_siam_fort` solver will find the ground state and\nwrite input files, *hopping_i.in*, *hopping_n.in*, *coulomb_i.in*, *coulomb_n.in*\nfor following XAS (or RIXS) calculation. We need to specify :code:`siam_type=0`\nwhich says that we will pass *imp_mat*, *bath_level* and *hyb*.\nWe need to specify :code:`do_ed = 1`. For this example, we cannot use\n:code:`do_ed = 0` for a ground state search as we have set the impurity and\nbath energy levels artificially, which means edrixs will have trouble to know\nwhich subspace to search to find the ground state.\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "if __name__ == '__main__':\n do_ed = 1\n eval_i, denmat, noccu_gs = edrixs.ed_siam_fort(\n comm, shell_name, nbath, siam_type=0, imp_mat=imp_mat, imp_mat_n=imp_mat_n,\n bath_level=bath_level, bath_level_n=bath_level_n, hyb=hyb, c_level=c_level,\n c_soc=c_soc, slater=slater, ext_B=ext_B,\n on_which=on_which, trans_c2n=trans_c2n, v_noccu=v_noccu, do_ed=do_ed,\n ed_solver=2, neval=50, nvector=3, ncv=100, idump=True)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Let's check that we have all the electrons we think we have and print how\nthe electron are distributed between the Ni (impurity) and O (bath).\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "if __name__ == '__main__':\n assert np.abs(noccu_gs - v_noccu) < 1e-6\n impurity_occupation = np.sum(denmat[0].diagonal()[0:norb_d]).real\n bath_occupation = np.sum(denmat[0].diagonal()[norb_d:]).real\n print('Impurity occupation = {:.6f}\\n'.format(impurity_occupation))\n print('Bath occupation = {:.6f}\\n'.format(bath_occupation))" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We see that 0.18 electrons move from the O to the Ni in the ground state.\n\nWe can now construct the XAS spectrum edrixs by applying a transition\noperator to create the excited state. We need to be careful to specify how\nmany of the low energy states are thermally populated. In this case\n:code:`num_gs=3`. This can be determined by inspecting the function output.\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "if __name__ == '__main__':\n xas, xas_poles = edrixs.xas_siam_fort(\n comm, shell_name, nbath, ominc_xas, gamma_c=gamma_c, v_noccu=v_noccu, thin=thin,\n phi=phi, num_gs=3, nkryl=200, pol_type=poltype_xas, temperature=temperature\n )" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Let's plot the data and save it just in case\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "if __name__ == '__main__':\n fig, ax = plt.subplots()\n\n ax.plot(ominc_xas, xas)\n ax.set_xlabel('Energy (eV)')\n ax.set_ylabel('XAS intensity')\n ax.set_title('Anderson impurity model for NiO')\n plt.show()\n\n np.savetxt('xas.dat', np.concatenate((np.array([ominc_xas]).T, xas), axis=1))" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - ".. rubric:: Footnotes\n\n.. [1] Maurits Haverkort et al\n [Phys. Rev. B 85, 165113 (2012)](https://doi.org/10.1103/PhysRevB.85.165113).\n.. [2] A. E. Bocquet et al.,\n [Phys. Rev. B 53, 1161 (1996)](https://doi.org/10.1103/PhysRevB.53.1161).\n.. [3] Arata Tanaka, and Takeo Jo,\n [J. Phys. Soc. Jpn. 63, 2788-2807(1994)](https://doi.org/10.1143/JPSJ.63.2788).\n\n" - ] - } - ], - "metadata": { - "kernelspec": { - "display_name": "Python 3", - "language": "python", - "name": "python3" - }, - "language_info": { - "codemirror_mode": { - "name": "ipython", - "version": 3 - }, - "file_extension": ".py", - "mimetype": "text/x-python", - "name": "python", - "nbconvert_exporter": "python", - "pygments_lexer": "ipython3", - "version": "3.10.16" - } - }, - "nbformat": 4, - "nbformat_minor": 0 -} \ No newline at end of file diff --git a/edrixs/_downloads/38cb8f43c9613932eb75463a927ea53c/example_4_GS_analysis.py b/edrixs/_downloads/38cb8f43c9613932eb75463a927ea53c/example_4_GS_analysis.py deleted file mode 100644 index 8ed75d13f0..0000000000 --- a/edrixs/_downloads/38cb8f43c9613932eb75463a927ea53c/example_4_GS_analysis.py +++ /dev/null @@ -1,170 +0,0 @@ -#!/usr/bin/env python -""" -Ground state analysis for NiO -================================================================================ -This example follows the :ref:`sphx_glr_auto_examples_example_3_AIM_XAS.py` -example and considers the same model. This time we show how to analyze -the wavevectors in terms of a -:math:`\\alpha |d^8L^{10}> + \\beta |d^9L^9> \\gamma |d^{10}L^8>` -representation. - -In doing this we will go through the exercise of building and diagonalizing -the Hamiltonian in a way that hopefully clarifies how to analyze other -properties of the model. -""" -import edrixs -import numpy as np -import matplotlib.pyplot as plt -import scipy -import example_3_AIM_XAS -import importlib -_ = importlib.reload(example_3_AIM_XAS) - -################################################################################ -# Hamiltonian -# ------------------------------------------------------------------------------ -# edrixs builds model Hamiltonians based on two fermion and four fermion terms. -# The four fermion terms come from Coulomb interactions and will be -# assigned to :code:`umat`. All other interactions contribute to two fermion -# terms in :code:`emat`. -# -# .. math:: -# \begin{equation} -# \hat{H}_{i} = \sum_{\alpha,\beta} t_{\alpha,\beta} -# \hat{f}^{\dagger}_{\alpha} \hat{f}_{\beta} -# + \sum_{\alpha,\beta,\gamma,\delta} U_{\alpha,\beta,\gamma,\delta} -# \hat{f}^{\dagger}_{\alpha}\hat{f}^{\dagger}_{\beta} -# \hat{f}_{\gamma}\hat{f}_{\delta}, -# \end{equation} -# - -################################################################################ -# Import parameters -# ------------------------------------------------------------------------------ -# Let's get the parammeters we need from the -# :ref:`sphx_glr_auto_examples_example_3_AIM_XAS.py` example. We need to -# consider :code:`ntot=20` spin-orbitals -# for this problem. - -from example_3_AIM_XAS import (F0_dd, F2_dd, F4_dd, - nd, norb_d, norb_bath, v_noccu, - imp_mat, bath_level, - hyb, ext_B, trans_c2n) -ntot = 20 -################################################################################ -# Four fermion matrix -# ------------------------------------------------------------------------------ -# The Coulomb interactions in the :math:`d` shell of this problem are described -# by a :math:`10\times10\times10\times10` matrix. We -# need to specify a :math:`20\times20\times20\times 20` matrix since we need to -# include the full problem with :code:`ntot=20` spin-orbitals. The edrixs -# convention is to put the :math:`d` orbitals first, so we assign them to the -# first :math:`10\times10\times10\times 10` indices of the matrix. edrixs -# creates this matrix in the complex harmmonic basis by default. -umat_delectrons = edrixs.get_umat_slater('d', F0_dd, F2_dd, F4_dd) -umat = np.zeros((ntot, ntot, ntot, ntot), dtype=complex) -umat[:norb_d, :norb_d, :norb_d, :norb_d] += umat_delectrons - - -################################################################################ -# Two fermion matrix -# ------------------------------------------------------------------------------ -# Previously we made a :math:`10\times10` two-fermion matrix describing the -# :math:`d`-shell interactions. Keep in mind we did this in the real harmonic -# basis. We need to specify the two-fermion matrix for -# the full problem :math:`20\times20` spin-orbitals in size. -emat_rhb = np.zeros((ntot, ntot), dtype='complex') -emat_rhb[0:norb_d, 0:norb_d] += imp_mat - -################################################################################ -# The :code:`bath_level` energies need to be applied to the diagonal of the -# last :math:`10\times10` region of the matrix. -indx = np.arange(norb_d, norb_d*2) -emat_rhb[indx, indx] += bath_level[0] - -################################################################################ -# The :code:`hyb` terms mix the impurity and bath states and are therefore -# applied to the off-diagonal terms of :code:`emat`. -indx1 = np.arange(norb_d) -indx2 = np.arange(norb_d, norb_d*2) -emat_rhb[indx1, indx2] += hyb[0] -emat_rhb[indx2, indx1] += np.conj(hyb[0]) - -################################################################################ -# We now need to transform into the complex harmonic basis. We assign -# the two diagonal blocks of a :math:`20\times20` matrix to the -# conjugate transpose of the transition matrix. -tmat = np.eye(ntot, dtype=complex) -for i in range(2): - off = i * norb_d - tmat[off:off+norb_d, off:off+norb_d] = np.conj(np.transpose(trans_c2n)) - -emat_chb = edrixs.cb_op(emat_rhb, tmat) - -################################################################################ -# The spin exchange is built from the spin operators and the effective field -# is applied to the :math:`d`-shell region of the matrix. -v_orbl = 2 -sx = edrixs.get_sx(v_orbl) -sy = edrixs.get_sy(v_orbl) -sz = edrixs.get_sz(v_orbl) -zeeman = ext_B[0] * (2 * sx) + ext_B[1] * (2 * sy) + ext_B[2] * (2 * sz) -emat_chb[0:norb_d, 0:norb_d] += zeeman - -################################################################################ -# Build the Fock basis and Hamiltonain and Diagonalize -# ------------------------------------------------------------------------------ -# We create the fock basis and build the Hamiltonian using the full set of -# :math:`20` spin orbitals, specifying that they are occuplied by :math:`18` -# electrons. See the :ref:`sphx_glr_auto_examples_example_0_ed_calculator.py` -# example for more details if needed. We also set the ground state to zero -# energy. -basis = np.array(edrixs.get_fock_bin_by_N(ntot, v_noccu)) -H = (edrixs.build_opers(2, emat_chb, basis) - + edrixs.build_opers(4, umat, basis)) - -e, v = scipy.linalg.eigh(H) -e -= e[0] - -################################################################################ -# State analysis -# ------------------------------------------------------------------------------ -# Now we have all the eigenvectors in the Fock basis, we need to pick out the -# states that have 8, 9 and 10 :math:`d`-electrons, respectively. -# The modulus squared of these coeffcients need to be added up to get -# :math:`\alpha`, :math:`\beta`, and :math:`\gamma`. - -num_d_electrons = basis[:, :norb_d].sum(1) - -alphas = np.sum(np.abs(v[num_d_electrons==8, :])**2, axis=0) -betas = np.sum(np.abs(v[num_d_electrons==9, :])**2, axis=0) -gammas = np.sum(np.abs(v[num_d_electrons==10, :])**2, axis=0) - -################################################################################ -# The ground state is the first entry. -message = "Ground state\nalpha={:.3f}\tbeta={:.3f}\tgamma={:.3f}" -print(message.format(alphas[0], betas[0], gammas[0])) - -################################################################################ -# Plot -# ------------------------------------------------------------------------------ -# Let's look how :math:`\alpha`, :math:`\beta`, and :math:`\gamma` vary with -# energy. - -fig, ax = plt.subplots() - -ax.plot(e, alphas, label=r'$\alpha$ $d^8L^{10}$') -ax.plot(e, betas, label=r'$\beta$ $d^9L^{9}$') -ax.plot(e, gammas, label=r'$\gamma$ $d^{10}L^{8}$') - -ax.set_xlabel('Energy (eV)') -ax.set_ylabel('Population') -ax.set_title('NiO') -ax.legend() -plt.show() - -################################################################################ -# We see that the ligand states are mixed into the ground state, but the -# majority of the weight for the :math:`L^9` and :math:`L^8` states -# live at :math:`\Delta` and :math:`2\Delta`. With a lot of additional -# structure from the other interactions. diff --git a/edrixs/_downloads/3c5626e47a8dbeace785112f693da4eb/example_1_crystal_field.ipynb b/edrixs/_downloads/3c5626e47a8dbeace785112f693da4eb/example_1_crystal_field.ipynb deleted file mode 100644 index 816163e15e..0000000000 --- a/edrixs/_downloads/3c5626e47a8dbeace785112f693da4eb/example_1_crystal_field.ipynb +++ /dev/null @@ -1,305 +0,0 @@ -{ - "cells": [ - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n# Crystal fields\nThis example explains how to implement crystal fields in edrixs.\n" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We need to import these modules.\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "import edrixs\nimport numpy as np\nimport scipy\n\nnp.set_printoptions(precision=2, suppress=True, linewidth=90)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Crystal field matrices\nLet us start by considering the common case of a $d$ atomic shell in a\ncubic crystal field. This is controlled by parameter $10D_q$ and is\ndescribed in terms of a matrix which we will assign to :code:`cfmat`. edrixs\ncan make this matrix via.\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "ten_dq = 10\ncfmat = edrixs.angular_momentum.cf_cubic_d(ten_dq)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Note this matrix is in a complex harmonic basis $Y^m_l$ where $m$\ngoes from $-l,-l+1,...,l-1, l$. There is an up spin and a down\nspin for each $Y^m_l$. This matrix is not diagonal in the complex\nharmonic basis, but it would be diagonal in the real harmonic basis\n$d_{3z^2-r^2}, d_{xz}, d_{yz}, d_{x^2-y^2}, d_{xy}$.\nLet us diagonalize this matrix as a check and print out the energies\nand their degeneracies.\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "e, v = scipy.linalg.eigh(cfmat)\ne = e.round(decimals=6)\nunique_e = np.unique(e)\ndegeneracies = [sum(evalue == e) for evalue in unique_e]\n\nprint(\"E \\tDegeneracy\")\nfor evalue, degenvalue in zip(unique_e, degeneracies):\n print(\"{:.1f}\\t{:.0f}\".format(evalue, degenvalue))\nprint(\"{} distinct energies\".format(len(unique_e)))" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "This makes sense! We see two different energies split by $10D_q=10$. Let\nus look at the six columns corresponding to the lower energy eigenvalues.\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "print(v[:, :6].real)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "These are the set of so-called $t_{2g}$ orbitals, composed of\n$Y^2_2, Y^{-2}_2, Y^{1}_2, Y^{-1}_2$. The rest of the eigenvectors\n(the last four) are\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "print(v[:, 6:].real)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "These are the set of so-called $e_{g}$ orbitals, composed of\n$Y^2_2, Y^{-2}_2, Y^{0}_2$. We can use edrixs to prove that\n:code:`cfmat` would be diagonal in the real\nharmonic basis. An operator $\\hat{O}$ can be transformed into an\noperator in another basis $\\hat{O}^{\\prime}$ using a unitary\ntransformation matrix $T$ as\n\n .. math::\n\n \\hat{O}^{\\prime} = (T)^{\\dagger} \\hat{O} (T).\n\nThis is computed as follows\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "cfmat_rhb = edrixs.cb_op(cfmat, edrixs.tmat_c2r('d', ispin=True))\nprint(cfmat_rhb.real)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where :code:`edrixs.tmat_c2r('d', ispin=True)` is the transformation matrix.\nWe needed to tell edrixs that we are working with a $d$-shell and that it\nshould include spin. We could also have transformed :code:`v` to see how these\neignevectors are composed of the real harmonic basis. We will see an example\nof this later.\n\n" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Crystal field on an atom\nTo simulate the solid state, we need to combine the crystal field with Coulomb\ninteractions. Let us choose an atomic model for Ni.\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "l = 2\nnorb = 10\nnoccu = 8\nbasis = edrixs.get_fock_bin_by_N(norb, noccu)\nslater = edrixs.get_atom_data('Ni', '3d', noccu, edge='L3')['slater_i']" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Let us implement a tetragonal crystal field, for which we need to pass\n:code:`d1` the splitting of $d_{yz}/d_{zx}$ and $d_{xy}$ and\n:code:`d3` the splitting of $d_{3z^2-r^2}$ and $d_{x^2-y^2}$.\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "ten_dq, d1, d3 = 2.5, 0.9, .2" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "To determine the eigenvalues and eigenvectors we need to transform both our\nCoulomb matrix and our crystal field matrix into the same basis. See the\nexample on exact diagonalization if needed. In this case, we put this\nprocedure into a function, with the option to scale the Coulomb interactions.\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "def diagonlize(scaleU=1):\n umat = edrixs.get_umat_slater('d',\n slater[0][1]*scaleU,\n slater[1][1]*scaleU,\n slater[2][1]*scaleU)\n cfmat = edrixs.angular_momentum.cf_tetragonal_d(ten_dq=ten_dq, d1=d1, d3=d3)\n H = edrixs.build_opers(2, cfmat, basis) + edrixs.build_opers(4, umat, basis)\n e, v = scipy.linalg.eigh(H)\n e = e - np.min(e) # define ground state as zero energy\n return e, v" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Let us look what happens when we run the function with the Coulomb\ninteractions switched off and check the degeneracy of the output. Look at this\npython\n[string formatting tutorial](https://realpython.com/python-formatted-output/)\nif the code is confusing.\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "e, v = diagonlize(scaleU=0)\ne = e.round(decimals=6)\nunique_e = np.unique(e)\ndegeneracies = [sum(evalue == e) for evalue in unique_e]\n\nprint(\"E \\tDegeneracy\")\nfor evalue, degenvalue in zip(unique_e, degeneracies):\n print(\"{:.1f}\\t{:.0f}\".format(evalue, degenvalue))\nprint(\"{} distinct energies\".format(len(unique_e)))" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We see 10 distinct energies, which is the number of ways one can arrange\ntwo holes among 4 energy levels -- which makes sense as the tetragonal field\ninvolves four levels $zx/zy, xy, 3z^2-r^2, x^2-y^2$. To see what is going\non in more detail, we can also calculate the expectation\nvalues of the occupancy number of the orbitals\n$3z^2-r^2, zx, zy, x^2-y^2, xy$.\nTo create the operator, first write the matrix in the real harmonics basis\n$|3z^2-r^2,\\uparrow>$, $|3z^2-r^2,\\downarrow>$,\n$|zx,\\uparrow>$, $|zx,\\downarrow>$, etc.\nIn this basis, they take a simple form: only the diagonal terms have element\n1. We therefore make a 3D empty array and assign the diagonal as 1. Check\nout the\n[numpy indexing notes](https://numpy.org/doc/stable/reference/arrays.indexing.html)\nif needed.\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "nd_real_harmoic_basis = np.zeros((norb, norb, norb), dtype=complex)\nindx = np.arange(norb)\nnd_real_harmoic_basis[indx, indx, indx] = 1" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Recalling the necessity to put everything in the same basis, we transform\ninto the complex harmonic basis and then transform into our Fock basis\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "nd_complex_harmoic_basis = edrixs.cb_op(nd_real_harmoic_basis,\n edrixs.tmat_r2c('d', True))\nnd_op = edrixs.build_opers(2, nd_complex_harmoic_basis, basis)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We apply the operator and print out as follows. Check the\n[numpy docs](https://numpy.org/doc/1.18/reference/generated/numpy.reshape.html)\nif the details of how the spin pairs have been added up is not immediately\ntransparent.\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "nd_expt = np.array([edrixs.cb_op(nd_vec, v).diagonal().real for nd_vec in nd_op])\n\nmessage = \"{:>3s}\" + \"\\t{:>6s}\"*5\nprint(message.format(*\"E 3z2-r2 zx zy x2-y2 xy\".split(\" \")))\n\nmessage = \"{:>3.1f}\" + \"\\t{:>6.1f}\"*5\nfor evalue, row in zip(e, nd_expt.T):\n spin_pairs = row.reshape(-1, 2).sum(1)\n print(message.format(evalue, *spin_pairs))" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The lowest energy state involves putting both holes in the $x^2-y^2$\norbital, which makes sense. Now, let us redo the proceedure including Coulomb\nrepulsion, which imposes an energy cost to putting multiple electrons in the\nsame orbital.\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "e, v = diagonlize(scaleU=1)\n\nnd_expt = np.array([edrixs.cb_op(nd_vec, v).diagonal().real for nd_vec in nd_op])\n\nmessage = \"{:>3s}\" + \"\\t{:>6s}\"*5\nprint(message.format(*\"E 3z2-r2 zx zy x2-y2 xy\".split(\" \")))\n\nmessage = \"{:>3.1f}\" + \"\\t{:>6.1f}\"*5\nfor evalue, row in zip(e, nd_expt.T):\n spin_pairs = row.reshape(-1, 2).sum(1)\n print(message.format(evalue, *spin_pairs))" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Now the lowest energy state involves splitting the holes between the two\nhighest energy $x^2-y^2$ and $3z^2-r^2$ orbitals. i.e. we have\ngone from low-spin to high spin. Working out the balance between these two\nstates is often one of the first things one wants to determine upon the\ndiscovery of an interesting new material [1]_.\n\n" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - ".. rubric:: Footnotes\n\n.. [1] M. Rossi et al., [arXiv:2011.00595 (2021)](https://arxiv.org/abs/2011.00595).\n\n" - ] - } - ], - "metadata": { - "kernelspec": { - "display_name": "Python 3", - "language": "python", - "name": "python3" - }, - "language_info": { - "codemirror_mode": { - "name": "ipython", - "version": 3 - }, - "file_extension": ".py", - "mimetype": "text/x-python", - "name": "python", - "nbconvert_exporter": "python", - "pygments_lexer": "ipython3", - "version": "3.10.16" - } - }, - "nbformat": 4, - "nbformat_minor": 0 -} \ No newline at end of file diff --git a/edrixs/_downloads/3d289608165bb541cc547b18d44b95f6/example_8_Coulomb.zip b/edrixs/_downloads/3d289608165bb541cc547b18d44b95f6/example_8_Coulomb.zip deleted file mode 100644 index db96ac6f963954f075d6e9439193693a59986016..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 45176 zcmeHQO>-N`c~;^i2f8XJSLNEFP%F|RA%4l;WtoSScIB1VAF7qs&Xz_32f%=s6)=OH z8Hj{Px+F zPrm%Y-~H$o{rvgg|M8nY{^bvU`r&VH-J<^~H8}Byqe0O6xb-j|58`3FIhx%5=X>Kc zxz`S(dqMPCjV4(?j#gGzS66f%6-HT*_?;|_qjcq9mY0VW?vwJPuKN75 zuO9!)Q_3{*lPsi%NmKnUP<}XE$zs*P_pmoXw*u9Rlc9=x{CVhSN!SU}4ekh@PN-q3 z#2*HD^d{(TtWX2gPb%C_0{>MQ9Z?18s2`xTKTzppNZBXoSftY9FzfVbaDKNNQXi2c zkb$HtS>Mm_3^NfP^)nU483uqJha*1=n(8;>#7-M#u9c90zh0YvCH)~h)PQl8m% zppKI;%ka=oS8k&YrJ7CZVt;?1zbWOlgQGAy4Za(*c$}MLO#A5<=juRtJxKHE_W9|K zeFhO%U5~11_fC^@<&_d%dZQ#92BxZ~;nC1UY|)OE*<)~MjADi zK*GW#=%OMqIRnh!NvH2eM}gB2PY?;Mlrmz;Lq)$}1AC~!SCHv6PR#UveYmi6r{f_+ z(;tlb{swd*!@o!V5OV5ug8_XLsjU^-uT4pliOLsp>dH!KT(hJZNoIlToxeQwy8h8o zkenB*o|diBXel#l;t>@(ajqN0L$^0G>jzI5+Ew#s5)eGA**W5kHUT^9&SCCK@!_X5EI86Y+9vF zCutT8S$O-Oa)yyq||(i#={YucB-P0NlM{NC*}*P9S!1a5E9yIR8=4RGVbf~x&a+zs{*O;-=K6IkuE`>SM>qApEmg1PU+ zgso0g-%o~Y8-X-Jz#v&*?GL&^&nN54&uIMwW;Ivd#>k!s<|2bmcDVpM-vav3h*2Xw zb|c~h-G*t*TC%@BRIfu{J>{k5-W?_W05fT8NKMv9FHc{cA2tc#z*!^#6=ClIAWMbF z=<}Mz<1`16sOk&77%(9ze9&M(zaI{QcG~H0{ya{)une6zAg34t6=S)6?vH_83Tdc^ z0PF9WvfVBMC_oz;^~w z2e|ZcPfg-68!Dcl0JerfIjOG2^F6}JD5xj58^cVnoWh=UxdL+a5LJfR(Vc7^tqIIf z`}Ar5u<2;jA(?Nu=QK;kolKU;_+4W!Xi@T6fc?e1s9q8eQPfmPih4vdMT-Jd=?|c; z5iBjNq+LGk{4w`{O;8W01sN$UqUwc73cG=H8tqqp$M5ImtkSSEK#3`sH1dlWPeJN5X@z5!;n`fZ$ z(2pl*(e?((M>9&nHEB-v`)YegQ`u>(X0WQsKEBlU|x3Zf^bFE1V1xK*9ZqDhz*0biMwxJ z?f~(_WuCm*P!s%rf`kbtOn{acNx;?>R|Rbw05pd7CGj!#P~G^L)-gU7tbrAR&NbC9 zgX|96b&#+$vMDL{n`|8_WSdi%V+IeOf9K>4{X5}*^%QXOrWWQ+a9 z`I-K%Yjd_fR9W42$Ae;`c~h>scP3SW3qVv%;JB4xAfhm0r-t1g{3|>Q_LZGjq>hC4 zI83QxgUWMwUceRDUT}trxxM|3m})4p2mVk3{ukq$;fA&h(m>w>yiAUI@<-LzE>0C0bB>`!2S+QH;f1X2wDEhDm_4$wFHp~~_g z#fAi13egB{%t~oa)B=G>r!psEQ6FH*ThppvXtY?F(@K3e*vOQ+mN#%-0*Q7WS!TA$ z6_n)krb5h^~{JCFPCB*zIn{oz9r{{xBO_dq^Dx{fkLJZG-TY z4TV&sK&ApN#<+~zj1fvfupqDVy=#_C=blM_Y zAnGdHFtQ{Rw?w}9vaqDZ;#yGz_siDhQwq7OyK3hF{*k2Jhm>XkKNX@akvzn%!IYq_ z!d-AL>oK{g%G znXZW--bdJV=K@Lm)<~lL0NI>X63sF^|Mq*6&Rs2?CIp?In79xF6SZ{l{M;PP2viJC zXrstMZnATP=H`<-N62bqaT#(4ILFZaBDs6U(Zw8q%D}9I8RI72X4&#)O&|pCJ(jn< zrg}oVmpax-{t_Kjj($+nt(3`7D#qOWoQOu1GL(q@9DwRR5OCLF+9JMA__vYN}W zWUfGobA7)h>ppD*lfd2KE!}2)3fxFL@)S2-Xit<2S@HfzNZ2t1!lAIby-R}CmOW-h z{lxkid(U3zG9YJRKLH@#@i-O_@j7Rhfk>CL9inf}Lx3I=#x6^BAz!Iju}-Iwiq-JeiJ?aET%`Njbzrw+)aF(H0z> z;7Ik`SA=b!KL^|epxwp4FE$v&(p*w1@Geq!+*D-5gK}P^N)20XUKE$Sg&L45^=`dD zfRvFppwNbTwV~)3T~W8{v9|>}uMG4VT*U*e+OaC-0`^xqBrU>Q5>% z2|?L&B}ya}XY&P#5=n&smh)2as#H@_wFauRl$WHOYbdlv8d0qmsWtUVD%bmNWnJa_ zQ4#N=@YCk>9mx9UIH9AnT{yTJVT2Ga{p9Q!?vdg6nr$!6(oO4RJ!^UZizH`IbajOC z1mO`P{}lVDkajwvD|5bVB?uwWo+2J+byew`p%mwd*kH?z9a|RFnbN-MjN5dwkl<1e z!VkSL7$A!6#DNQUfcPq?)Pq}-sK?^Y<_b=@THWtbvht!NB%?bJMVw_I*y|tB{Z!n@ zS(A(#9O%&d%>Wn=1@)O&r`?K2YvxO@#a(Q5b;0%AZ?GeF{;54v3n@Z%pikB)^A~8z z)ZQM4gDzSNl5`FIrP#}cTtivUE%sg0F*i39je2czOZH$xy+-YPF{nAfsm|c2iFNF+ zeId`2AR8wD$~aUHnm8y~LvThez1YFi9fDrmdDX(9P^&#@{d#SirgL5H1x0a-Wk)Lz z+KNWl%#S8~vbetT04a}l?(Kbo7ViJ}(cV3r^0VKg*b*Gxb7W7yOR}}y^&daDhcUR{ z;cF@Jg?L0hsw}nld=2^ib}FrHQp9Rq{Y-5Kn?HGhmbc(z!z0A{PDl)J`Kg(X+jJKy zU4wBO1d(1sy5WV4bU*80?PD~HGf(bi2Nrt+m(or)pgJT@>(uE0cdzJjWU%%o9N}SV zo~G+se*BYWAJZMAYZnR{&WthP2bjx~p?`9)U6vv~5IMLLM*wfwvc$d^Te zS-i^|X_cm{BC6bl?dE;mi5+~(#f$~nbfee1Q+Y%K%qD~2;LiTEP6EcavM?J22n{y2 zw?1oqcW^4RdEP)Ykeb0zDA!~V4%ePB#Q{2hH8}6)f)atrxm; zbVRjw$E|-Y`0YAYqpp%H9vw74USDCU!|fpI=#nq1(_z+c1WONso;m-AoWo+*g+EDl zYwzyM@fvi4uF7R0RC9PjVFTIG8U^t1!IMS zMx0JVBk7vpFywL2K?6jg)8}i;Sa{ykQ?*1-QQQw_&Ot6+_Ck#!3({6 zRd=U?gPNx`NnAxGy^=uL?xK_UIs1*iS7HFNd)@l*%%Wx-&&pk~Gn0{zZ1r6#sz_%= z@I`0L6P#Sq>fNcY-qbv;N#ZIh>6HY^X2z;}l<#*>-W<{-@Q)e5r}uX5KDhhv9&QSo zB%IzL5eGM$+dDY6Ika;CPsTa(j5|o#zE7EyzROWN`*y8sp4KESu&$M{VhgWFnlILe zI#wOQ-~pSp!B{tUWMNGbuj?k3bm#6audALkoh_dy&ce$Iv3Sqc^<+}Jt{>1?=E4?55SRLv8kjLs= zD0Kw+!<1nj>tD&b+E}@3hAnU)|627qclXFN?c!fT2?kFw;(!Vmnb;*1K`4ZN-kHxS zVO$%fzX;>*E)C;eWLE2}vQ1m4&`emk`0C-IStU>3{*@CYI-WVxM@AAcmfN`+6VF>n zaI;zaIWTiA3Z7b?Q1WFNYM)QR*4C5Zg9AQuk@qM}bh>*Ew=81LvR?m>)e}>QUH9hw z{Hs+Krvfu~RzNQQu#gBXRLIJmfhYDA1jk#@y9mzY*lE}_1D|2?s)YpiOrN}KHFt!Y zoPC{rb>ot`b7t0&8uw%E+}(9&uC9)2H+K&1%qP+rnVp6?#ICC7AD+&Vrq;+^z0TUX zb0*f6HTQF6!U;26CU&#ogrW^x$aF1KsBG_O?h2O9Or$eCb{cN%9YVtrSk&>xJ#%B? 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We aim\nto clarify the form of the matrices, how they are parametrized,\nand how the breaking of spherical symmetry can switch on additional elements\nthat one might not anticipate. Our example is based on a $d$ atomic shell.\n" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Create matrix\nThe Coulomb interaction between two particles can be written as\n\n .. math::\n \\begin{equation}\n \\hat{H} = \\frac{1}{2}\n \\int d\\mathbf{r} \\int d\\mathbf{r}^\\prime\n \\Sigma_{\\sigma, \\sigma^\\prime}\n |\\hat{\\psi}^\\sigma(\\mathbf{r})|^2 \\frac{e^2}{R}\n |\\hat{\\psi}^{\\sigma^\\prime}(\\mathbf{r})|^2,\n \\end{equation}\n\nwhere $\\hat{\\psi}^\\sigma(\\mathbf{r})$ is the electron wavefunction, with\nspin $\\sigma$, and $R=|r-r^\\prime|$ is the electron separation.\nSolving our problem in this form is difficult due to the need to symmeterize\nthe wavefunction to follow fermionic statistics.\nUsing second quantization, we can use operators to impose the required\nparticle exchange statistics and write the equation in terms of\na tensor $U$\n\n .. math::\n \\begin{equation}\n \\hat{H} = \\sum_{\\alpha,\\beta,\\gamma,\\delta,\\sigma,\\sigma^\\prime}\n U_{\\alpha\\sigma,\\beta\\sigma^\\prime,\\gamma\\sigma^\\prime,\\delta\\sigma}\n \\hat{f}^{\\dagger}_{\\alpha\\sigma}\n \\hat{f}^{\\dagger}_{\\beta\\sigma^\\prime}\n \\hat{f}_{t\\sigma^\\prime}\\hat{f}_{\\delta\\sigma},\n \\end{equation}\n\nwhere $\\alpha$, $\\beta$, $\\gamma$, $\\delta$ are\norbital indices and $\\hat{f}^{\\dagger}$\n($\\hat{f}$) are the creation (anihilation) operators.\nFor a $d$-electron system, we have $10$ distinct spin-orbitals\n($5$ orbitals each with $2$ spins), which makes matrix the\n$10\\times10\\times10\\times10$ in total size.\nIn EDRIXS the matrix can be created as follows:\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "import edrixs\nimport numpy as np\nimport scipy\nimport matplotlib.pyplot as plt\nimport itertools\n\nF0, F2, F4 = 6.94, 14.7, 4.41\numat_chb = edrixs.get_umat_slater('d', F0, F2, F4)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We stored this under variable :code:`umat_chb` where \"cbh\" stands for\ncomplex harmonic basis, which is the default basis in EDRIXS.\n\n" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Parameterizing interactions\nEDRIXS parameterizes the interactions in $U$ via Slater integral\nparameters $F^{k}$. These relate to integrals of various spherical\nHarmonics as well as Clebsch-Gordon coefficients, Gaunt coefficients,\nand Wigner 3J symbols. Textbooks such as [1]_ can be used for further\nreference. If you are interested in the details of how\nEDRIXS does this (and you probably aren't) function :func:`.umat_slater`,\nconstructs the required matrix via Gaunt coeficents from\n:func:`.get_gaunt`. Two alternative parameterizations are common.\nThe first are the Racah parameters, which are\n\n .. math::\n \\begin{eqnarray}\n A &=& F^0 - \\frac{49}{441} F^4 \\\\\n B &=& \\frac{1}{49}F^2 - \\frac{5}{441}F^4 \\\\\n C &=& \\frac{35}{441}F^4.\n \\end{eqnarray}\n\nor an alternative form for the Slater integrals\n\n .. math::\n \\begin{eqnarray}\n F_0 &=& F^0 \\\\\n F_2 &=& \\frac{1}{49}F^2 \\\\\n F_4 &=& \\frac{1}{441}F^4,\n \\end{eqnarray}\n\nwhich involves different normalization parameters.\n\n" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Basis transform\nIf we want to use the real harmonic basis, we can use a tensor\ntransformation, which imposes the following orbital order\n$3z^2-r^2, xz, yz, x^2-y^2, xy$, each of which involves\n$\\uparrow, \\downarrow$ spin pairs. Let's perform this transformation and\nstore a list of these orbitals.\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "umat = edrixs.transform_utensor(umat_chb, edrixs.tmat_c2r('d', True))\norbitals = ['3z^2-r^2', 'xz', 'yz', 'x^2-y^2', 'xy']" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Interactions\nTensor $U$ is a series of matrix\nelements\n\n .. math::\n \\begin{equation}\n \\langle\\psi_{\\gamma,\\delta}^{\\bar{\\sigma},\\bar{\\sigma}^\\prime}\n |\\hat{H}|\n \\psi_{\\alpha,\\beta}^{\\sigma,\\sigma^\\prime}\\rangle\n \\end{equation}\n\nthe combination of which defines the energetic cost of pairwise\nelectron-electron interactions between states $\\alpha,\\sigma$\nand $\\beta,\\sigma^\\prime$. In EDRIXS we follow the convention of\nsumming over all orbital pairs. Some other texts count each pair of\nindices only once. The matrix elements here will consequently\nbe half the magnitude of those in other references.\nWe can express the interactions in terms of\nthe orbitals involved. It is common to distinguish \"direct Coulomb\" and\n\"exchange\" interactions. The former come from electrons in the same orbital\nand the later involve swapping orbital labels for electrons. We will use\n$U_0$ and $J$ as a shorthand for distinguishing these.\n\nBefore we describe the different types of interactions, we note that since\nthe Coulomb interaction is real, and due to the spin symmmetry properties\nof the process $U$ always obeys\n\n .. math::\n \\begin{equation}\n U_{\\alpha\\sigma,\\beta\\sigma^\\prime,\\gamma\\sigma^\\prime,\\delta\\sigma} =\n U_{\\beta\\sigma,\\alpha\\sigma^\\prime,\\delta\\sigma^\\prime,\\gamma\\sigma} =\n U_{\\delta\\sigma,\\gamma\\sigma^\\prime,\\beta\\sigma^\\prime,\\alpha\\sigma} =\n U_{\\gamma\\sigma,\\delta\\sigma^\\prime,\\alpha\\sigma^\\prime,\\beta\\sigma}.\n \\end{equation}\n\n\n### 1. Intra orbital\nThe direct Coulomb energy cost to double-occupy an orbital comes from terms\nlike $U_{\\alpha\\sigma,\\alpha\\bar\\sigma,\\alpha\\bar\\sigma,\\alpha\\sigma}$.\nIn this notation, we use $\\sigma^\\prime$ to denote that the matrix\nelement is summed over all pairs and $\\bar{\\sigma}$ to denote sums\nover all opposite spin pairs. Due to rotational symmetry, all these\nelements are the same and equal to\n\n .. math::\n \\begin{eqnarray}\n U_0 &=& \\frac{A}{2} + 2B + \\frac{3C}{2}\\\\\n &=& \\frac{F_0}{2} + 2F_2 + 18F_4\n \\end{eqnarray}\n\nLet's print these to demonstrate where these live in the array\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "for i in range(0, 5):\n val = umat[i*2, i*2 + 1, i*2 + 1, i*2].real\n print(f\"{orbitals[i]:<8} \\t {val:.3f}\")" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "### 2. Inter orbital Coulomb interactions\nDirect Coulomb repulsion between different orbitals depends on terms like\n$U_{\\alpha\\sigma,\\beta\\sigma^\\prime,\\beta\\sigma^\\prime,\\alpha\\sigma}$.\nExpresions for these parameters are provided in column $U$ in\n`table_2_orbital`. We can print the values from :code:`umat`\nlike this:\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "for i, j in itertools.combinations(range(5), 2):\n val = umat[i*2, j*2 + 1, j*2 + 1, i*2].real\n print(f\"{orbitals[i]:<8} \\t {orbitals[j]:<8} \\t {val:.3f}\")" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "### 3. Inter-orbital exchange interactions\nExchange terms exist with the form\n$U_{\\alpha\\sigma,\\beta\\sigma^\\prime,\\alpha\\sigma^\\prime,\\beta\\sigma}$.\nExpresions for these parameters are provided in column $J$ of\n`table_2_orbital`. These come from terms like this in the matrix:\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "for i, j in itertools.combinations(range(5), 2):\n val = umat[i*2, j*2 + 1, i*2 + 1, j*2].real\n print(f\"{orbitals[i]:<8} \\t {orbitals[j]:<8} \\t {val:.3f}\")" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "### 4. Pair hopping term\nTerms that swap pairs of electrons exist as\n$(1-\\delta_{\\sigma\\sigma'})U_{\\alpha\\sigma,\\alpha\\bar\\sigma,\\beta\\bar\\sigma,\\beta\\sigma}$\nand depend on exchange interactions column $J$ from\n`table_2_orbital`\nand here in the matrix.\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "for i, j in itertools.combinations(range(5), 2):\n val = umat[i*2, i*2 + 1, j*2 + 1, j*2].real\n print(f\"{orbitals[i]:<8} \\t {orbitals[j]:<8} \\t {val:.3f}\")" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "### 5. Three orbital\nAnother set of terms that one might not immediately anticipate involve three\norbitals like\n\n .. math::\n \\begin{equation}\n U_{\\alpha\\sigma, \\gamma\\sigma', \\beta\\sigma', \\gamma\\sigma} \\\\\n U_{\\alpha\\sigma, \\gamma\\sigma', \\gamma\\sigma', \\beta\\sigma} \\\\\n (1-\\delta_{\\sigma\\sigma'})\n U_{\\alpha\\sigma, \\beta\\sigma', \\gamma\\sigma', \\gamma\\sigma}\n \\end{equation}\n\nfor $\\alpha=3z^2-r^2, \\beta=x^2-y^2, \\gamma=xz/yz$.\nThese are needed to maintain the rotational symmetry of the interations.\nSee `table_3_orbital` for the expressions. We can print some of\nthese via:\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "ijkl = [[0, 1, 3, 1],\n [0, 2, 3, 2],\n [1, 0, 3, 1],\n [1, 1, 3, 0],\n [2, 0, 3, 2],\n [2, 2, 3, 0]]\n\nfor i, j, k, l in ijkl:\n val = umat[i*2, j*2 + 1, k*2 + 1, l*2].real\n print(f\"{orbitals[i]:<8} \\t {orbitals[j]:<8} \\t\"\n f\"{orbitals[k]:<8} \\t {orbitals[l]:<8} \\t {val:.3f}\")" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "### 6. Four orbital\nFuther multi-orbital terms include\n$U_{\\alpha\\sigma,\\beta\\sigma^\\prime,\\gamma\\sigma^\\prime,\\delta\\sigma}$.\nWe can find these here in the matrix:\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "ijkl = [[0, 1, 2, 4],\n [0, 1, 4, 2],\n [0, 2, 1, 4],\n [0, 2, 4, 1],\n [0, 4, 1, 2],\n [0, 4, 2, 1],\n [3, 1, 4, 2],\n [3, 2, 4, 1],\n [3, 4, 1, 2],\n [3, 4, 2, 1]]\n\nfor i, j, k, l in ijkl:\n val = umat[i*2, j*2 + 1, k*2 + 1, l*2].real\n print(f\"{orbitals[i]:<8} \\t {orbitals[j]:<8} \\t {orbitals[k]:<8}\"\n f\"\\t {orbitals[l]:<8} \\t {val:.3f}\")" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Effects of multi-orbital terms\nTo test the effects of the multi-orbital terms, let's plot the eigenenergy\nspectra with and without multi-orbital terms switched on for system with and\nwithout a cubic crystal field. We will use a $d$-shell with two\nelectrons.\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "ten_dqs = [0, 2, 4, 12]\n\ndef diagonalize(ten_dq, umat):\n emat = edrixs.cb_op(edrixs.cf_cubic_d(ten_dq),\n edrixs.tmat_c2r('d', ispin=True))\n H = (edrixs.build_opers(4, umat, basis)\n + edrixs.build_opers(2, emat, basis))\n e, v = scipy.linalg.eigh(H)\n return e - e.min()\n\nbasis = edrixs.get_fock_bin_by_N(10, 2)\numat_no_multiorbital = np.copy(umat)\nB = F2/49 - 5*F4/441\nfor val in [np.sqrt(3)*B/2, np.sqrt(3)*B, 3*B/2]:\n umat_no_multiorbital[(np.abs(umat)- val) < 1e-6] = 0\n\nfig, axs = plt.subplots(1, len(ten_dqs), figsize=(8, 3))\n\nfor cind, (ax, ten_dq) in enumerate(zip(axs, ten_dqs)):\n ax.hlines(diagonalize(ten_dq, umat), xmin=0, xmax=1,\n label='on', color=f'C{cind}')\n ax.hlines(diagonalize(ten_dq, umat_no_multiorbital),\n xmin=1.5, xmax=2.5,\n label='off',\n linestyle=':', color=f'C{cind}')\n ax.set_title(f\"$10D_q={ten_dq}$\")\n ax.set_ylim([-.5, 20])\n ax.set_xticks([])\n ax.legend()\n\nfig.suptitle(\"Eigenvalues with 3&4-orbital effects on/off\")\nfig.subplots_adjust(wspace=.3)\naxs[0].set_ylabel('Eigenvalues (eV)')\nfig.subplots_adjust(top=.8)\nplt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "On the left of the plot Coulomb interactions in spherical symmetry cause\nsubstantial mxing between $t_{2g}$ and $e_{g}$ orbitals in the\neigenstates and 3 & 4 orbital orbital terms are crucial for obtaining the\nthe right eigenenergies. As $10D_q$ get large, this mixing is switched\noff and the spectra start to become independent of whether the 3 & 4 orbital\norbital terms are included or not.\n\n\n\n.. table:: Table of 2 orbital interactions\n\n +-----------------------------+------------------+-----------------------+----------------+--------------------+\n |Orbitals $\\alpha,\\beta$|$U_0$ Racah | $U_0$ Slater |$J$ Racah |$J$ Slater |\n +=============================+==================+=======================+================+====================+\n |$3z^2-r^2, xz$ |$A/2+B+C/2$ |$F_0/2+F_2-12F_4$| $B/2+C/2$|$F_2/2+15F_4$ |\n +-----------------------------+------------------+-----------------------+----------------+--------------------+\n |$3z^2-r^2, yz$ |$A/2+B+C/2$ |$F_0/2+F_2-12F_4$| $B/2+C/2$|$F_2/2+15F_4$ |\n +-----------------------------+------------------+-----------------------+----------------+--------------------+\n |$3z^2-r^2, x^2-y^2$ |$A/2-2B+C/2$|$F_0/2-2F_2+3F_4$|$2B+C/2$ |$2F_2+15F_4/2$|\n +-----------------------------+------------------+-----------------------+----------------+--------------------+\n |$3z^2-r^2, xy$ |$A/2-2B+C/2$|$F_0/2-2F_2+3F_4$|$2B+C/2$ |$2F_2+15F_4/2$|\n +-----------------------------+------------------+-----------------------+----------------+--------------------+\n |$xz, yz$ |$A/2-B+C/2$ |$F_0/2-F_2-12F_4$|$3B/2+C/2$|$3F_2/2+10F_4$|\n +-----------------------------+------------------+-----------------------+----------------+--------------------+\n |$xz, x^2-y^2$ |$A/2-B+C/2$ |$F_0/2-F_2-12F_4$|$3B/2+C/2$|$3F_2/2+10F_4$|\n +-----------------------------+------------------+-----------------------+----------------+--------------------+\n |$xz, xy$ |$A/2-B+C/2$ |$F_0/2-F_2-12F_4$|$3B/2+C/2$|$3F_2/2+10F_4$|\n +-----------------------------+------------------+-----------------------+----------------+--------------------+\n |$yz, x^2-y^2$ |$A/2-B+C/2$ |$F_0/2-F_2-12F_4$|$3B/2+C/2$|$3F_2/2+10F_4$|\n +-----------------------------+------------------+-----------------------+----------------+--------------------+\n |$yz, xy$ |$A/2-B+C/2$ |$F_0/2-F_2-12F_4$|$3B/2+C/2$|$3F_2/2+10F_4$|\n +-----------------------------+------------------+-----------------------+----------------+--------------------+\n |$x^2-y^2, xy$ |$A/2+2B+C/2$|$F_0+4F_2-34F_4$ | $C/2$ |$35F_4/2$ |\n +-----------------------------+------------------+-----------------------+----------------+--------------------+\n\n\n.. table:: Table of 3 orbital interactions\n\n +-----------------------------+-------------+----------------------------------------------------+-----------------------------------------------------+\n |Orbitals $\\alpha,\\beta,\\gamma,\\delta$|$\\langle\\alpha\\beta|\\gamma\\delta\\rangle$ Racah|$\\langle\\alpha\\beta|\\gamma\\delta\\rangle$ Slater|\n +=============================+=============+====================================================+=====================================================+\n |$3z^2-r^2, xz, x^2-y^2, xz$ | $\\sqrt{3}B/2$ | $\\sqrt{3}F_2/2-5\\sqrt{3}F_4/2$ |\n +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+\n |$3z^2-r^2, yz, x^2-y^2, yz$ | $-\\sqrt{3}B/2$ | $-\\sqrt{3}F_2/2+5\\sqrt{3}F_4/2$ |\n +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+\n |$xz, 3z^2-r^2, x^2-y^2, xz$ | $-\\sqrt{3}B$ | $-\\sqrt{3}F_2+5\\sqrt{3}F_4$ |\n +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+\n |$xz, xz, x^2-y^2, 3z^2-r^2$ | $\\sqrt{3}B/2$ | $\\sqrt{3}F_2/2-5\\sqrt{3}F_4/2$ |\n +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+\n |$yz, 3z^2-r^2, x^2-y^2, yz$ | $\\sqrt{3}B$ | $\\sqrt{3}F_2-5\\sqrt{3}F_4$ |\n +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+\n |$yz, yz, x^2-y^2, 3z^2-r^2$ | $-\\sqrt{3}B/2$ | $-\\sqrt{3}F_2/2+5\\sqrt{3}F_4/2$ |\n +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+\n\n\n.. table:: Table of 4 orbital interactions\n\n +-----------------------------+-------------+----------------------------------------------------+-----------------------------------------------------+\n |Orbitals $\\alpha,\\beta,\\gamma,\\delta$|$\\langle\\alpha\\beta|\\gamma\\delta\\rangle$ Racah|$\\langle\\alpha\\beta|\\gamma\\delta\\rangle$ Slater|\n +=============================+=============+====================================================+=====================================================+\n |$3z^2-r^2, xz, yz, xy$ | $-\\sqrt{3}B$ | $-\\sqrt{3}F_2+5\\sqrt{3}F_4$ |\n +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+\n |$3z^2-r^2, xz, xy, yz$ | $\\sqrt{3}B/2$ | $\\sqrt{3}F_2/2-5\\sqrt{3}F_4/2$ |\n +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+\n |$3z^2-r^2, yz, xz, xy$ | $-\\sqrt{3}B$ | $-\\sqrt{3}F_2+5\\sqrt{3}F_4$ |\n +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+\n |$3z^2-r^2, yz, xy, xz$ | $\\sqrt{3}B/2$ | $\\sqrt{3}F_2/2-5\\sqrt{3}F_4/2$ |\n +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+\n |$3z^2-r^2, xy, xz, yz$ | $\\sqrt{3}B/2$ | $\\sqrt{3}F_2/2-5\\sqrt{3}F_4/2$ |\n +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+\n |$3z^2-r^2, xy, yz, xz$ | $\\sqrt{3}B/2$ | $\\sqrt{3}F_2/2-5\\sqrt{3}F_4/2$ |\n +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+\n |$x^2-y^2 , xz, xy, yz$ | $-3B/2$ | $-3F_2/2+15F_4/2$ |\n +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+\n |$x^2-y^2 , yz, xy, xz$ | $3B/2$ | $3F_2/2-15F_4/2$ |\n +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+\n |$x^2-y^2 , xy, xz, yz$ | $-3B/2$ | $-3F_2/2+15F_4/2$ |\n +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+\n |$x^2-y^2 , xy, yz, xz$ | $3B/2$ | $3F_2/2-15F_4/2$ |\n +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+\n\n\n.. rubric:: Footnotes\n\n.. [1] MSugano S, Tanabe Y and Kamimura H. 1970. Multiplets of\n Transition-Metal Ions in Crystals. Academic Press, New York and London.\n\n" - ] - } - ], - "metadata": { - "kernelspec": { - "display_name": "Python 3", - "language": "python", - "name": "python3" - }, - "language_info": { - "codemirror_mode": { - "name": "ipython", - "version": 3 - }, - "file_extension": ".py", - "mimetype": "text/x-python", - "name": "python", - "nbconvert_exporter": "python", - "pygments_lexer": "ipython3", - "version": "3.8.12" - } - }, - "nbformat": 4, - "nbformat_minor": 0 -} \ No newline at end of file diff --git a/edrixs/_downloads/459c0482b0e43beaf4cd18b2100f6bba/helloworld.py b/edrixs/_downloads/459c0482b0e43beaf4cd18b2100f6bba/helloworld.py deleted file mode 100644 index 7eb3ecd6b9..0000000000 --- a/edrixs/_downloads/459c0482b0e43beaf4cd18b2100f6bba/helloworld.py +++ /dev/null @@ -1,122 +0,0 @@ -#!/usr/bin/env python - -import numpy as np -import matplotlib.pyplot as plt -import matplotlib as mpl -import edrixs - -# Setup parameters -# ---------------- -# Number of occupancy of 3d shell -noccu = 8 - -res = edrixs.get_atom_data('Ni', v_name='3d', v_noccu=noccu, edge='L23') -name_i, slat_i = [list(i) for i in zip(*res['slater_i'])] -name_n, slat_n = [list(i) for i in zip(*res['slater_n'])] - -# Slater integrals for initial Hamiltonian without core-hole -si = edrixs.rescale(slat_i, ([1, 2], [0.65]*2)) -si[0] = edrixs.get_F0('d', si[1], si[2]) # F0_dd - -# Slater integrals for intermediate Hamiltonian with core-hole -sn = edrixs.rescale(slat_n, ([1, 2, 4, 5, 6], [0.65, 0.65, 0.95, 0.7, 0.7])) -sn[0] = edrixs.get_F0('d', sn[1], sn[2]) # F0_dd -sn[3] = edrixs.get_F0('dp', sn[5], sn[6]) # F0_dp - -slater = (si, sn) - -# Spin-orbit coupling strengths -zeta_d_i = res['v_soc_i'][0] # valence 3d electron without core-hole -zeta_d_n = res['v_soc_n'][0] # valence 3d electron with core-hole -# E_{L2} - E_{L3} = 1.5 * zeta_p -zeta_p_n = (res['edge_ene'][0] - res['edge_ene'][1]) / 1.5 # core 2p electron - -# Tetragonal crystal field -cf = edrixs.cf_tetragonal_d(ten_dq=1.3, d1=0.05, d3=0.2) - -# Level shift of the core shell -off = 857.4 - -# Life time broadening -gamma_c = res['gamma_c'][1] # core hole -gamma_f = 0.1 # final states - -# Incident, scattered, azimuthal angles -# See Figure 1 of Y. Wang et al., -# `Computer Physics Communications 243, 151-165 (2019) -# `_ -# for the defintion of the scattering angles. -thin, thout, phi = 15 / 180.0 * np.pi, 75 / 180.0 * np.pi, 0.0 - -# Polarization types -poltype_xas = [('isotropic', 0.0)] # for XAS -poltype_rixs = [('linear', 0, 'linear', 0), ('linear', 0, 'linear', np.pi/2.0)] # for RIXS - -# Energy grid -ominc_xas = np.linspace(off - 10, off + 20, 1000) # for XAS -ominc_rixs_L3 = np.linspace(-5.9 + off, -0.9 + off, 100) # incident energy at L3 edge -ominc_rixs_L2 = np.linspace(10.4 + off, 14.9 + off, 100) # incident energy at L3 edge -eloss = np.linspace(-0.5, 5.0, 1000) # energy loss for RIXS - -# Run ED -eval_i, eval_n, trans_op = edrixs.ed_1v1c_py( - ('d', 'p'), shell_level=(0.0, -off), v_soc=(zeta_d_i, zeta_d_n), - c_soc=zeta_p_n, v_noccu=noccu, slater=slater, v_cfmat=cf -) - -# Run XAS -xas = edrixs.xas_1v1c_py( - eval_i, eval_n, trans_op, ominc_xas, gamma_c=gamma_c, thin=thin, phi=phi, - pol_type=poltype_xas, gs_list=[0, 1, 2], temperature=300 -) - -# Run RIXS at L3 edge -rixs_L3 = edrixs.rixs_1v1c_py( - eval_i, eval_n, trans_op, ominc_rixs_L3, eloss, gamma_c=gamma_c, gamma_f=gamma_f, - thin=thin, thout=thout, phi=phi, pol_type=poltype_rixs, gs_list=[0, 1, 2], - temperature=300 -) - -# Run RIXS at L2 edge -rixs_L2 = edrixs.rixs_1v1c_py( - eval_i, eval_n, trans_op, ominc_rixs_L2, eloss, gamma_c=gamma_c, gamma_f=gamma_f, - thin=thin, thout=thout, phi=phi, pol_type=poltype_rixs, gs_list=[0, 1, 2], - temperature=300 -) - -# Plot -fig = plt.figure(figsize=(16, 14)) -mpl.rcParams['font.size'] = 20 - -ax1 = plt.subplot(2, 2, 1) -plt.grid() -plt.plot(range(len(eval_i)), eval_i - min(eval_i), '-o') -plt.xlabel(r'Multiplets') -plt.ylabel(r'Energy (eV)') -plt.title(r'(a) Energy of multiplets') - -ax2 = plt.subplot(2, 2, 2) -plt.grid() -plt.plot(ominc_xas, xas[:, 0], '-') -plt.xlabel(r'Incident Energy (eV)') -plt.ylabel(r'XAS Intensity (a.u.)') -plt.title(r'(b) Isotropic XAS') - -ax3 = plt.subplot(2, 2, 3) -plt.imshow(np.sum(rixs_L3, axis=2), - extent=[min(eloss), max(eloss), min(ominc_rixs_L3), max(ominc_rixs_L3)], - origin='lower', aspect='auto', cmap='rainbow', interpolation='gaussian') -plt.xlabel(r'Energy loss (eV)') -plt.ylabel(r'Incident Energy (eV)') -plt.title(r'(c) RIXS map at $L_3$ edge (LH)') - -ax4 = plt.subplot(2, 2, 4) -plt.imshow(np.sum(rixs_L2, axis=2), - extent=[min(eloss), max(eloss), min(ominc_rixs_L2), max(ominc_rixs_L2)], - origin='lower', aspect='auto', cmap='rainbow', interpolation='gaussian') -plt.xlabel(r'Energy loss (eV)') -plt.ylabel(r'Incident Energy (eV)') -plt.title(r'(d) RIXS map at $L_2$ edge (LH)') - -plt.subplots_adjust(left=0.1, right=0.9, bottom=0.1, top=0.9, wspace=0.2, hspace=0.3) -plt.show() diff --git a/edrixs/_downloads/4bf727945bfb7ec7db12f60a40244241/example_7_transitions.py b/edrixs/_downloads/4bf727945bfb7ec7db12f60a40244241/example_7_transitions.py deleted file mode 100644 index c346e469db..0000000000 --- a/edrixs/_downloads/4bf727945bfb7ec7db12f60a40244241/example_7_transitions.py +++ /dev/null @@ -1,198 +0,0 @@ -#!/usr/bin/env python -""" -X-ray transitions -================================================================================ -This example explains how to calculate x-ray transition amplitudes between -specific orbital and spin states. We take the case of a cuprate which has one -hole in the :math:`d_{x^2-y^2}` orbital and a spin ordering direction along the -in-plane diagaonal direction and compute the angular dependence of spin-flip -and non-spin-flip processes. - -This case was chosen because the eigenvectors in question are simple enough -for us to write them out more-or-less by hand, so this example helps the reader -to understand what happens under the hood in more complex cases. - -Some of the code here is credited to Yao Shen who used this approach for the -analysis of a low valence nickelate material [1]_. The task performed repeats -analysis done by many researchers e.g. Luuk Ament et al [2]_ as well as -several other groups. -""" -import edrixs -import numpy as np -import matplotlib.pyplot as plt -import scipy - -################################################################################ -# Eigenvectors -# ------------------------------------------------------------------------------ -# Let us start by determining the eigenvectors involved in the transitions. -# The spin direction can be set using a vector -# :math:`\vec{B}` to represent a magnetic field in terms of generalized spin -# operator :math:`\tilde{\sigma}=\vec{B}\cdot\sigma` based on the Pauli matrices -# :math:`\sigma`. Let's put the spin along the :math:`[1, 1, 0]` direction -# and formuate the problem in the hole basis. -# For one particle, we know that the Hamiltonian will be diagonal in the real -# harmonic basis. -# We can generate the required eigenvectors by making a diagonal -# matrix, transforming it to the required -# complex harmonic basis (as is standard for EDRIXS) and diagonalizing it. -# As long as the crystal field splitting is much larger than the magnetic -# field, the eigenvectors will be independent of the exact values of both -# these parameters. - -B = 1e-3*np.array([1, 1, 0]) -cf_splitting = 1e3 -zeeman = sum(s*b for s, b in zip(edrixs.get_spin_momentum(2), B)) -dd_levels = np.array([energy for dd_level in cf_splitting*np.arange(5) - for energy in [dd_level]*2], dtype=complex) -emat_rhb = np.diag(dd_levels) -emat = edrixs.cb_op(emat_rhb, edrixs.tmat_r2c('d', ispin=True)) + zeeman -_, eigenvectors = np.linalg.eigh(emat) - -def get_eigenvector(orbital_index, spin_index): - return eigenvectors[:, 2*orbital_index + spin_index] - - -################################################################################ -# Let's examine the :math:`d_{x^2-y^2}` orbital first. Recall from the -# :ref:`sphx_glr_auto_examples_example_1_crystal_field.py` -# example that edrixs uses the standard orbital order of -# :math:`d_{3z^2-r^2}, d_{xz}, d_{yz}, d_{x^2-y^2}, d_{xy}`. So we want -# :code:`orbital_index = 3` element. Using this, we can build spin-up and -down -# eigenvectors. -orbital_index = 3 - -groundstate_vector = get_eigenvector(orbital_index, 0) -excitedstate_vector = get_eigenvector(orbital_index, 1) - -################################################################################ -# Transition operators and scattering matrix -# ------------------------------------------------------------------------------ -# Here we are considering the :math:`L_3`-edge. This means -# a :math:`2p_{3/2} \rightarrow 3d` -# absoprtion transition and a :math:`2p_{3/2} \rightarrow 3d` -# emission transition. We can read the relevant matrix from the edrixs database, -# keeping in mind that there are in fact three operations for -# :math:`x, y,` & :math:`z` directions. Note that edrixs provides operators -# in electron notation. If we define :math:`D` as the transition operator in -# electron language, :math:`D^\dagger` is the operator in the hole language -# we are using for this example. -# The angular dependence of a RIXS transition can be conveniently described -# using the scattering matrix, which is a :math:`3\times3` element object that -# specifies the transition amplitude for each incoming and outgoing x-ray -# polarization. Correspondingly, we have -# -# .. math:: -# \begin{equation} -# \mathcal{F}=\sum_n\langle f|D|n\rangle\langle n|D^{\dagger}|g\rangle -# \end{equation}. -# -# In matrix form this is -# -# .. math:: -# \begin{equation} -# \mathcal{F}(m,n)=\{f^{\dagger} \cdot D(m)\} \cdot \{D^{\dagger}(n) \cdot g\} -# \end{equation}. - -D_Tmat = edrixs.get_trans_oper('dp32') - -def get_F(vector_i, vector_f): - F = np.zeros((3, 3), dtype=complex) - for i in range(3): - for j in range(3): - F[i, j] = np.dot(np.dot(np.conj(vector_f.T), D_Tmat[i]), - np.dot(np.conj(D_Tmat[j].T), vector_i)) - return F - -################################################################################ -# Using this function, we can obtain non-spin-flip (NSF) and spin-flip (SF) -# scattering matrices by choosing whether we return to the ground state or -# whether we access the excited state with the spin flipped. -F_NSF = get_F(groundstate_vector, groundstate_vector) -F_SF = get_F(groundstate_vector, excitedstate_vector) - -################################################################################ -# Angular dependence -# ------------------------------------------------------------------------------ -# Let's consider the common case of fixing the total scattering angle at -# :code:`two_theta = 90` and choosing a series of incident angles :code:`thins`. -# Since the detector does not resolve polarization, we need to add both outgoing -# polarizations. It is then convenient to use function :func:`.dipole_polvec_rixs` -# to obtain the incoming and outgoing polarization vectors. -thins = np.linspace(0, 90) -two_theta = 90 -phi = 0 - - -def get_I(thin, alpha, F): - intensity = 0 - for beta in [0, np.pi/2]: - thout = two_theta - thin - ei, ef = edrixs.dipole_polvec_rixs(thin*np.pi/180, thout*np.pi/180, - phi*np.pi/180, alpha, beta) - intensity += np.abs(np.dot(ef, np.dot(F, ei)))**2 - return intensity - - -################################################################################ -# Plot -# ------------------------------------------------------------------------------ -# We now run through a few configurations specified in terms of incoming -# polarization angle :math:`\alpha` (defined in radians w.r.t. the scattering -# plane), :math:`F`, plotting label, and plotting color. -fig, ax = plt.subplots() - -config = [[0, F_NSF, r'$\pi$ NSF', 'C0'], - [np.pi/2, F_NSF, r'$\sigma$ NSF', 'C1'], - [0, F_SF, r'$\pi$ SF', 'C2'], - [np.pi/2, F_SF, r'$\sigma$ SF', 'C3']] - -for alpha, F, label, color in config: - Is = np.array([get_I(thin, alpha, F) for thin in thins]) - ax.plot(thins, Is, label=label, color=color) - -ax.legend() -ax.set_xlabel(r'Theta ($^\circ$)') -ax.set_ylabel('Relative intensity') -plt.show() - -################################################################################ -# Run through orbitals -# ------------------------------------------------------------------------------ -# For completeness, let's look at transitions from :math:`x^2-y^2` to all other -# orbitals. -fig, axs = plt.subplots(5, 1, figsize=(7, 7), - sharex=True, sharey=True) - -orbitals = ['$d_{3z^2-r^2}$', '$d_{xz}$', '$d_{yz}$', - '$d_{x^2-y^2}$', '$d_{xy}$'] -orbital_order = [4, 1, 2, 0, 3] - -plot_index = 0 -for ax, orbital_index in zip(axs, orbital_order): - for spin_index, spin_label in zip([0, 1], ['NSF', 'SF']): - excitedstate_vector = get_eigenvector(orbital_index, spin_index) - F = get_F(groundstate_vector, excitedstate_vector) - for alpha, pol_label in zip([0, np.pi/2], [r'$\pi$', r'$\sigma$']): - Is = np.array([get_I(thin, alpha, F) for thin in thins]) - ax.plot(thins, Is*10, label=f'{pol_label} {spin_label}', - color=f'C{plot_index%4}') - plot_index += 1 - ax.legend(title=orbitals[orbital_index], bbox_to_anchor=(1.1, 1), - loc="upper left", fontsize=8) - - -axs[-1].set_xlabel(r'$\theta$ ($^\circ$)') -axs[2].set_ylabel('Scattering intensity') - -fig.subplots_adjust(hspace=0, left=.3, right=.6) -plt.show() - -############################################################################## -# -# .. rubric:: Footnotes -# -# .. [1] Yao Shen et al., -# `arXiv:2110.08937 (2022) `_. -# .. [2] Luuk J. P. Ament et al., -# `Phys. Rev. Lett. 103, 117003 (2009) `_ diff --git a/edrixs/_downloads/4c17545c9c531bbea8f391a8442d3b7a/example_5_transitions.ipynb b/edrixs/_downloads/4c17545c9c531bbea8f391a8442d3b7a/example_5_transitions.ipynb deleted file mode 100644 index 51f5f3e423..0000000000 --- a/edrixs/_downloads/4c17545c9c531bbea8f391a8442d3b7a/example_5_transitions.ipynb +++ /dev/null @@ -1,176 +0,0 @@ -{ - "cells": [ - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n# X-ray transitions\nThis example explains how to calculate x-ray transition amplitudes between\nspecific orbital and spin states. We take the case of a cuprate which has one\nhole in the $d_{x^2-y^2}$ orbital and a spin ordering direction along the\nin-plane diagaonal direction and compute the angular dependence of spin-flip\nand non-spin-flip processes.\n\nThis case was chosen because the eigenvectors in question are simple enough\nfor us to write them out more-or-less by hand, so this example helps the reader\nto understand what happens under the hood in more complex cases.\n\nSome of the code here is credited to Yao Shen who used this approach for the\nanalysis of a low valence nickelate material [1]_. The task performed repeats\nanalysis done by many researchers e.g. Luuk Ament et al [2]_ as well as\nseveral other groups.\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "import edrixs\nimport numpy as np\nimport matplotlib.pyplot as plt\nimport scipy" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Eigenvectors\nLet us start by determining the eigenvectors involved in the transitions.\nThe spin direction can be set using a vector\n$\\vec{B}$ to represent a magnetic field in terms of generalized spin\noperator $\\tilde{\\sigma}=\\vec{B}\\cdot\\sigma$ based on the Pauli matrices\n$\\sigma$. Let's put the spin along the $[1, 1, 0]$ direction\nand formuate the problem in the hole basis.\nFor one particle, we know that the Hamiltonian will be diagonal in the real\nharmonic basis.\nWe can generate the required eigenvectors by making a diagonal\nmatrix, transforming it to the required\ncomplex harmonic basis (as is standard for EDRIXS) and diagonalizing it.\nAs long as the crystal field splitting is much larger than the magnetic\nfield, the eigenvectors will be independent of the exact values of both\nthese parameters.\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "B = 1e-3*np.array([1, 1, 0])\ncf_splitting = 1e3\nzeeman = sum(s*b for s, b in zip(edrixs.get_spin_momentum(2), B))\ndd_levels = np.array([energy for dd_level in cf_splitting*np.arange(5)\n for energy in [dd_level]*2], dtype=complex)\nemat_rhb = np.diag(dd_levels)\nemat = edrixs.cb_op(emat_rhb, edrixs.tmat_r2c('d', ispin=True)) + zeeman\n_, eigenvectors = np.linalg.eigh(emat)\n\ndef get_eigenvector(orbital_index, spin_index):\n return eigenvectors[:, 2*orbital_index + spin_index]" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Let's examine the $d_{x^2-y^2}$ orbital first. Recall from the\n`sphx_glr_auto_examples_example_1_crystal_field.py`\nexample that edrixs uses the standard orbital order of\n$d_{3z^2-r^2}, d_{xz}, d_{yz}, d_{x^2-y^2}, d_{xy}$. So we want\n:code:`orbital_index = 3` element. Using this, we can build spin-up and -down\neigenvectors.\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "orbital_index = 3\n\ngroundstate_vector = get_eigenvector(orbital_index, 0)\nexcitedstate_vector = get_eigenvector(orbital_index, 1)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Transition operators and scattering matrix\nHere we are considering the $L_3$-edge. This means\na $2p_{3/2} \\rightarrow 3d$\nabsoprtion transition and a $2p_{3/2} \\rightarrow 3d$\nemission transition. We can read the relevant matrix from the edrixs database,\nkeeping in mind that there are in fact three operations for\n$x, y,$ & $z$ directions. Note that edrixs provides operators\nin electron notation. If we define $D$ as the transition operator in\nelectron language, $D^\\dagger$ is the operator in the hole language\nwe are using for this example.\nThe angular dependence of a RIXS transition can be conveniently described\nusing the scattering matrix, which is a $3\\times3$ element object that\nspecifies the transition amplitude for each incoming and outgoing x-ray\npolarization. Correspondingly, we have\n\n .. math::\n \\begin{equation}\n \\mathcal{F}=\\sum_n\\langle f|D|n\\rangle\\langle n|D^{\\dagger}|g\\rangle\n \\end{equation}.\n\nIn matrix form this is\n\n .. math::\n \\begin{equation}\n \\mathcal{F}(m,n)=\\{f^{\\dagger} \\cdot D(m)\\} \\cdot \\{D^{\\dagger}(n) \\cdot g\\}\n \\end{equation}.\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "D_Tmat = edrixs.get_trans_oper('dp32')\n\ndef get_F(vector_i, vector_f):\n F = np.zeros((3, 3), dtype=complex)\n for i in range(3):\n for j in range(3):\n F[i, j] = np.dot(np.dot(np.conj(vector_f.T), D_Tmat[i]),\n np.dot(np.conj(D_Tmat[j].T), vector_i))\n return F" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Using this function, we can obtain non-spin-flip (NSF) and spin-flip (SF)\nscattering matrices by choosing whether we return to the ground state or\nwhether we access the excited state with the spin flipped.\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "F_NSF = get_F(groundstate_vector, groundstate_vector)\nF_SF = get_F(groundstate_vector, excitedstate_vector)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Angular dependence\nLet's consider the common case of fixing the total scattering angle at\n:code:`two_theta = 90` and choosing a series of incident angles :code:`thins`.\nSince the detector does not resolve polarization, we need to add both outgoing\npolarizations. It is then convenient to use function :func:`.dipole_polvec_rixs`\nto obtain the incoming and outgoing polarization vectors.\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "thins = np.linspace(0, 90)\ntwo_theta = 90\nphi = 0\n\n\ndef get_I(thin, alpha, F):\n intensity = 0\n for beta in [0, np.pi/2]:\n thout = two_theta - thin\n ei, ef = edrixs.dipole_polvec_rixs(thin*np.pi/180, thout*np.pi/180,\n phi*np.pi/180, alpha, beta)\n intensity += np.abs(np.dot(ef, np.dot(F, ei)))**2\n return intensity" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Plot\nWe now run through a few configurations specified in terms of incoming\npolarization angle $\\alpha$ (defined in radians w.r.t. the scattering\nplane), $F$, plotting label, and plotting color.\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "fig, ax = plt.subplots()\n\nconfig = [[0, F_NSF, r'$\\pi$ NSF', 'C0'],\n [np.pi/2, F_NSF, r'$\\sigma$ NSF', 'C1'],\n [0, F_SF, r'$\\pi$ SF', 'C2'],\n [np.pi/2, F_SF, r'$\\sigma$ SF', 'C3']]\n\nfor alpha, F, label, color in config:\n Is = np.array([get_I(thin, alpha, F) for thin in thins])\n ax.plot(thins, Is, label=label, color=color)\n\nax.legend()\nax.set_xlabel(r'Theta ($^\\circ$)')\nax.set_ylabel('Relative intensity')\nplt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Run through orbitals\nFor completeness, let's look at transitions from $x^2-y^2$ to all other\norbitals.\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "fig, axs = plt.subplots(5, 1, figsize=(7, 7),\n sharex=True, sharey=True)\n\norbitals = ['$d_{3z^2-r^2}$', '$d_{xz}$', '$d_{yz}$',\n '$d_{x^2-y^2}$', '$d_{xy}$']\norbital_order = [4, 1, 2, 0, 3]\n\nplot_index = 0\nfor ax, orbital_index in zip(axs, orbital_order):\n for spin_index, spin_label in zip([0, 1], ['NSF', 'SF']):\n excitedstate_vector = get_eigenvector(orbital_index, spin_index)\n F = get_F(groundstate_vector, excitedstate_vector)\n for alpha, pol_label in zip([0, np.pi/2], [r'$\\pi$', r'$\\sigma$']):\n Is = np.array([get_I(thin, alpha, F) for thin in thins])\n ax.plot(thins, Is*10, label=f'{pol_label} {spin_label}',\n color=f'C{plot_index%4}')\n plot_index += 1\n ax.legend(title=orbitals[orbital_index], bbox_to_anchor=(1.1, 1),\n loc=\"upper left\", fontsize=8)\n\n\naxs[-1].set_xlabel(r'$\\theta$ ($^\\circ$)')\naxs[2].set_ylabel('Scattering intensity')\n\nfig.subplots_adjust(hspace=0, left=.3, right=.6)\nplt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - ".. rubric:: Footnotes\n\n.. [1] Yao Shen et al.,\n [arXiv:2110.08937 (2022)](https://arxiv.org/abs/2110.08937).\n.. [2] Luuk J. P. Ament et al.,\n [Phys. Rev. Lett. 103, 117003 (2009)](https://doi.org/10.1103/PhysRevLett.103.117003)\n\n" - ] - } - ], - "metadata": { - "kernelspec": { - "display_name": "Python 3", - "language": "python", - "name": "python3" - }, - "language_info": { - "codemirror_mode": { - "name": "ipython", - "version": 3 - }, - "file_extension": ".py", - "mimetype": "text/x-python", - "name": "python", - "nbconvert_exporter": "python", - "pygments_lexer": "ipython3", - "version": "3.10.14" - } - }, - "nbformat": 4, - "nbformat_minor": 0 -} \ No newline at end of file diff --git a/edrixs/_downloads/549f42a520207f2312c42f9bc1883308/example_2_single_atom_RIXS.zip b/edrixs/_downloads/549f42a520207f2312c42f9bc1883308/example_2_single_atom_RIXS.zip deleted file mode 100644 index 4cf0e068042a6d38a2ce5a49443b4066ea0cd894..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 23188 zcmdU1-E!kbcAj{XT;O_dQ`OO^O5`pDk<^cE)pED&wdEPx z_LooS=jwyzzx><3PFnxbeDZ|;{R6JaOMjZivfu6Jks9NLUnJB1AD;j5rJG(o{l~po zp6v}IwI|iNNUw@XqMWU*Er%M2z>kAj>=#j@axqFW;VVIpM}e3op^Tm0IzP@cnMrXW zMLtO`1pO-#5hT-eR!G79uxO#jhd;Pn??qfh#Y6kq zh*TjnKj87Y;-94`{8#ujk(fy?MKW?;W}Z0CXT#$`ci@R1o@f7L-yaQ}QI<@_ALPgt zr=7FD@N<52>O*Mc$B-_5oXMP|yW(32^`j|dqEVC!qh^7ZMdCzpJj;s=PqK(KO9Cmw z8Pq@n$MkXKJPM(eiC>UvCP^yk*;y_}vsn1yc@#)Yfo?~+bCE?wAr-YyNg<}vR|Sa~ zLZHZG8vB8qN?apNX3`I_)ZBBJ1Q-Y2<$9#j8O@YdPz#+$0TfG%8ejNVIg1MdnddST zi6Q~cK!#a#nPZGYUYOrZiXzRA_x4mC=etqljcm+MlU{=uZ%7N(h{oU_K!Ovf?3A${9(KJo6LTI^;d8cO6^orI-rS=k5 zD2xNw}O8~$;gikGPjq7g;_EuO=$60@l1eoT|bG=~xFG)L3jjs8&I zH(0MH#01m!#WB<|IUXE_e>xbHJ_3&o5;cm(GZ-ZdnCuV&VR;wEl&r67i#0;N?(aiJ zIEL?H+C?^-ipYgsW-?C{1YsH3Ieo}7k95=ioBh4++ksZiB#Ft6(GPq=z=N@%FoQ@1 z@rtlkRtSn$CG&B|$L5 zjf2Noci^DRMCxbg9KG@*?=J>PvJaldQkf;jcH}f zf_Sk%9fm&I!8*b}1DE_^EaxWzD1p@g4CG}Bup=m`7L7m@T{4W4wbPQ;hf|3XjG)x; z8P++IQk*2gD*=}sLSf?b&lv4trVxkmMnx_{AHxASGooU)3OH$y4F-EIvjXo4H2o~| z)fjI8@(B+n$wHYyQYL?#MH%kEu5=f!^IZr-=sI6M!=JX>J#?T1etFP!PTKvjB~H5h z`=C!PHN?2zg^#rN5)VWC zHip<@zv=CKy$|F3qSFskh#*O)J2{(DU+tWo9-bl8IA83qxP8R8X~6Wf86|an(r)a8 zDSBhc_(Kw!4ozxlKL;$xF+E2RcDksBGa#Z*QBVTDACN zekz$M%c+mJ*60h zfrQT@e+(Fmqt}e7FvX1m{w@RpIFn)F!&%6y##sU|V0YkDXDY*patydMe#>Yq75>Y{ zk%X&LqJd|Sg(yUBU_Qhl%K-i-@Wu1UCznTT7m!h(hAX~FIB!Af!9HIlfg(&GnKF=9 zD!G6>zF%Sj_kD$hNGPyaPu7^jO?3mjG@^AFA&jT!0C|3zN*`$^$A}50z$iqW2{}!p zEJBpxaB)olG9z5;bh@2GA-^Y4Jbzq6nyK@RWR{ijknvQ*hz&DCdLl@2fRaB-?Er1WL}JS!63JUk zvh+fP2``ylj}&8sb`T3fHVb67#{shWi;34u>ckA=Q&4$w<}@BPUrjX-e^EOA<$8MdnK#lh?)m5UBz3rWSIAoc4JT zW{fUQg1&zl_^P5KAG5LyU{oki@n(BZ&XKqg*HuJS0z z1ujfy3IIi;$VK;Hza=_{o!!pSp=fm5osXM%tlT|?N#rKkc(2oTJMGTV-k;qd4P0vG zwh!EP=OgTx^d{3R4<62?1~N+X_2ObQNtNhfxoG=m(>_eA@Oxew?R5tt^C{i8F@c|k zR*@6UN!Yz)5ooz6WD@n-16RDT*{@xx9P>poY*x^$6|FI^775bOa_Q$5EO?=pEMsk* zMli~pQaaap3K`$~*nd;bYJ{ap+%K+DIVhJCfF|N7FKDq!yUbKE+K24>{5`y_^l4oXs}xs0f_)+*ajJnT?o9TtpOg4t&d?>t_Zt*Mb3cal-gtC zgMGguqt*I1VOsl??eq0umwZA`-=xjh=EI5cs4h$7D2K=?oX{i%ItQ4Mb3qxf73cz; zz{Z6MyPO|zGuqBa;|ZnK7}lprT)duQlr+F>3Xd6KbBiPa-BXyyPVp&ogyZ5TB6$$@DMLdiu??gWy7hdqaZ#~RIN}DL!-r^ zLVEP{5$ zFtClM5x=*YJ4}{$-sF(5QlJ#7tkVdnNLYmgqa}vO>~twdjO0ZvIV3McKm%y$%2BS} z$U25GR;)HzvO$_Sp%fDUM*gpRCcvYO0|AqP!t0)*hk@G?nFER~Byt2d!*F?rV?7G8 z5a_tWvR;8kw4I2tJ=f(e^3t+e6TP0tDt5)ufVIwo*7QLNN8e&poG)-Ztw#n6cXCSQ zT0jFu*eNP52(!q=k;at3i7YmOEF|}wk1s^FKGj?~P_hb4$<}TftPl5LuA_juU#LBff@DfbFP77zzcU3_pdsBHA2QNmH(qK|77PDdHhoN7nVZq^u%A zl%vfBhLe!th+zWwTKaH+J)M#<QI64ju)iU(kU9 zZNkW*9t*0!mIYEdKD)riIm00cytXRAQex_tCnjcJPuX@R=@x+m<=NVGC~ccB;jn>H z0kzb##uwU2tfS%^R6ak(96K#4_oh^RKyfK}#h`$i2nQY>XmOF$a`a^|fy5KXd}Ck~ z4xV~+lH;PXLCVP+NE`7#?&g5>UZX>~PrFG08XF~6U_UAvc(BX~U@52#5OA~vO&iBg z`o+{wd*1-Os6seIv4UPVkJ)AL{Av5lr7pF-wFJbR9wSRZK|(j@-_3e+#J2t=UvBJt zPIb5|(U9LaQU9d^hmlE*u1urnIIpDQ+|~6aMO4JHu>-rqz4G~v8Pno)^oT}VZ}`RA zt*S>nv1C{bqYGm*EpHl|X`;tA+TqBKG?ik3qN5r+RHTEXX(*Ut4228QtyV3o0*637 zR!A2U3X&Mx1DQ{(s%x`@2=`mW$_XE7_ICVPkr-u>ATkYpeCTHl966B5^hnX&a+!7( zmCOU+CUcs3ZZ>8%hpB02;R&lDv&5v!+K_bbEZVFgs|gb!?QpE<6J(K3^jJ+W?rCGU z({x~&eLgeA4a6Wn8zc8`kS6UDWPz+5@f!iv(SscrzcFhguWVe(Lm(qlHlE>Ts5r(A zz^BQYZ}gCQ_f3b57%pe6Dw<+^*h%*G^*wCt;5XXFaj!_y+8sQV$tg0>1~kE5jbaPE z-tZ|cC21!!#E?)IBg<{1E`o17&wfd$M|*QW3y`+vI(AvE0P;Y>XD%M6C}hCpal%fi zA|HL~8mSC+VO-iA%8K5QmHqLb+R6}in2QP1X~ z>b#3EX7T4ys^M_}yLXHAQ!9L7yYe&$!)Y)LA}EzP08!8AXxyG*0;p;|!Rm zgA31ZRkh$}&b0xEtFj+mH=E0T4uJnPaK9PUb)K-ZtKFXnB zv!b&(qq1tQ5-!_wh)&6OC>WZID3&<43=s$DJz_-p{*gn(C$zJqLlKE{emZ3XU%?kKWnc?mh#NZNUmAZGXhUN;+@xuEZYK{Oj;dvJUVhDcIiu5|cGA=6i2UcH!_Hxw2lUZe z1NzX_kMMwAlw{5~Aanlo@BjN>|Fiex3H=i{lR2aGN)6ut*R}$g)X!dpIO*E5(Ps-sWIl0OaQDsI@@kIbyFix0Sdgcl>Y|>l&hvp6X^gv&G0QJ*T@BvHq1o_jjKkb#H0pxFr|LG}IFg8g5_0;AVpqS^KD4?D?)u|LP zk~yGfjeBuGo2c1(ii*4IQ9wPlWlZ;s{~DuQZqJmLR<>+bbW7~^UemN_bH2pwYuBUG1<>LO0#9Te-wV1hRdLuQLR@O3EG4or#2>>Y!?Aa?>?c*`>||2wJDMTx;322>lD}TFwBCU3O^zLm$@)RjNO;>Dm=k$AUXZ!^mdSuIwDn7d5f-6ra;MOs5QHOx`w{%-GH z@^e7a+Ji*b-M%wjw_L{BNx+@hx`b^HBI|kzN2B-Q>SFR~v^_`Dg^%DcPM37=%hE-? 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We aim -to clarify the form of the matrices, how they are parametrized, -and how the breaking of spherical symmetry can switch on additional elements -that one might not anticipate. Our example is based on a :math:`d` atomic shell. -""" - -################################################################################ -# Create matrix -# ------------------------------------------------------------------------------ -# The Coulomb interaction between two particles can be written as -# -# .. math:: -# \begin{equation} -# \hat{H} = \frac{1}{2} -# \int d\mathbf{r} \int d\mathbf{r}^\prime -# \Sigma_{\sigma, \sigma^\prime} -# |\hat{\psi}^\sigma(\mathbf{r})|^2 \frac{e^2}{R} -# |\hat{\psi}^{\sigma^\prime}(\mathbf{r^\prime})|^2, -# \end{equation} -# -# where :math:`\hat{\psi}^\sigma(\mathbf{r})` is the electron wavefunction, with -# spin :math:`\sigma`, and :math:`R=|r-r^\prime|` is the electron separation. -# Solving our problem in this form is difficult due to the need to symmeterize -# the wavefunction to follow fermionic statistics. -# Using second quantization, we can use operators to impose the required -# particle exchange statistics and write the equation in terms of -# a tensor :math:`U` -# -# .. math:: -# \begin{equation} -# \hat{H} = \sum_{\alpha,\beta,\gamma,\delta,\sigma,\sigma^\prime} -# U_{\alpha\sigma,\beta\sigma^\prime,\gamma\sigma^\prime,\delta\sigma} -# \hat{f}^{\dagger}_{\alpha\sigma} -# \hat{f}^{\dagger}_{\beta\sigma^\prime} -# \hat{f}_{\gamma\sigma^\prime}\hat{f}_{\delta\sigma}, -# \end{equation} -# -# where :math:`\alpha`, :math:`\beta`, :math:`\gamma`, :math:`\delta` are -# orbital indices and :math:`\hat{f}^{\dagger}` -# (:math:`\hat{f}`) are the creation (anihilation) operators. -# For a :math:`d`-electron system, we have :math:`10` distinct spin-orbitals -# (:math:`5` orbitals each with :math:`2` spins), which makes matrix the -# :math:`10\times10\times10\times10` in total size. -# In EDRIXS the matrix can be created as follows: -import edrixs -import numpy as np -import scipy -import matplotlib.pyplot as plt -import itertools - -F0, F2, F4 = 6.94, 14.7, 4.41 -umat_chb = edrixs.get_umat_slater('d', F0, F2, F4) -################################################################################ -# We stored this under variable :code:`umat_chb` where "cbh" stands for -# complex harmonic basis, which is the default basis in EDRIXS. - -################################################################################ -# Parameterizing interactions -# ------------------------------------------------------------------------------ -# EDRIXS parameterizes the interactions in :math:`U` via Slater integral -# parameters :math:`F^{k}`. These relate to integrals of various spherical -# Harmonics as well as Clebsch-Gordon coefficients, Gaunt coefficients, -# and Wigner 3J symbols. Textbooks such as [1]_ can be used for further -# reference. If you are interested in the details of how -# EDRIXS does this (and you probably aren't) function :func:`.umat_slater`, -# constructs the required matrix via Gaunt coeficents from -# :func:`.get_gaunt`. Two alternative parameterizations are common. -# The first are the Racah parameters, which are -# -# .. math:: -# \begin{eqnarray} -# A &=& F^0 - \frac{49}{441} F^4 \\ -# B &=& \frac{1}{49}F^2 - \frac{5}{441}F^4 \\ -# C &=& \frac{35}{441}F^4. -# \end{eqnarray} -# -# or an alternative form for the Slater integrals -# -# .. math:: -# \begin{eqnarray} -# F_0 &=& F^0 \\ -# F_2 &=& \frac{1}{49}F^2 \\ -# F_4 &=& \frac{1}{441}F^4, -# \end{eqnarray} -# -# which involves different normalization parameters. - -################################################################################ -# Basis transform -# ------------------------------------------------------------------------------ -# If we want to use the real harmonic basis, we can use a tensor -# transformation, which imposes the following orbital order -# :math:`3z^2-r^2, xz, yz, x^2-y^2, xy`, each of which involves -# :math:`\uparrow, \downarrow` spin pairs. Let's perform this transformation and -# store a list of these orbitals. -umat = edrixs.transform_utensor(umat_chb, edrixs.tmat_c2r('d', True)) -orbitals = ['3z^2-r^2', 'xz', 'yz', 'x^2-y^2', 'xy'] - -################################################################################ -# Interactions -# ------------------------------------------------------------------------------ -# Tensor :math:`U` is a series of matrix -# elements -# -# .. math:: -# \begin{equation} -# \langle\psi_{\gamma,\delta}^{\bar{\sigma},\bar{\sigma}^\prime} -# |\hat{H}| -# \psi_{\alpha,\beta}^{\sigma,\sigma^\prime}\rangle -# \end{equation} -# -# the combination of which defines the energetic cost of pairwise -# electron-electron interactions between states :math:`\alpha,\sigma` -# and :math:`\beta,\sigma^\prime`. In EDRIXS we follow the convention of -# summing over all orbital pairs. Some other texts count each pair of -# indices only once. The matrix elements here will consequently -# be half the magnitude of those in other references. -# We can express the interactions in terms of -# the orbitals involved. It is common to distinguish "direct Coulomb" and -# "exchange" interactions. The former come from electrons in the same orbital -# and the later involve swapping orbital labels for electrons. We will use -# :math:`U_0` and :math:`J` as a shorthand for distinguishing these. -# -# Before we describe the different types of interactions, we note that since -# the Coulomb interaction is real, and due to the spin symmmetry properties -# of the process :math:`U` always obeys -# -# .. math:: -# \begin{equation} -# U_{\alpha\sigma,\beta\sigma^\prime,\gamma\sigma^\prime,\delta\sigma} = -# U_{\beta\sigma,\alpha\sigma^\prime,\delta\sigma^\prime,\gamma\sigma} = -# U_{\delta\sigma,\gamma\sigma^\prime,\beta\sigma^\prime,\alpha\sigma} = -# U_{\gamma\sigma,\delta\sigma^\prime,\alpha\sigma^\prime,\beta\sigma}. -# \end{equation} -# -# -# 1. Intra orbital -# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -# The direct Coulomb energy cost to double-occupy an orbital comes from terms -# like :math:`U_{\alpha\sigma,\alpha\bar\sigma,\alpha\bar\sigma,\alpha\sigma}`. -# In this notation, we use :math:`\sigma^\prime` to denote that the matrix -# element is summed over all pairs and :math:`\bar{\sigma}` to denote sums -# over all opposite spin pairs. Due to rotational symmetry, all these -# elements are the same and equal to -# -# .. math:: -# \begin{eqnarray} -# U_0 &=& \frac{A}{2} + 2B + \frac{3C}{2}\\ -# &=& \frac{F_0}{2} + 2F_2 + 18F_4 -# \end{eqnarray} -# -# Let's print these to demonstrate where these live in the array -for i in range(0, 5): - val = umat[i*2, i*2 + 1, i*2 + 1, i*2].real - print(f"{orbitals[i]:<8} \t {val:.3f}") - -################################################################################ -# 2. Inter orbital Coulomb interactions -# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -# Direct Coulomb repulsion between different orbitals depends on terms like -# :math:`U_{\alpha\sigma,\beta\sigma^\prime,\beta\sigma^\prime,\alpha\sigma}`. -# Expresions for these parameters are provided in column :math:`U` in -# :ref:`table_2_orbital`. We can print the values from :code:`umat` -# like this: -for i, j in itertools.combinations(range(5), 2): - val = umat[i*2, j*2 + 1, j*2 + 1, i*2].real - print(f"{orbitals[i]:<8} \t {orbitals[j]:<8} \t {val:.3f}") - -################################################################################ -# 3. Inter-orbital exchange interactions -# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -# Exchange terms exist with the form -# :math:`U_{\alpha\sigma,\beta\sigma^\prime,\alpha\sigma^\prime,\beta\sigma}`. -# Expresions for these parameters are provided in column :math:`J` of -# :ref:`table_2_orbital`. These come from terms like this in the matrix: -for i, j in itertools.combinations(range(5), 2): - val = umat[i*2, j*2 + 1, i*2 + 1, j*2].real - print(f"{orbitals[i]:<8} \t {orbitals[j]:<8} \t {val:.3f}") - -################################################################################ -# 4. Pair hopping term -# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -# Terms that swap pairs of electrons exist as -# :math:`(1-\delta_{\sigma\sigma'})U_{\alpha\sigma,\alpha\bar\sigma,\beta\bar\sigma,\beta\sigma}` -# and depend on exchange interactions column :math:`J` from -# :ref:`table_2_orbital` -# and here in the matrix. -for i, j in itertools.combinations(range(5), 2): - val = umat[i*2, i*2 + 1, j*2 + 1, j*2].real - print(f"{orbitals[i]:<8} \t {orbitals[j]:<8} \t {val:.3f}") - -################################################################################ -# 5. Three orbital -# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -# Another set of terms that one might not immediately anticipate involve three -# orbitals like -# -# .. math:: -# \begin{equation} -# U_{\alpha\sigma, \gamma\sigma', \beta\sigma', \gamma\sigma} \\ -# U_{\alpha\sigma, \gamma\sigma', \gamma\sigma', \beta\sigma} \\ -# (1-\delta_{\sigma\sigma'}) -# U_{\alpha\sigma, \beta\sigma', \gamma\sigma', \gamma\sigma} -# \end{equation} -# -# for :math:`\alpha=3z^2-r^2, \beta=x^2-y^2, \gamma=xz/yz`. -# These are needed to maintain the rotational symmetry of the interations. -# See :ref:`table_3_orbital` for the expressions. We can print some of -# these via: -ijkl = [[0, 1, 3, 1], - [0, 2, 3, 2], - [1, 0, 3, 1], - [1, 1, 3, 0], - [2, 0, 3, 2], - [2, 2, 3, 0]] - -for i, j, k, l in ijkl: - val = umat[i*2, j*2 + 1, k*2 + 1, l*2].real - print(f"{orbitals[i]:<8} \t {orbitals[j]:<8} \t" - f"{orbitals[k]:<8} \t {orbitals[l]:<8} \t {val:.3f}") - -################################################################################ -# 6. Four orbital -# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -# Futher multi-orbital terms include -# :math:`U_{\alpha\sigma,\beta\sigma^\prime,\gamma\sigma^\prime,\delta\sigma}`. -# We can find these here in the matrix: -ijkl = [[0, 1, 2, 4], - [0, 1, 4, 2], - [0, 2, 1, 4], - [0, 2, 4, 1], - [0, 4, 1, 2], - [0, 4, 2, 1], - [3, 1, 4, 2], - [3, 2, 4, 1], - [3, 4, 1, 2], - [3, 4, 2, 1]] - -for i, j, k, l in ijkl: - val = umat[i*2, j*2 + 1, k*2 + 1, l*2].real - print(f"{orbitals[i]:<8} \t {orbitals[j]:<8} \t {orbitals[k]:<8}" - f"\t {orbitals[l]:<8} \t {val:.3f}") - -################################################################################ -# Effects of multi-orbital terms -# ------------------------------------------------------------------------------ -# To test the effects of the multi-orbital terms, let's plot the eigenenergy -# spectra with and without multi-orbital terms switched on for system with and -# without a cubic crystal field. We will use a :math:`d`-shell with two -# electrons. -ten_dqs = [0, 2, 4, 12] - -def diagonalize(ten_dq, umat): - emat = edrixs.cb_op(edrixs.cf_cubic_d(ten_dq), - edrixs.tmat_c2r('d', ispin=True)) - H = (edrixs.build_opers(4, umat, basis) - + edrixs.build_opers(2, emat, basis)) - e, v = scipy.linalg.eigh(H) - return e - e.min() - -basis = edrixs.get_fock_bin_by_N(10, 2) -umat_no_multiorbital = np.copy(umat) -B = F2/49 - 5*F4/441 -for val in [np.sqrt(3)*B/2, np.sqrt(3)*B, 3*B/2]: - umat_no_multiorbital[(np.abs(umat)- val) < 1e-6] = 0 - -fig, axs = plt.subplots(1, len(ten_dqs), figsize=(8, 3)) - -for cind, (ax, ten_dq) in enumerate(zip(axs, ten_dqs)): - ax.hlines(diagonalize(ten_dq, umat), xmin=0, xmax=1, - label='on', color=f'C{cind}') - ax.hlines(diagonalize(ten_dq, umat_no_multiorbital), - xmin=1.5, xmax=2.5, - label='off', - linestyle=':', color=f'C{cind}') - ax.set_title(f"$10D_q={ten_dq}$") - ax.set_ylim([-.5, 20]) - ax.set_xticks([]) - ax.legend() - -fig.suptitle("Eigenvalues with 3&4-orbital effects on/off") -fig.subplots_adjust(wspace=.3) -axs[0].set_ylabel('Eigenvalues (eV)') -fig.subplots_adjust(top=.8) -plt.show() - -################################################################################ -# On the left of the plot Coulomb interactions in spherical symmetry cause -# substantial mxing between :math:`t_{2g}` and :math:`e_{g}` orbitals in the -# eigenstates and 3 & 4 orbital orbital terms are crucial for obtaining the -# the right eigenenergies. As :math:`10D_q` get large, this mixing is switched -# off and the spectra start to become independent of whether the 3 & 4 orbital -# orbital terms are included or not. -# -# -# -# .. _table_2_orbital: -# .. table:: Table of 2 orbital interactions -# -# +-----------------------------+------------------+-----------------------+----------------+--------------------+ -# |Orbitals :math:`\alpha,\beta`|:math:`U_0` Racah | :math:`U_0` Slater |:math:`J` Racah |:math:`J` Slater | -# +=============================+==================+=======================+================+====================+ -# |:math:`3z^2-r^2, xz` |:math:`A/2+B+C/2` |:math:`F_0/2+F_2-12F_4`| :math:`B/2+C/2`|:math:`F_2/2+15F_4` | -# +-----------------------------+------------------+-----------------------+----------------+--------------------+ -# |:math:`3z^2-r^2, yz` |:math:`A/2+B+C/2` |:math:`F_0/2+F_2-12F_4`| :math:`B/2+C/2`|:math:`F_2/2+15F_4` | -# +-----------------------------+------------------+-----------------------+----------------+--------------------+ -# |:math:`3z^2-r^2, x^2-y^2` |:math:`A/2-2B+C/2`|:math:`F_0/2-2F_2+3F_4`|:math:`2B+C/2` |:math:`2F_2+15F_4/2`| -# +-----------------------------+------------------+-----------------------+----------------+--------------------+ -# |:math:`3z^2-r^2, xy` |:math:`A/2-2B+C/2`|:math:`F_0/2-2F_2+3F_4`|:math:`2B+C/2` |:math:`2F_2+15F_4/2`| -# +-----------------------------+------------------+-----------------------+----------------+--------------------+ -# |:math:`xz, yz` |:math:`A/2-B+C/2` |:math:`F_0/2-F_2-12F_4`|:math:`3B/2+C/2`|:math:`3F_2/2+10F_4`| -# +-----------------------------+------------------+-----------------------+----------------+--------------------+ -# |:math:`xz, x^2-y^2` |:math:`A/2-B+C/2` |:math:`F_0/2-F_2-12F_4`|:math:`3B/2+C/2`|:math:`3F_2/2+10F_4`| -# +-----------------------------+------------------+-----------------------+----------------+--------------------+ -# |:math:`xz, xy` |:math:`A/2-B+C/2` |:math:`F_0/2-F_2-12F_4`|:math:`3B/2+C/2`|:math:`3F_2/2+10F_4`| -# +-----------------------------+------------------+-----------------------+----------------+--------------------+ -# |:math:`yz, x^2-y^2` |:math:`A/2-B+C/2` |:math:`F_0/2-F_2-12F_4`|:math:`3B/2+C/2`|:math:`3F_2/2+10F_4`| -# +-----------------------------+------------------+-----------------------+----------------+--------------------+ -# |:math:`yz, xy` |:math:`A/2-B+C/2` |:math:`F_0/2-F_2-12F_4`|:math:`3B/2+C/2`|:math:`3F_2/2+10F_4`| -# +-----------------------------+------------------+-----------------------+----------------+--------------------+ -# |:math:`x^2-y^2, xy` |:math:`A/2+2B+C/2`|:math:`F_0+4F_2-34F_4` | :math:`C/2` |:math:`35F_4/2` | -# +-----------------------------+------------------+-----------------------+----------------+--------------------+ -# -# -# .. _table_3_orbital: -# .. table:: Table of 3 orbital interactions -# -# +-----------------------------+-------------+----------------------------------------------------+-----------------------------------------------------+ -# |Orbitals :math:`\alpha,\beta,\gamma,\delta`|:math:`\langle\alpha\beta|\gamma\delta\rangle` Racah|:math:`\langle\alpha\beta|\gamma\delta\rangle` Slater| -# +=============================+=============+====================================================+=====================================================+ -# |:math:`3z^2-r^2, xz, x^2-y^2, xz` | :math:`\sqrt{3}B/2` | :math:`\sqrt{3}F_2/2-5\sqrt{3}F_4/2` | -# +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ -# |:math:`3z^2-r^2, yz, x^2-y^2, yz` | :math:`-\sqrt{3}B/2` | :math:`-\sqrt{3}F_2/2+5\sqrt{3}F_4/2` | -# +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ -# |:math:`xz, 3z^2-r^2, x^2-y^2, xz` | :math:`-\sqrt{3}B` | :math:`-\sqrt{3}F_2+5\sqrt{3}F_4` | -# +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ -# |:math:`xz, xz, x^2-y^2, 3z^2-r^2` | :math:`\sqrt{3}B/2` | :math:`\sqrt{3}F_2/2-5\sqrt{3}F_4/2` | -# +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ -# |:math:`yz, 3z^2-r^2, x^2-y^2, yz` | :math:`\sqrt{3}B` | :math:`\sqrt{3}F_2-5\sqrt{3}F_4` | -# +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ -# |:math:`yz, yz, x^2-y^2, 3z^2-r^2` | :math:`-\sqrt{3}B/2` | :math:`-\sqrt{3}F_2/2+5\sqrt{3}F_4/2` | -# +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ -# -# -# .. _table_4_orbital: -# .. table:: Table of 4 orbital interactions -# -# +-----------------------------+-------------+----------------------------------------------------+-----------------------------------------------------+ -# |Orbitals :math:`\alpha,\beta,\gamma,\delta`|:math:`\langle\alpha\beta|\gamma\delta\rangle` Racah|:math:`\langle\alpha\beta|\gamma\delta\rangle` Slater| -# +=============================+=============+====================================================+=====================================================+ -# |:math:`3z^2-r^2, xz, yz, xy` | :math:`-\sqrt{3}B` | :math:`-\sqrt{3}F_2+5\sqrt{3}F_4` | -# +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ -# |:math:`3z^2-r^2, xz, xy, yz` | :math:`\sqrt{3}B/2` | :math:`\sqrt{3}F_2/2-5\sqrt{3}F_4/2` | -# +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ -# |:math:`3z^2-r^2, yz, xz, xy` | :math:`-\sqrt{3}B` | :math:`-\sqrt{3}F_2+5\sqrt{3}F_4` | -# +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ -# |:math:`3z^2-r^2, yz, xy, xz` | :math:`\sqrt{3}B/2` | :math:`\sqrt{3}F_2/2-5\sqrt{3}F_4/2` | -# +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ -# |:math:`3z^2-r^2, xy, xz, yz` | :math:`\sqrt{3}B/2` | :math:`\sqrt{3}F_2/2-5\sqrt{3}F_4/2` | -# +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ -# |:math:`3z^2-r^2, xy, yz, xz` | :math:`\sqrt{3}B/2` | :math:`\sqrt{3}F_2/2-5\sqrt{3}F_4/2` | -# +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ -# |:math:`x^2-y^2 , xz, xy, yz` | :math:`-3B/2` | :math:`-3F_2/2+15F_4/2` | -# +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ -# |:math:`x^2-y^2 , yz, xy, xz` | :math:`3B/2` | :math:`3F_2/2-15F_4/2` | -# +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ -# |:math:`x^2-y^2 , xy, xz, yz` | :math:`-3B/2` | :math:`-3F_2/2+15F_4/2` | -# +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ -# |:math:`x^2-y^2 , xy, yz, xz` | :math:`3B/2` | :math:`3F_2/2-15F_4/2` | -# +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ -# -# -# .. rubric:: Footnotes -# -# .. [1] MSugano S, Tanabe Y and Kamimura H. 1970. Multiplets of -# Transition-Metal Ions in Crystals. Academic Press, New York and London. diff --git a/edrixs/_downloads/66f2a6da335f3f194c2adc5e82be4b27/example_8_Hunds_interactions.ipynb b/edrixs/_downloads/66f2a6da335f3f194c2adc5e82be4b27/example_8_Hunds_interactions.ipynb deleted file mode 100644 index 59ddcab40b..0000000000 --- a/edrixs/_downloads/66f2a6da335f3f194c2adc5e82be4b27/example_8_Hunds_interactions.ipynb +++ /dev/null @@ -1,193 +0,0 @@ -{ - "cells": [ - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n# Hund's Interactions in charge transfer insulators\nIn this exercise we will solve a toy model relevant to cubic $d^8$ charge transfer insulators\nsuch as NiO or NiPS\\ :sub:`3`. We are interested in better understanding the interplay between the\nHund's interactions and the charge transfer energy in terms of the energy of the triplet-singlet\nexcitations of this model. These seem to act against each other in that the Hund's interactions\nimpose a energy cost for the triplet-singlet excitations whenever there are two holes on\nthe Ni $d$ orbitals. The charge transfer physics, on the other hand, will promote a\n$d^9\\underline{L}$ ground state in which the Hund's interactions are not active.\n\nThe simplest model that captures this physics requires four Ni spin-orbitals, representing the Ni\n$e_g$ manifold. We will represent the ligand states in the same way as the Anderson impurity\nmodel in terms of one effective ligand spin-orbital per Ni spin-orbital. We assume these effective\norbitals have been constructed so that each Ni orbital only bonds to one sister orbital. For\nsimplicity, we will treat all Ni and all ligand orbitals as equivalent, even though a more\nrealistic model would account for the different Coulomb and hopping of the $d_{3z^2-r^2}$\nand $d_{x^2-y^2}$ orbitals. We therefore simply connect Ni and ligand orbitals via a constant\nhopping $t$. We also include the ligand energy parameter $e_L$.\n\nThe easiest way to implement the requried Coulomb interactions is to use the so-called Kanamori\nHamiltonian, which is a simplfied form for the interactions, which treats all orbitals as\nequivalent. Daniel Khomskii's book provides a great explanation of this physics [1]_. We\nparameterize the interactions via Coulomb repulsion parameter $U$ and Hund's exchange\n$J_H$. EDRIXS provides this functionality via the more general\n:func:`.get_umat_kanamori` function.\n\nIt's also easiest to consider this problem in hole langauge, which means our eight spin-orbitals\nare populated by two fermions.\n" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Setup\nWe start by loading the necessary modules, and defining the total number of\norbitals and electrons.\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "import edrixs\nimport scipy\nimport numpy as np\nimport matplotlib.pyplot as plt\n\nnorb = 8\nnoccu = 2" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Diagonalization\nLet's write a function to diagonalize our model in a similar way to\nthe `sphx_glr_auto_examples_example_6_Hubbard_dimer.py` example.\nWithin this function, we also create operators to count the number of\n$d$ holes and operators to calculate expectation values for\n$S^2$ and $S_z$. For the latter to make sense, we also include a\nsmall effective spin interaction along $z$.\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "def diagonalize(U, JH, t, eL, n=1):\n # Setup Coulomb matrix\n umat = np.zeros((norb, norb, norb, norb), dtype=complex)\n uNi = edrixs.get_umat_kanamori(norb//2, U, JH)\n umat[:norb//2, :norb//2, :norb//2, :norb//2] = uNi\n\n # Setup hopping matrix\n emat = np.zeros((norb, norb), dtype=complex)\n ind = np.arange(norb//2)\n emat[ind, ind + norb//2] = t\n emat[ind+norb//2, ind] = np.conj(t) # conj is not needed, but is good practise.\n ind = np.arange(norb//2, norb)\n emat[ind, ind] += eL\n\n # Spin operator\n spin_mom = np.zeros((3, norb, norb), dtype=complex)\n spin_mom[:, :2, :2] = edrixs.get_spin_momentum(0)\n spin_mom[:, 2:4, 2:4] = edrixs.get_spin_momentum(0)\n spin_mom[:, 4:6, 4:6] = edrixs.get_spin_momentum(0)\n spin_mom[:, 6:8, 6:8] = edrixs.get_spin_momentum(0)\n\n # add small effective field along z\n emat += 1e-6*spin_mom[2]\n\n # Diagonalize\n basis = edrixs.get_fock_bin_by_N(norb, noccu)\n H = edrixs.build_opers(2, emat, basis) + edrixs.build_opers(4, umat, basis)\n e, v = scipy.linalg.eigh(H)\n e -= e[0] # Define ground state as zero energy\n\n # Operator for holes on Ni\n basis = np.array(basis)\n num_d_electrons = basis[:, :4].sum(1)\n d0 = np.sum(np.abs(v[num_d_electrons == 0, :])**2, axis=0)\n d1 = np.sum(np.abs(v[num_d_electrons == 1, :])**2, axis=0)\n d2 = np.sum(np.abs(v[num_d_electrons == 2, :])**2, axis=0)\n\n # S^2 and Sz operators\n opS = edrixs.build_opers(2, spin_mom, basis)\n S_squared_op = np.dot(opS[0], opS[0]) + np.dot(opS[1], opS[1]) + np.dot(opS[2], opS[2])\n S_squared_exp = edrixs.cb_op(S_squared_op, v).diagonal().real\n S_z_exp = edrixs.cb_op(opS[2], v).diagonal().real\n\n return e[:n], d0[:n], d1[:n], d2[:n], S_squared_exp[:n], S_z_exp[:n]" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## The atomic limit\nFor simplicity, let's start in the atomic limit with $e_L \\gg t \\gg U$\nwhere all holes are on nickel. In this case, there are six ways to distribute\ntwo holes on the four Ni spin-orbitals. Let's examine the expectation values\nof the $S^2$ and $S_z$ operators.\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "U = 10\nJH = 2\nt = 100\neL = 1e10\n\ne, d0, d1, d2, S_squared_exp, S_z_exp = diagonalize(U, JH, t, eL, n=6)\n\nprint(\"Ground state\\nE\\t\")\nfor i in range(3):\n print(f\"{e[i]:.2f}\\t{S_squared_exp[i]:.2f}\\t{S_z_exp[i]:.2f}\")\n\nprint(\"\\nExcited state\\nE\\t\")\nfor i in range(3, 6):\n print(f\"{e[i]:.2f}\\t{S_squared_exp[i]:.2f}\\t{S_z_exp[i]:.2f}\")" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The ground state is a high-spin triplet. The fourth and fifth\nstates (the first excited state) are low-spin singlet excitons at\n$2 J_H$. These have one hole on each orbital in the antisymmetric\ncombination of $|\\uparrow\\downarrow>-|\\downarrow\\uparrow>$.\nThe state at $3 J_H$ also has one hole on each orbital in the symmetric\n$|\\uparrow\\downarrow>+|\\downarrow\\uparrow>$ configuration.\n\n" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Where are the holes for large hopping\nAs discussed at the start, we are interested to see interplay between Hund's\nand charge-transfer physics, which will obviously depend strongly on whether\nthe holes are on Ni or the ligand. Let's see what happens as $e_L$ is\nreduced while observing the location of the ground state and exciton holes.\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "U = 10\nJH = 2\nt = 100\n\neLs = np.linspace(0, 1000, 30)\n\nfig, axs = plt.subplots(1, 2, figsize=(8, 4))\n\nfor ax, ind in zip(axs.ravel(), [0, 3]):\n ds = np.array([diagonalize(U, JH, t, eL, n=6)\n for eL in eLs])\n\n ax.plot(eLs, ds[:, 1, ind], 'o', label='$d^0$')\n ax.plot(eLs, ds[:, 2, ind], 's', label='$d^1$')\n ax.plot(eLs, ds[:, 3, ind], '^', label='$d^2$')\n ax.set_xlabel(\"Energy of ligands $e_L$\")\n ax.set_ylabel(\"Number of electrons\")\n ax.legend()\n\naxs[0].set_title(\"Location of ground state holes\")\naxs[1].set_title(\"Location of exciton holes\")\n\nplt.tight_layout()\nplt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "For large $e_L$, we see that both holes are on nickel as expected. In\nthe opposite limit of $|e_L| \\ll t$ and $U \\ll t$ the holes are\nshared in the ratio 0.25:0.5:0.25 as there are two ways to have one hole on\nNi. In the limit of large $e_L$, all holes move onto Ni. Since\n$t$ is large, this applies equally to both the ground state and the\nexciton.\n\n" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Connecton between atomic and charge transfer limits\nWe now examine the quantum numbers during cross over between the two limits\nwith $e_L$. Let's first look at the how $$ changes for the\nground state and exciton and then examine how the exciton energy changes.\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "U = 10\nJH = 2\nt = 100\n\neLs = np.linspace(0, 1000, 30)\n\ninfo = np.array([diagonalize(U, JH, t, eL, n=6)\n for eL in eLs])\n\nfig, axs = plt.subplots(1, 2, figsize=(8, 4))\n\n\naxs[0].plot(eLs, info[:, 4, 0], label='Ground state')\naxs[0].plot(eLs, info[:, 4, 3], label='Exciton')\naxs[0].set_xlabel(\"Energy of ligands $e_L$\")\naxs[0].set_ylabel('$$')\naxs[0].set_title('Quantum numbers')\naxs[0].legend()\n\naxs[1].plot(eLs, info[:, 0, 3], '+', color='C0')\naxs[1].set_xlabel(\"Energy of ligands $e_L$\")\naxs[1].set_ylabel('Exciton energy', color='C0')\naxr = axs[1].twinx()\naxr.plot(eLs, info[:, 3, 5], 'x', color='C1')\naxr.set_ylabel('$d^2$ fraction', color='C1')\n\nfor ax, color in zip([axs[1], axr], ['C0', 'C1']):\n for tick in ax.get_yticklabels():\n tick.set_color(color)\n\naxs[1].set_ylim(0, 2*JH)\naxr.set_ylim(0, 1)\n\naxs[1].set_title('Exciton energy vs. $d^2$ character')\nplt.tight_layout()\nplt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "In the left panel, we see that the two limits are adiabatically connected\nas they preseve the same quantum numbers. This is because there is always\nan appreciable double occupancy under conditions where the\n$d^9\\underline{L}$ character is maximized and this continues to favor\nthe high spin ground state. Other interactions such as strong tetragonal\ncrystal field would be needed to overcome the Hund's interactions and break\nthis paradigm. In the right panel, we see that the exciton energy simply\nscales with the double occupancy. Overall, even though\nHund's interactions are irrelevant for the $d^9\\underline{L}$\nelectronic configuration, whenever $t$ is appreciable there is a\nstrong mixing with the $d^8$ component is always present, which\ndominates the energy of the exciton.\n\n" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Charge transfer excitons\nAnother limiting case of the model is where $t$ is smaller than the\nCoulomb interactions. This, however, tends to produce\nground state and exciton configurations that correspond to those of distinct\natomic models. Let's look at the $e_L$ dependence in this case.\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "U = 10\nJH = 2\nt = .5\neL = 7\n\neLs = np.linspace(0, 20, 30)\n\nfig, axs = plt.subplots(1, 2, figsize=(8, 4))\n\nfor ax, ind in zip(axs.ravel(), [0, 3]):\n ds = np.array([diagonalize(U, JH, t, eL, n=6)\n for eL in eLs])\n\n ax.plot(eLs, ds[:, 1, ind], 'o', label='$d^0$')\n ax.plot(eLs, ds[:, 2, ind], 's', label='$d^1$')\n ax.plot(eLs, ds[:, 3, ind], '^', label='$d^2$')\n ax.set_xlabel(\"Energy of ligands $e_L$\")\n ax.set_ylabel(\"Number of electrons\")\n ax.legend()\n\naxs[0].axvline(x=eL, linestyle=':', color='k')\naxs[1].axvline(x=eL, linestyle=':', color='k')\n\naxs[0].set_title(\"Location of ground state holes\")\naxs[1].set_title(\"Location of exciton holes\")\n\nplt.tight_layout()\nplt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Around $e_L = 7$ the plot shows that the excition is primairly a\n$d^2 \\rightarrow d^1$ transition or a\n$d^8 \\rightarrow d^{9}\\underline{L}$ transition in electron language.\nLet's examine the energy and quantum numbers.\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "e, d0, d1, d2, S_squared_exp, S_z_exp = diagonalize(U, JH, t, eL, n=6)\n\nprint(\"Ground state\\nE\\t\")\nfor i in range(3):\n print(f\"{e[i]:.2f}\\t{S_squared_exp[i]:.2f}\\t{S_z_exp[i]:.2f}\")\n\nprint(\"\\nExcited state\\nE\\t\")\nfor i in range(3, 6):\n print(f\"{e[i]:.2f}\\t{S_squared_exp[i]:.2f}\\t{S_z_exp[i]:.2f}\")" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We once again see the same quantum numbers, despite the differences in mixing\nin the ground state and exciton.\n\n" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - ".. rubric:: Footnotes\n\n.. [1] D. Khomskii, Transition Metal Compounds, Cambridge University Press (2014)\n\n" - ] - } - ], - "metadata": { - "kernelspec": { - "display_name": "Python 3", - "language": "python", - "name": "python3" - }, - "language_info": { - "codemirror_mode": { - "name": "ipython", - "version": 3 - }, - "file_extension": ".py", - "mimetype": "text/x-python", - "name": "python", - "nbconvert_exporter": "python", - "pygments_lexer": "ipython3", - "version": "3.10.16" - } - }, - "nbformat": 4, - "nbformat_minor": 0 -} \ No newline at end of file diff --git a/edrixs/_downloads/687fb58743d637976afd57fbd8a78ef9/example_9_Coulomb.ipynb b/edrixs/_downloads/687fb58743d637976afd57fbd8a78ef9/example_9_Coulomb.ipynb deleted file mode 100644 index d05c2ff880..0000000000 --- a/edrixs/_downloads/687fb58743d637976afd57fbd8a78ef9/example_9_Coulomb.ipynb +++ /dev/null @@ -1,215 +0,0 @@ -{ - "cells": [ - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n# Coulomb interactions\nIn this example we provide more details on how Coulomb interactions are\nimplemented in multiplet calculations and EDRIXS in particular. We aim\nto clarify the form of the matrices, how they are parametrized,\nand how the breaking of spherical symmetry can switch on additional elements\nthat one might not anticipate. Our example is based on a $d$ atomic shell.\n" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Create matrix\nThe Coulomb interaction between two particles can be written as\n\n .. math::\n \\begin{equation}\n \\hat{H} = \\frac{1}{2}\n \\int d\\mathbf{r} \\int d\\mathbf{r}^\\prime\n \\Sigma_{\\sigma, \\sigma^\\prime}\n |\\hat{\\psi}^\\sigma(\\mathbf{r})|^2 \\frac{e^2}{R}\n |\\hat{\\psi}^{\\sigma^\\prime}(\\mathbf{r^\\prime})|^2,\n \\end{equation}\n\nwhere $\\hat{\\psi}^\\sigma(\\mathbf{r})$ is the electron wavefunction, with\nspin $\\sigma$, and $R=|r-r^\\prime|$ is the electron separation.\nSolving our problem in this form is difficult due to the need to symmeterize\nthe wavefunction to follow fermionic statistics.\nUsing second quantization, we can use operators to impose the required\nparticle exchange statistics and write the equation in terms of\na tensor $U$\n\n .. math::\n \\begin{equation}\n \\hat{H} = \\sum_{\\alpha,\\beta,\\gamma,\\delta,\\sigma,\\sigma^\\prime}\n U_{\\alpha\\sigma,\\beta\\sigma^\\prime,\\gamma\\sigma^\\prime,\\delta\\sigma}\n \\hat{f}^{\\dagger}_{\\alpha\\sigma}\n \\hat{f}^{\\dagger}_{\\beta\\sigma^\\prime}\n \\hat{f}_{\\gamma\\sigma^\\prime}\\hat{f}_{\\delta\\sigma},\n \\end{equation}\n\nwhere $\\alpha$, $\\beta$, $\\gamma$, $\\delta$ are\norbital indices and $\\hat{f}^{\\dagger}$\n($\\hat{f}$) are the creation (anihilation) operators.\nFor a $d$-electron system, we have $10$ distinct spin-orbitals\n($5$ orbitals each with $2$ spins), which makes matrix the\n$10\\times10\\times10\\times10$ in total size.\nIn EDRIXS the matrix can be created as follows:\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "import edrixs\nimport numpy as np\nimport scipy\nimport matplotlib.pyplot as plt\nimport itertools\n\nF0, F2, F4 = 6.94, 14.7, 4.41\numat_chb = edrixs.get_umat_slater('d', F0, F2, F4)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We stored this under variable :code:`umat_chb` where \"cbh\" stands for\ncomplex harmonic basis, which is the default basis in EDRIXS.\n\n" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Parameterizing interactions\nEDRIXS parameterizes the interactions in $U$ via Slater integral\nparameters $F^{k}$. These relate to integrals of various spherical\nHarmonics as well as Clebsch-Gordon coefficients, Gaunt coefficients,\nand Wigner 3J symbols. Textbooks such as [1]_ can be used for further\nreference. If you are interested in the details of how\nEDRIXS does this (and you probably aren't) function :func:`.umat_slater`,\nconstructs the required matrix via Gaunt coeficents from\n:func:`.get_gaunt`. Two alternative parameterizations are common.\nThe first are the Racah parameters, which are\n\n .. math::\n \\begin{eqnarray}\n A &=& F^0 - \\frac{49}{441} F^4 \\\\\n B &=& \\frac{1}{49}F^2 - \\frac{5}{441}F^4 \\\\\n C &=& \\frac{35}{441}F^4.\n \\end{eqnarray}\n\nor an alternative form for the Slater integrals\n\n .. math::\n \\begin{eqnarray}\n F_0 &=& F^0 \\\\\n F_2 &=& \\frac{1}{49}F^2 \\\\\n F_4 &=& \\frac{1}{441}F^4,\n \\end{eqnarray}\n\nwhich involves different normalization parameters.\n\n" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Basis transform\nIf we want to use the real harmonic basis, we can use a tensor\ntransformation, which imposes the following orbital order\n$3z^2-r^2, xz, yz, x^2-y^2, xy$, each of which involves\n$\\uparrow, \\downarrow$ spin pairs. Let's perform this transformation and\nstore a list of these orbitals.\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "umat = edrixs.transform_utensor(umat_chb, edrixs.tmat_c2r('d', True))\norbitals = ['3z^2-r^2', 'xz', 'yz', 'x^2-y^2', 'xy']" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Interactions\nTensor $U$ is a series of matrix\nelements\n\n .. math::\n \\begin{equation}\n \\langle\\psi_{\\gamma,\\delta}^{\\bar{\\sigma},\\bar{\\sigma}^\\prime}\n |\\hat{H}|\n \\psi_{\\alpha,\\beta}^{\\sigma,\\sigma^\\prime}\\rangle\n \\end{equation}\n\nthe combination of which defines the energetic cost of pairwise\nelectron-electron interactions between states $\\alpha,\\sigma$\nand $\\beta,\\sigma^\\prime$. In EDRIXS we follow the convention of\nsumming over all orbital pairs. Some other texts count each pair of\nindices only once. The matrix elements here will consequently\nbe half the magnitude of those in other references.\nWe can express the interactions in terms of\nthe orbitals involved. It is common to distinguish \"direct Coulomb\" and\n\"exchange\" interactions. The former come from electrons in the same orbital\nand the later involve swapping orbital labels for electrons. We will use\n$U_0$ and $J$ as a shorthand for distinguishing these.\n\nBefore we describe the different types of interactions, we note that since\nthe Coulomb interaction is real, and due to the spin symmmetry properties\nof the process $U$ always obeys\n\n .. math::\n \\begin{equation}\n U_{\\alpha\\sigma,\\beta\\sigma^\\prime,\\gamma\\sigma^\\prime,\\delta\\sigma} =\n U_{\\beta\\sigma,\\alpha\\sigma^\\prime,\\delta\\sigma^\\prime,\\gamma\\sigma} =\n U_{\\delta\\sigma,\\gamma\\sigma^\\prime,\\beta\\sigma^\\prime,\\alpha\\sigma} =\n U_{\\gamma\\sigma,\\delta\\sigma^\\prime,\\alpha\\sigma^\\prime,\\beta\\sigma}.\n \\end{equation}\n\n\n### 1. Intra orbital\nThe direct Coulomb energy cost to double-occupy an orbital comes from terms\nlike $U_{\\alpha\\sigma,\\alpha\\bar\\sigma,\\alpha\\bar\\sigma,\\alpha\\sigma}$.\nIn this notation, we use $\\sigma^\\prime$ to denote that the matrix\nelement is summed over all pairs and $\\bar{\\sigma}$ to denote sums\nover all opposite spin pairs. Due to rotational symmetry, all these\nelements are the same and equal to\n\n .. math::\n \\begin{eqnarray}\n U_0 &=& \\frac{A}{2} + 2B + \\frac{3C}{2}\\\\\n &=& \\frac{F_0}{2} + 2F_2 + 18F_4\n \\end{eqnarray}\n\nLet's print these to demonstrate where these live in the array\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "for i in range(0, 5):\n val = umat[i*2, i*2 + 1, i*2 + 1, i*2].real\n print(f\"{orbitals[i]:<8} \\t {val:.3f}\")" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "### 2. Inter orbital Coulomb interactions\nDirect Coulomb repulsion between different orbitals depends on terms like\n$U_{\\alpha\\sigma,\\beta\\sigma^\\prime,\\beta\\sigma^\\prime,\\alpha\\sigma}$.\nExpresions for these parameters are provided in column $U$ in\n`table_2_orbital`. We can print the values from :code:`umat`\nlike this:\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "for i, j in itertools.combinations(range(5), 2):\n val = umat[i*2, j*2 + 1, j*2 + 1, i*2].real\n print(f\"{orbitals[i]:<8} \\t {orbitals[j]:<8} \\t {val:.3f}\")" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "### 3. Inter-orbital exchange interactions\nExchange terms exist with the form\n$U_{\\alpha\\sigma,\\beta\\sigma^\\prime,\\alpha\\sigma^\\prime,\\beta\\sigma}$.\nExpresions for these parameters are provided in column $J$ of\n`table_2_orbital`. These come from terms like this in the matrix:\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "for i, j in itertools.combinations(range(5), 2):\n val = umat[i*2, j*2 + 1, i*2 + 1, j*2].real\n print(f\"{orbitals[i]:<8} \\t {orbitals[j]:<8} \\t {val:.3f}\")" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "### 4. Pair hopping term\nTerms that swap pairs of electrons exist as\n$(1-\\delta_{\\sigma\\sigma'})U_{\\alpha\\sigma,\\alpha\\bar\\sigma,\\beta\\bar\\sigma,\\beta\\sigma}$\nand depend on exchange interactions column $J$ from\n`table_2_orbital`\nand here in the matrix.\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "for i, j in itertools.combinations(range(5), 2):\n val = umat[i*2, i*2 + 1, j*2 + 1, j*2].real\n print(f\"{orbitals[i]:<8} \\t {orbitals[j]:<8} \\t {val:.3f}\")" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "### 5. Three orbital\nAnother set of terms that one might not immediately anticipate involve three\norbitals like\n\n .. math::\n \\begin{equation}\n U_{\\alpha\\sigma, \\gamma\\sigma', \\beta\\sigma', \\gamma\\sigma} \\\\\n U_{\\alpha\\sigma, \\gamma\\sigma', \\gamma\\sigma', \\beta\\sigma} \\\\\n (1-\\delta_{\\sigma\\sigma'})\n U_{\\alpha\\sigma, \\beta\\sigma', \\gamma\\sigma', \\gamma\\sigma}\n \\end{equation}\n\nfor $\\alpha=3z^2-r^2, \\beta=x^2-y^2, \\gamma=xz/yz$.\nThese are needed to maintain the rotational symmetry of the interations.\nSee `table_3_orbital` for the expressions. We can print some of\nthese via:\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "ijkl = [[0, 1, 3, 1],\n [0, 2, 3, 2],\n [1, 0, 3, 1],\n [1, 1, 3, 0],\n [2, 0, 3, 2],\n [2, 2, 3, 0]]\n\nfor i, j, k, l in ijkl:\n val = umat[i*2, j*2 + 1, k*2 + 1, l*2].real\n print(f\"{orbitals[i]:<8} \\t {orbitals[j]:<8} \\t\"\n f\"{orbitals[k]:<8} \\t {orbitals[l]:<8} \\t {val:.3f}\")" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "### 6. Four orbital\nFuther multi-orbital terms include\n$U_{\\alpha\\sigma,\\beta\\sigma^\\prime,\\gamma\\sigma^\\prime,\\delta\\sigma}$.\nWe can find these here in the matrix:\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "ijkl = [[0, 1, 2, 4],\n [0, 1, 4, 2],\n [0, 2, 1, 4],\n [0, 2, 4, 1],\n [0, 4, 1, 2],\n [0, 4, 2, 1],\n [3, 1, 4, 2],\n [3, 2, 4, 1],\n [3, 4, 1, 2],\n [3, 4, 2, 1]]\n\nfor i, j, k, l in ijkl:\n val = umat[i*2, j*2 + 1, k*2 + 1, l*2].real\n print(f\"{orbitals[i]:<8} \\t {orbitals[j]:<8} \\t {orbitals[k]:<8}\"\n f\"\\t {orbitals[l]:<8} \\t {val:.3f}\")" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Effects of multi-orbital terms\nTo test the effects of the multi-orbital terms, let's plot the eigenenergy\nspectra with and without multi-orbital terms switched on for system with and\nwithout a cubic crystal field. We will use a $d$-shell with two\nelectrons.\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "ten_dqs = [0, 2, 4, 12]\n\ndef diagonalize(ten_dq, umat):\n emat = edrixs.cb_op(edrixs.cf_cubic_d(ten_dq),\n edrixs.tmat_c2r('d', ispin=True))\n H = (edrixs.build_opers(4, umat, basis)\n + edrixs.build_opers(2, emat, basis))\n e, v = scipy.linalg.eigh(H)\n return e - e.min()\n\nbasis = edrixs.get_fock_bin_by_N(10, 2)\numat_no_multiorbital = np.copy(umat)\nB = F2/49 - 5*F4/441\nfor val in [np.sqrt(3)*B/2, np.sqrt(3)*B, 3*B/2]:\n umat_no_multiorbital[(np.abs(umat)- val) < 1e-6] = 0\n\nfig, axs = plt.subplots(1, len(ten_dqs), figsize=(8, 3))\n\nfor cind, (ax, ten_dq) in enumerate(zip(axs, ten_dqs)):\n ax.hlines(diagonalize(ten_dq, umat), xmin=0, xmax=1,\n label='on', color=f'C{cind}')\n ax.hlines(diagonalize(ten_dq, umat_no_multiorbital),\n xmin=1.5, xmax=2.5,\n label='off',\n linestyle=':', color=f'C{cind}')\n ax.set_title(f\"$10D_q={ten_dq}$\")\n ax.set_ylim([-.5, 20])\n ax.set_xticks([])\n ax.legend()\n\nfig.suptitle(\"Eigenvalues with 3&4-orbital effects on/off\")\nfig.subplots_adjust(wspace=.3)\naxs[0].set_ylabel('Eigenvalues (eV)')\nfig.subplots_adjust(top=.8)\nplt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "On the left of the plot Coulomb interactions in spherical symmetry cause\nsubstantial mxing between $t_{2g}$ and $e_{g}$ orbitals in the\neigenstates and 3 & 4 orbital orbital terms are crucial for obtaining the\nthe right eigenenergies. As $10D_q$ get large, this mixing is switched\noff and the spectra start to become independent of whether the 3 & 4 orbital\norbital terms are included or not.\n\n\n\n.. table:: Table of 2 orbital interactions\n\n +-----------------------------+------------------+-----------------------+----------------+--------------------+\n |Orbitals $\\alpha,\\beta$|$U_0$ Racah | $U_0$ Slater |$J$ Racah |$J$ Slater |\n +=============================+==================+=======================+================+====================+\n |$3z^2-r^2, xz$ |$A/2+B+C/2$ |$F_0/2+F_2-12F_4$| $B/2+C/2$|$F_2/2+15F_4$ |\n +-----------------------------+------------------+-----------------------+----------------+--------------------+\n |$3z^2-r^2, yz$ |$A/2+B+C/2$ |$F_0/2+F_2-12F_4$| $B/2+C/2$|$F_2/2+15F_4$ |\n +-----------------------------+------------------+-----------------------+----------------+--------------------+\n |$3z^2-r^2, x^2-y^2$ |$A/2-2B+C/2$|$F_0/2-2F_2+3F_4$|$2B+C/2$ |$2F_2+15F_4/2$|\n +-----------------------------+------------------+-----------------------+----------------+--------------------+\n |$3z^2-r^2, xy$ |$A/2-2B+C/2$|$F_0/2-2F_2+3F_4$|$2B+C/2$ |$2F_2+15F_4/2$|\n +-----------------------------+------------------+-----------------------+----------------+--------------------+\n |$xz, yz$ |$A/2-B+C/2$ |$F_0/2-F_2-12F_4$|$3B/2+C/2$|$3F_2/2+10F_4$|\n +-----------------------------+------------------+-----------------------+----------------+--------------------+\n |$xz, x^2-y^2$ |$A/2-B+C/2$ |$F_0/2-F_2-12F_4$|$3B/2+C/2$|$3F_2/2+10F_4$|\n +-----------------------------+------------------+-----------------------+----------------+--------------------+\n |$xz, xy$ |$A/2-B+C/2$ |$F_0/2-F_2-12F_4$|$3B/2+C/2$|$3F_2/2+10F_4$|\n +-----------------------------+------------------+-----------------------+----------------+--------------------+\n |$yz, x^2-y^2$ |$A/2-B+C/2$ |$F_0/2-F_2-12F_4$|$3B/2+C/2$|$3F_2/2+10F_4$|\n +-----------------------------+------------------+-----------------------+----------------+--------------------+\n |$yz, xy$ |$A/2-B+C/2$ |$F_0/2-F_2-12F_4$|$3B/2+C/2$|$3F_2/2+10F_4$|\n +-----------------------------+------------------+-----------------------+----------------+--------------------+\n |$x^2-y^2, xy$ |$A/2+2B+C/2$|$F_0+4F_2-34F_4$ | $C/2$ |$35F_4/2$ |\n +-----------------------------+------------------+-----------------------+----------------+--------------------+\n\n\n.. table:: Table of 3 orbital interactions\n\n +-----------------------------+-------------+----------------------------------------------------+-----------------------------------------------------+\n |Orbitals $\\alpha,\\beta,\\gamma,\\delta$|$\\langle\\alpha\\beta|\\gamma\\delta\\rangle$ Racah|$\\langle\\alpha\\beta|\\gamma\\delta\\rangle$ Slater|\n +=============================+=============+====================================================+=====================================================+\n |$3z^2-r^2, xz, x^2-y^2, xz$ | $\\sqrt{3}B/2$ | $\\sqrt{3}F_2/2-5\\sqrt{3}F_4/2$ |\n +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+\n |$3z^2-r^2, yz, x^2-y^2, yz$ | $-\\sqrt{3}B/2$ | $-\\sqrt{3}F_2/2+5\\sqrt{3}F_4/2$ |\n +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+\n |$xz, 3z^2-r^2, x^2-y^2, xz$ | $-\\sqrt{3}B$ | $-\\sqrt{3}F_2+5\\sqrt{3}F_4$ |\n +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+\n |$xz, xz, x^2-y^2, 3z^2-r^2$ | $\\sqrt{3}B/2$ | $\\sqrt{3}F_2/2-5\\sqrt{3}F_4/2$ |\n +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+\n |$yz, 3z^2-r^2, x^2-y^2, yz$ | $\\sqrt{3}B$ | $\\sqrt{3}F_2-5\\sqrt{3}F_4$ |\n +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+\n |$yz, yz, x^2-y^2, 3z^2-r^2$ | $-\\sqrt{3}B/2$ | $-\\sqrt{3}F_2/2+5\\sqrt{3}F_4/2$ |\n +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+\n\n\n.. table:: Table of 4 orbital interactions\n\n +-----------------------------+-------------+----------------------------------------------------+-----------------------------------------------------+\n |Orbitals $\\alpha,\\beta,\\gamma,\\delta$|$\\langle\\alpha\\beta|\\gamma\\delta\\rangle$ Racah|$\\langle\\alpha\\beta|\\gamma\\delta\\rangle$ Slater|\n +=============================+=============+====================================================+=====================================================+\n |$3z^2-r^2, xz, yz, xy$ | $-\\sqrt{3}B$ | $-\\sqrt{3}F_2+5\\sqrt{3}F_4$ |\n +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+\n |$3z^2-r^2, xz, xy, yz$ | $\\sqrt{3}B/2$ | $\\sqrt{3}F_2/2-5\\sqrt{3}F_4/2$ |\n +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+\n |$3z^2-r^2, yz, xz, xy$ | $-\\sqrt{3}B$ | $-\\sqrt{3}F_2+5\\sqrt{3}F_4$ |\n +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+\n |$3z^2-r^2, yz, xy, xz$ | $\\sqrt{3}B/2$ | $\\sqrt{3}F_2/2-5\\sqrt{3}F_4/2$ |\n +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+\n |$3z^2-r^2, xy, xz, yz$ | $\\sqrt{3}B/2$ | $\\sqrt{3}F_2/2-5\\sqrt{3}F_4/2$ |\n +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+\n |$3z^2-r^2, xy, yz, xz$ | $\\sqrt{3}B/2$ | $\\sqrt{3}F_2/2-5\\sqrt{3}F_4/2$ |\n +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+\n |$x^2-y^2 , xz, xy, yz$ | $-3B/2$ | $-3F_2/2+15F_4/2$ |\n +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+\n |$x^2-y^2 , yz, xy, xz$ | $3B/2$ | $3F_2/2-15F_4/2$ |\n +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+\n |$x^2-y^2 , xy, xz, yz$ | $-3B/2$ | $-3F_2/2+15F_4/2$ |\n +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+\n |$x^2-y^2 , xy, yz, xz$ | $3B/2$ | $3F_2/2-15F_4/2$ |\n +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+\n\n\n.. rubric:: Footnotes\n\n.. [1] MSugano S, Tanabe Y and Kamimura H. 1970. Multiplets of\n Transition-Metal Ions in Crystals. Academic Press, New York and London.\n\n" - ] - } - ], - "metadata": { - "kernelspec": { - "display_name": "Python 3", - "language": "python", - "name": "python3" - }, - "language_info": { - "codemirror_mode": { - "name": "ipython", - "version": 3 - }, - "file_extension": ".py", - "mimetype": "text/x-python", - "name": "python", - "nbconvert_exporter": "python", - "pygments_lexer": "ipython3", - "version": "3.10.16" - } - }, - "nbformat": 4, - "nbformat_minor": 0 -} \ No newline at end of file diff --git a/edrixs/_downloads/6892534c439210f8e60d50aca5e6bbee/example_8_Coulomb.py b/edrixs/_downloads/6892534c439210f8e60d50aca5e6bbee/example_8_Coulomb.py deleted file mode 100644 index 2b8065f156..0000000000 --- a/edrixs/_downloads/6892534c439210f8e60d50aca5e6bbee/example_8_Coulomb.py +++ /dev/null @@ -1,381 +0,0 @@ -#!/usr/bin/env python -""" -Coulomb interactions -===================================== -In this example we provide more details on how Coulomb interactions are -implemented in multiplet calculations and EDRIXS in particular. We aim -to clarify the form of the matrices, how they are parametrized, -and how the breaking of spherical symmetry can switch on additional elements -that one might not anticipate. Our example is based on a :math:`d` atomic shell. -""" - -################################################################################ -# Create matrix -# ------------------------------------------------------------------------------ -# The Coulomb interaction between two particles can be written as -# -# .. math:: -# \begin{equation} -# \hat{H} = \frac{1}{2} -# \int d\mathbf{r} \int d\mathbf{r}^\prime -# \Sigma_{\sigma, \sigma^\prime} -# |\hat{\psi}^\sigma(\mathbf{r})|^2 \frac{e^2}{R} -# |\hat{\psi}^{\sigma^\prime}(\mathbf{r^\prime})|^2, -# \end{equation} -# -# where :math:`\hat{\psi}^\sigma(\mathbf{r})` is the electron wavefunction, with -# spin :math:`\sigma`, and :math:`R=|r-r^\prime|` is the electron separation. -# Solving our problem in this form is difficult due to the need to symmeterize -# the wavefunction to follow fermionic statistics. -# Using second quantization, we can use operators to impose the required -# particle exchange statistics and write the equation in terms of -# a tensor :math:`U` -# -# .. math:: -# \begin{equation} -# \hat{H} = \sum_{\alpha,\beta,\gamma,\delta,\sigma,\sigma^\prime} -# U_{\alpha\sigma,\beta\sigma^\prime,\gamma\sigma^\prime,\delta\sigma} -# \hat{f}^{\dagger}_{\alpha\sigma} -# \hat{f}^{\dagger}_{\beta\sigma^\prime} -# \hat{f}_{\gamma\sigma^\prime}\hat{f}_{\delta\sigma}, -# \end{equation} -# -# where :math:`\alpha`, :math:`\beta`, :math:`\gamma`, :math:`\delta` are -# orbital indices and :math:`\hat{f}^{\dagger}` -# (:math:`\hat{f}`) are the creation (anihilation) operators. -# For a :math:`d`-electron system, we have :math:`10` distinct spin-orbitals -# (:math:`5` orbitals each with :math:`2` spins), which makes matrix the -# :math:`10\times10\times10\times10` in total size. -# In EDRIXS the matrix can be created as follows: -import edrixs -import numpy as np -import scipy -import matplotlib.pyplot as plt -import itertools - -F0, F2, F4 = 6.94, 14.7, 4.41 -umat_chb = edrixs.get_umat_slater('d', F0, F2, F4) -################################################################################ -# We stored this under variable :code:`umat_chb` where "cbh" stands for -# complex harmonic basis, which is the default basis in EDRIXS. - -################################################################################ -# Parameterizing interactions -# ------------------------------------------------------------------------------ -# EDRIXS parameterizes the interactions in :math:`U` via Slater integral -# parameters :math:`F^{k}`. These relate to integrals of various spherical -# Harmonics as well as Clebsch-Gordon coefficients, Gaunt coefficients, -# and Wigner 3J symbols. Textbooks such as [1]_ can be used for further -# reference. If you are interested in the details of how -# EDRIXS does this (and you probably aren't) function :func:`.umat_slater`, -# constructs the required matrix via Gaunt coeficents from -# :func:`.get_gaunt`. Two alternative parameterizations are common. -# The first are the Racah parameters, which are -# -# .. math:: -# \begin{eqnarray} -# A &=& F^0 - \frac{49}{441} F^4 \\ -# B &=& \frac{1}{49}F^2 - \frac{5}{441}F^4 \\ -# C &=& \frac{35}{441}F^4. -# \end{eqnarray} -# -# or an alternative form for the Slater integrals -# -# .. math:: -# \begin{eqnarray} -# F_0 &=& F^0 \\ -# F_2 &=& \frac{1}{49}F^2 \\ -# F_4 &=& \frac{1}{441}F^4, -# \end{eqnarray} -# -# which involves different normalization parameters. - -################################################################################ -# Basis transform -# ------------------------------------------------------------------------------ -# If we want to use the real harmonic basis, we can use a tensor -# transformation, which imposes the following orbital order -# :math:`3z^2-r^2, xz, yz, x^2-y^2, xy`, each of which involves -# :math:`\uparrow, \downarrow` spin pairs. Let's perform this transformation and -# store a list of these orbitals. -umat = edrixs.transform_utensor(umat_chb, edrixs.tmat_c2r('d', True)) -orbitals = ['3z^2-r^2', 'xz', 'yz', 'x^2-y^2', 'xy'] - -################################################################################ -# Interactions -# ------------------------------------------------------------------------------ -# Tensor :math:`U` is a series of matrix -# elements -# -# .. math:: -# \begin{equation} -# \langle\psi_{\gamma,\delta}^{\bar{\sigma},\bar{\sigma}^\prime} -# |\hat{H}| -# \psi_{\alpha,\beta}^{\sigma,\sigma^\prime}\rangle -# \end{equation} -# -# the combination of which defines the energetic cost of pairwise -# electron-electron interactions between states :math:`\alpha,\sigma` -# and :math:`\beta,\sigma^\prime`. In EDRIXS we follow the convention of -# summing over all orbital pairs. Some other texts count each pair of -# indices only once. The matrix elements here will consequently -# be half the magnitude of those in other references. -# We can express the interactions in terms of -# the orbitals involved. It is common to distinguish "direct Coulomb" and -# "exchange" interactions. The former come from electrons in the same orbital -# and the later involve swapping orbital labels for electrons. We will use -# :math:`U_0` and :math:`J` as a shorthand for distinguishing these. -# -# Before we describe the different types of interactions, we note that since -# the Coulomb interaction is real, and due to the spin symmmetry properties -# of the process :math:`U` always obeys -# -# .. math:: -# \begin{equation} -# U_{\alpha\sigma,\beta\sigma^\prime,\gamma\sigma^\prime,\delta\sigma} = -# U_{\beta\sigma,\alpha\sigma^\prime,\delta\sigma^\prime,\gamma\sigma} = -# U_{\delta\sigma,\gamma\sigma^\prime,\beta\sigma^\prime,\alpha\sigma} = -# U_{\gamma\sigma,\delta\sigma^\prime,\alpha\sigma^\prime,\beta\sigma}. -# \end{equation} -# -# -# 1. Intra orbital -# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -# The direct Coulomb energy cost to double-occupy an orbital comes from terms -# like :math:`U_{\alpha\sigma,\alpha\bar\sigma,\alpha\bar\sigma,\alpha\sigma}`. -# In this notation, we use :math:`\sigma^\prime` to denote that the matrix -# element is summed over all pairs and :math:`\bar{\sigma}` to denote sums -# over all opposite spin pairs. Due to rotational symmetry, all these -# elements are the same and equal to -# -# .. math:: -# \begin{eqnarray} -# U_0 &=& \frac{A}{2} + 2B + \frac{3C}{2}\\ -# &=& \frac{F_0}{2} + 2F_2 + 18F_4 -# \end{eqnarray} -# -# Let's print these to demonstrate where these live in the array -for i in range(0, 5): - val = umat[i*2, i*2 + 1, i*2 + 1, i*2].real - print(f"{orbitals[i]:<8} \t {val:.3f}") - -################################################################################ -# 2. Inter orbital Coulomb interactions -# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -# Direct Coulomb repulsion between different orbitals depends on terms like -# :math:`U_{\alpha\sigma,\beta\sigma^\prime,\beta\sigma^\prime,\alpha\sigma}`. -# Expresions for these parameters are provided in column :math:`U` in -# :ref:`table_2_orbital`. We can print the values from :code:`umat` -# like this: -for i, j in itertools.combinations(range(5), 2): - val = umat[i*2, j*2 + 1, j*2 + 1, i*2].real - print(f"{orbitals[i]:<8} \t {orbitals[j]:<8} \t {val:.3f}") - -################################################################################ -# 3. Inter-orbital exchange interactions -# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -# Exchange terms exist with the form -# :math:`U_{\alpha\sigma,\beta\sigma^\prime,\alpha\sigma^\prime,\beta\sigma}`. -# Expresions for these parameters are provided in column :math:`J` of -# :ref:`table_2_orbital`. These come from terms like this in the matrix: -for i, j in itertools.combinations(range(5), 2): - val = umat[i*2, j*2 + 1, i*2 + 1, j*2].real - print(f"{orbitals[i]:<8} \t {orbitals[j]:<8} \t {val:.3f}") - -################################################################################ -# 4. Pair hopping term -# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -# Terms that swap pairs of electrons exist as -# :math:`(1-\delta_{\sigma\sigma'})U_{\alpha\sigma,\alpha\bar\sigma,\beta\bar\sigma,\beta\sigma}` -# and depend on exchange interactions column :math:`J` from -# :ref:`table_2_orbital` -# and here in the matrix. -for i, j in itertools.combinations(range(5), 2): - val = umat[i*2, i*2 + 1, j*2 + 1, j*2].real - print(f"{orbitals[i]:<8} \t {orbitals[j]:<8} \t {val:.3f}") - -################################################################################ -# 5. Three orbital -# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -# Another set of terms that one might not immediately anticipate involve three -# orbitals like -# -# .. math:: -# \begin{equation} -# U_{\alpha\sigma, \gamma\sigma', \beta\sigma', \gamma\sigma} \\ -# U_{\alpha\sigma, \gamma\sigma', \gamma\sigma', \beta\sigma} \\ -# (1-\delta_{\sigma\sigma'}) -# U_{\alpha\sigma, \beta\sigma', \gamma\sigma', \gamma\sigma} -# \end{equation} -# -# for :math:`\alpha=3z^2-r^2, \beta=x^2-y^2, \gamma=xz/yz`. -# These are needed to maintain the rotational symmetry of the interations. -# See :ref:`table_3_orbital` for the expressions. We can print some of -# these via: -ijkl = [[0, 1, 3, 1], - [0, 2, 3, 2], - [1, 0, 3, 1], - [1, 1, 3, 0], - [2, 0, 3, 2], - [2, 2, 3, 0]] - -for i, j, k, l in ijkl: - val = umat[i*2, j*2 + 1, k*2 + 1, l*2].real - print(f"{orbitals[i]:<8} \t {orbitals[j]:<8} \t" - f"{orbitals[k]:<8} \t {orbitals[l]:<8} \t {val:.3f}") - -################################################################################ -# 6. Four orbital -# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -# Futher multi-orbital terms include -# :math:`U_{\alpha\sigma,\beta\sigma^\prime,\gamma\sigma^\prime,\delta\sigma}`. -# We can find these here in the matrix: -ijkl = [[0, 1, 2, 4], - [0, 1, 4, 2], - [0, 2, 1, 4], - [0, 2, 4, 1], - [0, 4, 1, 2], - [0, 4, 2, 1], - [3, 1, 4, 2], - [3, 2, 4, 1], - [3, 4, 1, 2], - [3, 4, 2, 1]] - -for i, j, k, l in ijkl: - val = umat[i*2, j*2 + 1, k*2 + 1, l*2].real - print(f"{orbitals[i]:<8} \t {orbitals[j]:<8} \t {orbitals[k]:<8}" - f"\t {orbitals[l]:<8} \t {val:.3f}") - -################################################################################ -# Effects of multi-orbital terms -# ------------------------------------------------------------------------------ -# To test the effects of the multi-orbital terms, let's plot the eigenenergy -# spectra with and without multi-orbital terms switched on for system with and -# without a cubic crystal field. We will use a :math:`d`-shell with two -# electrons. -ten_dqs = [0, 2, 4, 12] - -def diagonalize(ten_dq, umat): - emat = edrixs.cb_op(edrixs.cf_cubic_d(ten_dq), - edrixs.tmat_c2r('d', ispin=True)) - H = (edrixs.build_opers(4, umat, basis) - + edrixs.build_opers(2, emat, basis)) - e, v = scipy.linalg.eigh(H) - return e - e.min() - -basis = edrixs.get_fock_bin_by_N(10, 2) -umat_no_multiorbital = np.copy(umat) -B = F2/49 - 5*F4/441 -for val in [np.sqrt(3)*B/2, np.sqrt(3)*B, 3*B/2]: - umat_no_multiorbital[(np.abs(umat)- val) < 1e-6] = 0 - -fig, axs = plt.subplots(1, len(ten_dqs), figsize=(8, 3)) - -for cind, (ax, ten_dq) in enumerate(zip(axs, ten_dqs)): - ax.hlines(diagonalize(ten_dq, umat), xmin=0, xmax=1, - label='on', color=f'C{cind}') - ax.hlines(diagonalize(ten_dq, umat_no_multiorbital), - xmin=1.5, xmax=2.5, - label='off', - linestyle=':', color=f'C{cind}') - ax.set_title(f"$10D_q={ten_dq}$") - ax.set_ylim([-.5, 20]) - ax.set_xticks([]) - ax.legend() - -fig.suptitle("Eigenvalues with 3&4-orbital effects on/off") -fig.subplots_adjust(wspace=.3) -axs[0].set_ylabel('Eigenvalues (eV)') -fig.subplots_adjust(top=.8) -plt.show() - -################################################################################ -# On the left of the plot Coulomb interactions in spherical symmetry cause -# substantial mxing between :math:`t_{2g}` and :math:`e_{g}` orbitals in the -# eigenstates and 3 & 4 orbital orbital terms are crucial for obtaining the -# the right eigenenergies. As :math:`10D_q` get large, this mixing is switched -# off and the spectra start to become independent of whether the 3 & 4 orbital -# orbital terms are included or not. -# -# -# -# .. _table_2_orbital: -# .. table:: Table of 2 orbital interactions -# -# +-----------------------------+------------------+-----------------------+----------------+--------------------+ -# |Orbitals :math:`\alpha,\beta`|:math:`U_0` Racah | :math:`U_0` Slater |:math:`J` Racah |:math:`J` Slater | -# +=============================+==================+=======================+================+====================+ -# |:math:`3z^2-r^2, xz` |:math:`A/2+B+C/2` |:math:`F_0/2+F_2-12F_4`| :math:`B/2+C/2`|:math:`F_2/2+15F_4` | -# +-----------------------------+------------------+-----------------------+----------------+--------------------+ -# |:math:`3z^2-r^2, yz` |:math:`A/2+B+C/2` |:math:`F_0/2+F_2-12F_4`| :math:`B/2+C/2`|:math:`F_2/2+15F_4` | -# +-----------------------------+------------------+-----------------------+----------------+--------------------+ -# |:math:`3z^2-r^2, x^2-y^2` |:math:`A/2-2B+C/2`|:math:`F_0/2-2F_2+3F_4`|:math:`2B+C/2` |:math:`2F_2+15F_4/2`| -# +-----------------------------+------------------+-----------------------+----------------+--------------------+ -# |:math:`3z^2-r^2, xy` |:math:`A/2-2B+C/2`|:math:`F_0/2-2F_2+3F_4`|:math:`2B+C/2` |:math:`2F_2+15F_4/2`| -# +-----------------------------+------------------+-----------------------+----------------+--------------------+ -# |:math:`xz, yz` |:math:`A/2-B+C/2` |:math:`F_0/2-F_2-12F_4`|:math:`3B/2+C/2`|:math:`3F_2/2+10F_4`| -# +-----------------------------+------------------+-----------------------+----------------+--------------------+ -# |:math:`xz, x^2-y^2` |:math:`A/2-B+C/2` |:math:`F_0/2-F_2-12F_4`|:math:`3B/2+C/2`|:math:`3F_2/2+10F_4`| -# +-----------------------------+------------------+-----------------------+----------------+--------------------+ -# |:math:`xz, xy` |:math:`A/2-B+C/2` |:math:`F_0/2-F_2-12F_4`|:math:`3B/2+C/2`|:math:`3F_2/2+10F_4`| -# +-----------------------------+------------------+-----------------------+----------------+--------------------+ -# |:math:`yz, x^2-y^2` |:math:`A/2-B+C/2` |:math:`F_0/2-F_2-12F_4`|:math:`3B/2+C/2`|:math:`3F_2/2+10F_4`| -# +-----------------------------+------------------+-----------------------+----------------+--------------------+ -# |:math:`yz, xy` |:math:`A/2-B+C/2` |:math:`F_0/2-F_2-12F_4`|:math:`3B/2+C/2`|:math:`3F_2/2+10F_4`| -# +-----------------------------+------------------+-----------------------+----------------+--------------------+ -# |:math:`x^2-y^2, xy` |:math:`A/2+2B+C/2`|:math:`F_0+4F_2-34F_4` | :math:`C/2` |:math:`35F_4/2` | -# +-----------------------------+------------------+-----------------------+----------------+--------------------+ -# -# -# .. _table_3_orbital: -# .. table:: Table of 3 orbital interactions -# -# +-----------------------------+-------------+----------------------------------------------------+-----------------------------------------------------+ -# |Orbitals :math:`\alpha,\beta,\gamma,\delta`|:math:`\langle\alpha\beta|\gamma\delta\rangle` Racah|:math:`\langle\alpha\beta|\gamma\delta\rangle` Slater| -# +=============================+=============+====================================================+=====================================================+ -# |:math:`3z^2-r^2, xz, x^2-y^2, xz` | :math:`\sqrt{3}B/2` | :math:`\sqrt{3}F_2/2-5\sqrt{3}F_4/2` | -# +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ -# |:math:`3z^2-r^2, yz, x^2-y^2, yz` | :math:`-\sqrt{3}B/2` | :math:`-\sqrt{3}F_2/2+5\sqrt{3}F_4/2` | -# +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ -# |:math:`xz, 3z^2-r^2, x^2-y^2, xz` | :math:`-\sqrt{3}B` | :math:`-\sqrt{3}F_2+5\sqrt{3}F_4` | -# +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ -# |:math:`xz, xz, x^2-y^2, 3z^2-r^2` | :math:`\sqrt{3}B/2` | :math:`\sqrt{3}F_2/2-5\sqrt{3}F_4/2` | -# +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ -# |:math:`yz, 3z^2-r^2, x^2-y^2, yz` | :math:`\sqrt{3}B` | :math:`\sqrt{3}F_2-5\sqrt{3}F_4` | -# +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ -# |:math:`yz, yz, x^2-y^2, 3z^2-r^2` | :math:`-\sqrt{3}B/2` | :math:`-\sqrt{3}F_2/2+5\sqrt{3}F_4/2` | -# +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ -# -# -# .. _table_4_orbital: -# .. table:: Table of 4 orbital interactions -# -# +-----------------------------+-------------+----------------------------------------------------+-----------------------------------------------------+ -# |Orbitals :math:`\alpha,\beta,\gamma,\delta`|:math:`\langle\alpha\beta|\gamma\delta\rangle` Racah|:math:`\langle\alpha\beta|\gamma\delta\rangle` Slater| -# +=============================+=============+====================================================+=====================================================+ -# |:math:`3z^2-r^2, xz, yz, xy` | :math:`-\sqrt{3}B` | :math:`-\sqrt{3}F_2+5\sqrt{3}F_4` | -# +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ -# |:math:`3z^2-r^2, xz, xy, yz` | :math:`\sqrt{3}B/2` | :math:`\sqrt{3}F_2/2-5\sqrt{3}F_4/2` | -# +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ -# |:math:`3z^2-r^2, yz, xz, xy` | :math:`-\sqrt{3}B` | :math:`-\sqrt{3}F_2+5\sqrt{3}F_4` | -# +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ -# |:math:`3z^2-r^2, yz, xy, xz` | :math:`\sqrt{3}B/2` | :math:`\sqrt{3}F_2/2-5\sqrt{3}F_4/2` | -# +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ -# |:math:`3z^2-r^2, xy, xz, yz` | :math:`\sqrt{3}B/2` | :math:`\sqrt{3}F_2/2-5\sqrt{3}F_4/2` | -# +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ -# |:math:`3z^2-r^2, xy, yz, xz` | :math:`\sqrt{3}B/2` | :math:`\sqrt{3}F_2/2-5\sqrt{3}F_4/2` | -# +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ -# |:math:`x^2-y^2 , xz, xy, yz` | :math:`-3B/2` | :math:`-3F_2/2+15F_4/2` | -# +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ -# |:math:`x^2-y^2 , yz, xy, xz` | :math:`3B/2` | :math:`3F_2/2-15F_4/2` | -# +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ -# |:math:`x^2-y^2 , xy, xz, yz` | :math:`-3B/2` | :math:`-3F_2/2+15F_4/2` | -# +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ -# |:math:`x^2-y^2 , xy, yz, xz` | :math:`3B/2` | :math:`3F_2/2-15F_4/2` | -# +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ -# -# -# .. rubric:: Footnotes -# -# .. [1] MSugano S, Tanabe Y and Kamimura H. 1970. Multiplets of -# Transition-Metal Ions in Crystals. Academic Press, New York and London. diff --git a/edrixs/_downloads/6a1b3a453f5ca6a1cd55874996f58fad/example_2_single_atom_RIXS.py b/edrixs/_downloads/6a1b3a453f5ca6a1cd55874996f58fad/example_2_single_atom_RIXS.py deleted file mode 100644 index dc6bd57b00..0000000000 --- a/edrixs/_downloads/6a1b3a453f5ca6a1cd55874996f58fad/example_2_single_atom_RIXS.py +++ /dev/null @@ -1,248 +0,0 @@ -#!/usr/bin/env python -""" -RIXS calculations for an atomic model -===================================== -Here we show how to compute RIXS for a single site atomic model with crystal -field and electron-electron interactions. We take the case of -Sr\ :sub:`2`\ YIrO\ :sub:`6` -from Ref. [1]_ as the material in question. The aim of this example is to -illustrate the proceedure and to provide what we hope is useful advice. What is -written is not meant to be a replacement for reading the docstrings of the -functions, which can always be accessed on the -`edrixs website `_ or -by executing functions with ?? in IPython. -""" -import edrixs -import numpy as np -import matplotlib.pyplot as plt - -################################################################################ -# Specify active core and valence orbitals -# ------------------------------------------------------------------------------ -# Sr\ :sub:`2`\ YIrO\ :sub:`6`\ has a :math:`5d^4` electronic configuration and -# we want to calculate the :math:`L_3` edge spectrum i.e. resonating with a -# :math:`2p_{3/2}` core hole. We will start by including only the -# :math:`t_{2g}` valance orbitals. -shell_name = ('t2g', 'p32') -v_noccu = 4 - -################################################################################ -# Slater parameters -# ------------------------------------------------------------------------------ -# Here we want to use Hund's interaction -# :math:`J_H` and spin orbit coupling :math:`\lambda` as adjustable parameters -# to match experiment. We will take -# the core hole interaction parameter from the Hartree Fock numbers EDRIXS has -# in its database. These need to be converted and arranged into the order -# required by EDRIXS. -Ud = 2 -JH = 0.25 -lam = 0.42 -F0_d, F2_d, F4_d = edrixs.UdJH_to_F0F2F4(Ud, JH) -info = edrixs.utils.get_atom_data('Ir', '5d', v_noccu, edge='L3') -G1_dp = info['slater_n'][5][1] -G3_dp = info['slater_n'][6][1] -F0_dp = edrixs.get_F0('dp', G1_dp, G3_dp) -F2_dp = info['slater_n'][4][1] - -slater_i = [F0_d, F2_d, F4_d] # Fk for d -slater_n = [ - F0_d, F2_d, F4_d, # Fk for d - F0_dp, F2_dp, # Fk for dp - G1_dp, G3_dp, # Gk for dp - 0.0, 0.0 # Fk for p -] -slater = [slater_i, slater_n] -v_soc = (lam, lam) - -################################################################################ -# Diagonalization -# ------------------------------------------------------------------------------ -# We obtain the ground and intermediate state eigenenergies and the transition -# operators via matrix diagonalization. Note that the calculation does not know -# the core hole energy, so we need to adjust the energy that the resonance will -# appear at by hand. We know empirically that the resonance is at 11215 eV -# and that putting four electrons into the valance band costs about -# :math:`4 F^0_d\approx6` eV. In this case -# we are assuming a perfectly cubic crystal field, which we have already -# implemented when we specified the use of the :math:`t_{2g}` subshell only -# so we do not need to pass an additional :code:`v_cfmat` matrix. - -off = 11215 - 6 -out = edrixs.ed_1v1c_py(shell_name, shell_level=(0, -off), v_soc=v_soc, - c_soc=info['c_soc'], v_noccu=v_noccu, slater=slater) -eval_i, eval_n, trans_op = out - -################################################################################ -# Compute XAS -# ------------------------------------------------------------------------------ -# To calculate XAS we need to correctly specify the orientation of the x-rays -# with respect to the sample. By default, the :math:`x, y, z` coordinates -# of the sample's crystal field, will be aligned with our lab frame, passing -# :code:`loc_axis` to :code:`ed_1v1c_py` can be used to specify a different -# convention. The experimental geometry is specified following the angles -# shown in Figure 1 of Y. Wang et al., -# `Computer Physics Communications 243, 151-165 (2019) -# `_. The default -# setting has x-rays along :math:`z` for :math:`\theta=\pi/2` rad -# and the x-ray beam along :math:`-x` for -# :math:`\theta=\phi=0`. Parameter :code:`scatter_axis` can be passed to -# :code:`xas_1v1c_py` to specify a different geometry if desired. -# -# Variable :code:`pol_type` specifies a list of different x-ray -# polarizations to calculate. Here we will use so-called :math:`\pi`-polarization -# where the x-rays are parallel to the plane spanned by the incident -# beam and the sample :math:`z`-axis. -# -# EDRIXS represents the system's ground state using a set of -# low energy eigenstates weighted by Boltzmann thermal factors. -# These eigenstates are specified by :code:`gs_list`, -# which is of the form :math:`[0, 1, 2, 3, \dots]`. In this example, we -# calculate these states as those that have non-negligible thermal -# population. The function :code:`xas_1v1c_py` assumes that the spectral -# broadening is dominated by the inverse core hole lifetime :code:`gamma_c`, -# which is the Lorentzian half width at half maximum. - -ominc = np.linspace(11200, 11230, 50) -temperature = 300 # in K -prob = edrixs.boltz_dist(eval_i, temperature) -gs_list = [n for n, prob in enumerate(prob) if prob > 1e-6] - -thin = 30*np.pi/180 -phi = 0 -pol_type = [('linear', 0)] - -xas = edrixs.xas_1v1c_py( - eval_i, eval_n, trans_op, ominc, gamma_c=info['gamma_c'], - thin=thin, phi=phi, pol_type=pol_type, - gs_list=gs_list) - - -################################################################################ -# Compute RIXS -# ------------------------------------------------------------------------------ -# Calculating RIXS is overall similar to XAS, but with a few additional -# considerations. The spectral width in the energy loss axis of RIXS it -# not set by the core hole lifetime, but by either the final state lifetime -# or the experimental resolution and is parameterized by :code:`gamma_f` -# -- the Lorentzian half width at half maximum. -# -# The angle and polarization of the emitted beam must also be specified, so -# we pass :code:`pol_type_rixs` to the function, which specifies the -# includes the incoming and outgoing x-ray states. If, as is common in -# experiments, the emitted polarization is not resolved -# one needs to add both emitted polarization channels, which is what we will -# do later on in this example. - -eloss = np.linspace(-.5, 6, 400) -pol_type_rixs = [('linear', 0, 'linear', 0), ('linear', 0, 'linear', np.pi/2)] - -thout = 60*np.pi/180 -gamma_f = 0.02 - -rixs = edrixs.rixs_1v1c_py( - eval_i, eval_n, trans_op, ominc, eloss, - gamma_c=info['gamma_c'], gamma_f=gamma_f, - thin=thin, thout=thout, phi=phi, - pol_type=pol_type_rixs, gs_list=gs_list, - temperature=temperature -) - -################################################################################ -# The array :code:`xas` will have shape -# :code:`(len(ominc_xas), len(pol_type))` - -################################################################################ -# Plot XAS and RIXS -# ------------------------------------------------------------------------------ -# Let's plot everything. We will use a function so we can reuse the code later. -# Note that the rixs array :code:`rixs` has shape -# :code:`(len(ominc_xas), len(ominc_xas), len(pol_type))`. We will use some numpy -# tricks to sum over the two different emitted polarizations. - -fig, axs = plt.subplots(2, 2, figsize=(10, 10)) - - -def plot_it(axs, ominc, xas, eloss, rixscut, rixsmap=None, label=None): - axs[0].plot(ominc, xas[:, 0], label=label) - axs[0].set_xlabel('Energy (eV)') - axs[0].set_ylabel('Intensity') - axs[0].set_title('XAS') - - axs[1].plot(eloss, rixscut, label=f"{label}") - axs[1].set_xlabel('Energy loss (eV)') - axs[1].set_ylabel('Intensity') - axs[1].set_title(f'RIXS at resonance') - - if rixsmap is not None: - art = axs[2].pcolormesh(ominc, eloss, rixsmap.T, shading='auto') - plt.colorbar(art, ax=axs[2], label='Intensity') - axs[2].set_xlabel('Incident energy (eV)') - axs[2].set_ylabel('Energy loss') - axs[2].set_title('RIXS map') - - -rixs_pol_sum = rixs.sum(-1) -cut_index = np.argmax(rixs_pol_sum[:, eloss < 2].sum(1)) -rixscut = rixs_pol_sum[cut_index] - -plot_it(axs.ravel(), ominc, xas, eloss, rixscut, rixsmap=rixs_pol_sum) -axs[0, 1].set_xlim(right=3) -axs[1, 0].set_ylim(top=3) -axs[1, 1].remove() - -plt.show() - -################################################################################ -# Full d shell calculation -# ------------------------------------------------------------------------------ -# Some researchers have questioned the appropriateness of only including the -# :math:`t_{2g}` subshell for iridates [2]_. Let's test this. We specify that -# the full :math:`d` shell should be used and apply cubic crystal field matrix -# :code:`v_cfmat`. We shift the energy offset by :math:`\frac{2}{5}10D_q`, which -# is the amount the crystal field moves the :math:`t_{2g}` subshell. - -ten_dq = 3.5 -v_cfmat = edrixs.cf_cubic_d(ten_dq) -off = 11215 - 6 + ten_dq*2/5 -out = edrixs.ed_1v1c_py(('d', 'p32'), shell_level=(0, -off), v_soc=v_soc, - v_cfmat=v_cfmat, - c_soc=info['c_soc'], v_noccu=v_noccu, slater=slater) -eval_i, eval_n, trans_op = out - -xas_full_d_shell = edrixs.xas_1v1c_py( - eval_i, eval_n, trans_op, ominc, gamma_c=info['gamma_c'], - thin=thin, phi=phi, pol_type=pol_type, - gs_list=gs_list) - -rixs_full_d_shell = edrixs.rixs_1v1c_py( - eval_i, eval_n, trans_op, np.array([11215]), eloss, - gamma_c=info['gamma_c'], gamma_f=gamma_f, - thin=thin, thout=thout, phi=phi, - pol_type=pol_type_rixs, gs_list=gs_list, - temperature=temperature) - -fig, axs = plt.subplots(1, 2, figsize=(10, 4)) -plot_it(axs, ominc, xas, eloss, rixscut, label='$t_{2g}$ subshell') -rixscut = rixs_full_d_shell.sum((0, -1)) -plot_it(axs, ominc, xas_full_d_shell, eloss, rixscut, label='$d$ shell') - -axs[0].legend() -axs[1].legend() -plt.show() - -################################################################################ -# As expected, we see the appearance of excitations on the energy scale of -# :math:`10D_q` in the XAS and RIXS. The low energy manifold is qualitatively, -# but not quantiatively similar. This makes it clear that the parameterization -# of Sr\ :sub:`2`\ YIrO\ :sub:`6`\ is dependent on the model. - -############################################################################## -# -# .. rubric:: Footnotes -# -# .. [1] Bo Yuan et al., -# `Phys. Rev. B 95, 235114 (2017) `_. -# -# .. [2] Georgios L. Stamokostas and Gregory A. Fiete -# `Phys. Rev. B 97, 085150 (2018) `_. diff --git a/edrixs/_downloads/6c36c3d3d14c1014b0bd91c1f7e1e81f/example_8_Coulomb.ipynb b/edrixs/_downloads/6c36c3d3d14c1014b0bd91c1f7e1e81f/example_8_Coulomb.ipynb deleted file mode 100644 index b192071ea6..0000000000 --- a/edrixs/_downloads/6c36c3d3d14c1014b0bd91c1f7e1e81f/example_8_Coulomb.ipynb +++ /dev/null @@ -1,215 +0,0 @@ -{ - "cells": [ - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n# Coulomb interactions\nIn this example we provide more details on how Coulomb interactions are\nimplemented in multiplet calculations and EDRIXS in particular. We aim\nto clarify the form of the matrices, how they are parametrized,\nand how the breaking of spherical symmetry can switch on additional elements\nthat one might not anticipate. Our example is based on a $d$ atomic shell.\n" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Create matrix\nThe Coulomb interaction between two particles can be written as\n\n .. math::\n \\begin{equation}\n \\hat{H} = \\frac{1}{2}\n \\int d\\mathbf{r} \\int d\\mathbf{r}^\\prime\n \\Sigma_{\\sigma, \\sigma^\\prime}\n |\\hat{\\psi}^\\sigma(\\mathbf{r})|^2 \\frac{e^2}{R}\n |\\hat{\\psi}^{\\sigma^\\prime}(\\mathbf{r^\\prime})|^2,\n \\end{equation}\n\nwhere $\\hat{\\psi}^\\sigma(\\mathbf{r})$ is the electron wavefunction, with\nspin $\\sigma$, and $R=|r-r^\\prime|$ is the electron separation.\nSolving our problem in this form is difficult due to the need to symmeterize\nthe wavefunction to follow fermionic statistics.\nUsing second quantization, we can use operators to impose the required\nparticle exchange statistics and write the equation in terms of\na tensor $U$\n\n .. math::\n \\begin{equation}\n \\hat{H} = \\sum_{\\alpha,\\beta,\\gamma,\\delta,\\sigma,\\sigma^\\prime}\n U_{\\alpha\\sigma,\\beta\\sigma^\\prime,\\gamma\\sigma^\\prime,\\delta\\sigma}\n \\hat{f}^{\\dagger}_{\\alpha\\sigma}\n \\hat{f}^{\\dagger}_{\\beta\\sigma^\\prime}\n \\hat{f}_{\\gamma\\sigma^\\prime}\\hat{f}_{\\delta\\sigma},\n \\end{equation}\n\nwhere $\\alpha$, $\\beta$, $\\gamma$, $\\delta$ are\norbital indices and $\\hat{f}^{\\dagger}$\n($\\hat{f}$) are the creation (anihilation) operators.\nFor a $d$-electron system, we have $10$ distinct spin-orbitals\n($5$ orbitals each with $2$ spins), which makes matrix the\n$10\\times10\\times10\\times10$ in total size.\nIn EDRIXS the matrix can be created as follows:\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "import edrixs\nimport numpy as np\nimport scipy\nimport matplotlib.pyplot as plt\nimport itertools\n\nF0, F2, F4 = 6.94, 14.7, 4.41\numat_chb = edrixs.get_umat_slater('d', F0, F2, F4)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We stored this under variable :code:`umat_chb` where \"cbh\" stands for\ncomplex harmonic basis, which is the default basis in EDRIXS.\n\n" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Parameterizing interactions\nEDRIXS parameterizes the interactions in $U$ via Slater integral\nparameters $F^{k}$. These relate to integrals of various spherical\nHarmonics as well as Clebsch-Gordon coefficients, Gaunt coefficients,\nand Wigner 3J symbols. Textbooks such as [1]_ can be used for further\nreference. If you are interested in the details of how\nEDRIXS does this (and you probably aren't) function :func:`.umat_slater`,\nconstructs the required matrix via Gaunt coeficents from\n:func:`.get_gaunt`. Two alternative parameterizations are common.\nThe first are the Racah parameters, which are\n\n .. math::\n \\begin{eqnarray}\n A &=& F^0 - \\frac{49}{441} F^4 \\\\\n B &=& \\frac{1}{49}F^2 - \\frac{5}{441}F^4 \\\\\n C &=& \\frac{35}{441}F^4.\n \\end{eqnarray}\n\nor an alternative form for the Slater integrals\n\n .. math::\n \\begin{eqnarray}\n F_0 &=& F^0 \\\\\n F_2 &=& \\frac{1}{49}F^2 \\\\\n F_4 &=& \\frac{1}{441}F^4,\n \\end{eqnarray}\n\nwhich involves different normalization parameters.\n\n" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Basis transform\nIf we want to use the real harmonic basis, we can use a tensor\ntransformation, which imposes the following orbital order\n$3z^2-r^2, xz, yz, x^2-y^2, xy$, each of which involves\n$\\uparrow, \\downarrow$ spin pairs. Let's perform this transformation and\nstore a list of these orbitals.\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "umat = edrixs.transform_utensor(umat_chb, edrixs.tmat_c2r('d', True))\norbitals = ['3z^2-r^2', 'xz', 'yz', 'x^2-y^2', 'xy']" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Interactions\nTensor $U$ is a series of matrix\nelements\n\n .. math::\n \\begin{equation}\n \\langle\\psi_{\\gamma,\\delta}^{\\bar{\\sigma},\\bar{\\sigma}^\\prime}\n |\\hat{H}|\n \\psi_{\\alpha,\\beta}^{\\sigma,\\sigma^\\prime}\\rangle\n \\end{equation}\n\nthe combination of which defines the energetic cost of pairwise\nelectron-electron interactions between states $\\alpha,\\sigma$\nand $\\beta,\\sigma^\\prime$. In EDRIXS we follow the convention of\nsumming over all orbital pairs. Some other texts count each pair of\nindices only once. The matrix elements here will consequently\nbe half the magnitude of those in other references.\nWe can express the interactions in terms of\nthe orbitals involved. It is common to distinguish \"direct Coulomb\" and\n\"exchange\" interactions. The former come from electrons in the same orbital\nand the later involve swapping orbital labels for electrons. We will use\n$U_0$ and $J$ as a shorthand for distinguishing these.\n\nBefore we describe the different types of interactions, we note that since\nthe Coulomb interaction is real, and due to the spin symmmetry properties\nof the process $U$ always obeys\n\n .. math::\n \\begin{equation}\n U_{\\alpha\\sigma,\\beta\\sigma^\\prime,\\gamma\\sigma^\\prime,\\delta\\sigma} =\n U_{\\beta\\sigma,\\alpha\\sigma^\\prime,\\delta\\sigma^\\prime,\\gamma\\sigma} =\n U_{\\delta\\sigma,\\gamma\\sigma^\\prime,\\beta\\sigma^\\prime,\\alpha\\sigma} =\n U_{\\gamma\\sigma,\\delta\\sigma^\\prime,\\alpha\\sigma^\\prime,\\beta\\sigma}.\n \\end{equation}\n\n\n### 1. Intra orbital\nThe direct Coulomb energy cost to double-occupy an orbital comes from terms\nlike $U_{\\alpha\\sigma,\\alpha\\bar\\sigma,\\alpha\\bar\\sigma,\\alpha\\sigma}$.\nIn this notation, we use $\\sigma^\\prime$ to denote that the matrix\nelement is summed over all pairs and $\\bar{\\sigma}$ to denote sums\nover all opposite spin pairs. Due to rotational symmetry, all these\nelements are the same and equal to\n\n .. math::\n \\begin{eqnarray}\n U_0 &=& \\frac{A}{2} + 2B + \\frac{3C}{2}\\\\\n &=& \\frac{F_0}{2} + 2F_2 + 18F_4\n \\end{eqnarray}\n\nLet's print these to demonstrate where these live in the array\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "for i in range(0, 5):\n val = umat[i*2, i*2 + 1, i*2 + 1, i*2].real\n print(f\"{orbitals[i]:<8} \\t {val:.3f}\")" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "### 2. Inter orbital Coulomb interactions\nDirect Coulomb repulsion between different orbitals depends on terms like\n$U_{\\alpha\\sigma,\\beta\\sigma^\\prime,\\beta\\sigma^\\prime,\\alpha\\sigma}$.\nExpresions for these parameters are provided in column $U$ in\n`table_2_orbital`. We can print the values from :code:`umat`\nlike this:\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "for i, j in itertools.combinations(range(5), 2):\n val = umat[i*2, j*2 + 1, j*2 + 1, i*2].real\n print(f\"{orbitals[i]:<8} \\t {orbitals[j]:<8} \\t {val:.3f}\")" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "### 3. Inter-orbital exchange interactions\nExchange terms exist with the form\n$U_{\\alpha\\sigma,\\beta\\sigma^\\prime,\\alpha\\sigma^\\prime,\\beta\\sigma}$.\nExpresions for these parameters are provided in column $J$ of\n`table_2_orbital`. These come from terms like this in the matrix:\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "for i, j in itertools.combinations(range(5), 2):\n val = umat[i*2, j*2 + 1, i*2 + 1, j*2].real\n print(f\"{orbitals[i]:<8} \\t {orbitals[j]:<8} \\t {val:.3f}\")" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "### 4. Pair hopping term\nTerms that swap pairs of electrons exist as\n$(1-\\delta_{\\sigma\\sigma'})U_{\\alpha\\sigma,\\alpha\\bar\\sigma,\\beta\\bar\\sigma,\\beta\\sigma}$\nand depend on exchange interactions column $J$ from\n`table_2_orbital`\nand here in the matrix.\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "for i, j in itertools.combinations(range(5), 2):\n val = umat[i*2, i*2 + 1, j*2 + 1, j*2].real\n print(f\"{orbitals[i]:<8} \\t {orbitals[j]:<8} \\t {val:.3f}\")" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "### 5. Three orbital\nAnother set of terms that one might not immediately anticipate involve three\norbitals like\n\n .. math::\n \\begin{equation}\n U_{\\alpha\\sigma, \\gamma\\sigma', \\beta\\sigma', \\gamma\\sigma} \\\\\n U_{\\alpha\\sigma, \\gamma\\sigma', \\gamma\\sigma', \\beta\\sigma} \\\\\n (1-\\delta_{\\sigma\\sigma'})\n U_{\\alpha\\sigma, \\beta\\sigma', \\gamma\\sigma', \\gamma\\sigma}\n \\end{equation}\n\nfor $\\alpha=3z^2-r^2, \\beta=x^2-y^2, \\gamma=xz/yz$.\nThese are needed to maintain the rotational symmetry of the interations.\nSee `table_3_orbital` for the expressions. We can print some of\nthese via:\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "ijkl = [[0, 1, 3, 1],\n [0, 2, 3, 2],\n [1, 0, 3, 1],\n [1, 1, 3, 0],\n [2, 0, 3, 2],\n [2, 2, 3, 0]]\n\nfor i, j, k, l in ijkl:\n val = umat[i*2, j*2 + 1, k*2 + 1, l*2].real\n print(f\"{orbitals[i]:<8} \\t {orbitals[j]:<8} \\t\"\n f\"{orbitals[k]:<8} \\t {orbitals[l]:<8} \\t {val:.3f}\")" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "### 6. Four orbital\nFuther multi-orbital terms include\n$U_{\\alpha\\sigma,\\beta\\sigma^\\prime,\\gamma\\sigma^\\prime,\\delta\\sigma}$.\nWe can find these here in the matrix:\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "ijkl = [[0, 1, 2, 4],\n [0, 1, 4, 2],\n [0, 2, 1, 4],\n [0, 2, 4, 1],\n [0, 4, 1, 2],\n [0, 4, 2, 1],\n [3, 1, 4, 2],\n [3, 2, 4, 1],\n [3, 4, 1, 2],\n [3, 4, 2, 1]]\n\nfor i, j, k, l in ijkl:\n val = umat[i*2, j*2 + 1, k*2 + 1, l*2].real\n print(f\"{orbitals[i]:<8} \\t {orbitals[j]:<8} \\t {orbitals[k]:<8}\"\n f\"\\t {orbitals[l]:<8} \\t {val:.3f}\")" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Effects of multi-orbital terms\nTo test the effects of the multi-orbital terms, let's plot the eigenenergy\nspectra with and without multi-orbital terms switched on for system with and\nwithout a cubic crystal field. We will use a $d$-shell with two\nelectrons.\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "ten_dqs = [0, 2, 4, 12]\n\ndef diagonalize(ten_dq, umat):\n emat = edrixs.cb_op(edrixs.cf_cubic_d(ten_dq),\n edrixs.tmat_c2r('d', ispin=True))\n H = (edrixs.build_opers(4, umat, basis)\n + edrixs.build_opers(2, emat, basis))\n e, v = scipy.linalg.eigh(H)\n return e - e.min()\n\nbasis = edrixs.get_fock_bin_by_N(10, 2)\numat_no_multiorbital = np.copy(umat)\nB = F2/49 - 5*F4/441\nfor val in [np.sqrt(3)*B/2, np.sqrt(3)*B, 3*B/2]:\n umat_no_multiorbital[(np.abs(umat)- val) < 1e-6] = 0\n\nfig, axs = plt.subplots(1, len(ten_dqs), figsize=(8, 3))\n\nfor cind, (ax, ten_dq) in enumerate(zip(axs, ten_dqs)):\n ax.hlines(diagonalize(ten_dq, umat), xmin=0, xmax=1,\n label='on', color=f'C{cind}')\n ax.hlines(diagonalize(ten_dq, umat_no_multiorbital),\n xmin=1.5, xmax=2.5,\n label='off',\n linestyle=':', color=f'C{cind}')\n ax.set_title(f\"$10D_q={ten_dq}$\")\n ax.set_ylim([-.5, 20])\n ax.set_xticks([])\n ax.legend()\n\nfig.suptitle(\"Eigenvalues with 3&4-orbital effects on/off\")\nfig.subplots_adjust(wspace=.3)\naxs[0].set_ylabel('Eigenvalues (eV)')\nfig.subplots_adjust(top=.8)\nplt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "On the left of the plot Coulomb interactions in spherical symmetry cause\nsubstantial mxing between $t_{2g}$ and $e_{g}$ orbitals in the\neigenstates and 3 & 4 orbital orbital terms are crucial for obtaining the\nthe right eigenenergies. As $10D_q$ get large, this mixing is switched\noff and the spectra start to become independent of whether the 3 & 4 orbital\norbital terms are included or not.\n\n\n\n.. table:: Table of 2 orbital interactions\n\n +-----------------------------+------------------+-----------------------+----------------+--------------------+\n |Orbitals $\\alpha,\\beta$|$U_0$ Racah | $U_0$ Slater |$J$ Racah |$J$ Slater |\n +=============================+==================+=======================+================+====================+\n |$3z^2-r^2, xz$ |$A/2+B+C/2$ |$F_0/2+F_2-12F_4$| $B/2+C/2$|$F_2/2+15F_4$ |\n +-----------------------------+------------------+-----------------------+----------------+--------------------+\n |$3z^2-r^2, yz$ |$A/2+B+C/2$ |$F_0/2+F_2-12F_4$| $B/2+C/2$|$F_2/2+15F_4$ |\n +-----------------------------+------------------+-----------------------+----------------+--------------------+\n |$3z^2-r^2, x^2-y^2$ |$A/2-2B+C/2$|$F_0/2-2F_2+3F_4$|$2B+C/2$ |$2F_2+15F_4/2$|\n +-----------------------------+------------------+-----------------------+----------------+--------------------+\n |$3z^2-r^2, xy$ |$A/2-2B+C/2$|$F_0/2-2F_2+3F_4$|$2B+C/2$ |$2F_2+15F_4/2$|\n +-----------------------------+------------------+-----------------------+----------------+--------------------+\n |$xz, yz$ |$A/2-B+C/2$ |$F_0/2-F_2-12F_4$|$3B/2+C/2$|$3F_2/2+10F_4$|\n +-----------------------------+------------------+-----------------------+----------------+--------------------+\n |$xz, x^2-y^2$ |$A/2-B+C/2$ |$F_0/2-F_2-12F_4$|$3B/2+C/2$|$3F_2/2+10F_4$|\n +-----------------------------+------------------+-----------------------+----------------+--------------------+\n |$xz, xy$ |$A/2-B+C/2$ |$F_0/2-F_2-12F_4$|$3B/2+C/2$|$3F_2/2+10F_4$|\n +-----------------------------+------------------+-----------------------+----------------+--------------------+\n |$yz, x^2-y^2$ |$A/2-B+C/2$ |$F_0/2-F_2-12F_4$|$3B/2+C/2$|$3F_2/2+10F_4$|\n +-----------------------------+------------------+-----------------------+----------------+--------------------+\n |$yz, xy$ |$A/2-B+C/2$ |$F_0/2-F_2-12F_4$|$3B/2+C/2$|$3F_2/2+10F_4$|\n +-----------------------------+------------------+-----------------------+----------------+--------------------+\n |$x^2-y^2, xy$ |$A/2+2B+C/2$|$F_0+4F_2-34F_4$ | $C/2$ |$35F_4/2$ |\n +-----------------------------+------------------+-----------------------+----------------+--------------------+\n\n\n.. table:: Table of 3 orbital interactions\n\n +-----------------------------+-------------+----------------------------------------------------+-----------------------------------------------------+\n |Orbitals $\\alpha,\\beta,\\gamma,\\delta$|$\\langle\\alpha\\beta|\\gamma\\delta\\rangle$ Racah|$\\langle\\alpha\\beta|\\gamma\\delta\\rangle$ Slater|\n +=============================+=============+====================================================+=====================================================+\n |$3z^2-r^2, xz, x^2-y^2, xz$ | $\\sqrt{3}B/2$ | $\\sqrt{3}F_2/2-5\\sqrt{3}F_4/2$ |\n +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+\n |$3z^2-r^2, yz, x^2-y^2, yz$ | $-\\sqrt{3}B/2$ | $-\\sqrt{3}F_2/2+5\\sqrt{3}F_4/2$ |\n +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+\n |$xz, 3z^2-r^2, x^2-y^2, xz$ | $-\\sqrt{3}B$ | $-\\sqrt{3}F_2+5\\sqrt{3}F_4$ |\n +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+\n |$xz, xz, x^2-y^2, 3z^2-r^2$ | $\\sqrt{3}B/2$ | $\\sqrt{3}F_2/2-5\\sqrt{3}F_4/2$ |\n +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+\n |$yz, 3z^2-r^2, x^2-y^2, yz$ | $\\sqrt{3}B$ | $\\sqrt{3}F_2-5\\sqrt{3}F_4$ |\n +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+\n |$yz, yz, x^2-y^2, 3z^2-r^2$ | $-\\sqrt{3}B/2$ | $-\\sqrt{3}F_2/2+5\\sqrt{3}F_4/2$ |\n +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+\n\n\n.. table:: Table of 4 orbital interactions\n\n +-----------------------------+-------------+----------------------------------------------------+-----------------------------------------------------+\n |Orbitals $\\alpha,\\beta,\\gamma,\\delta$|$\\langle\\alpha\\beta|\\gamma\\delta\\rangle$ Racah|$\\langle\\alpha\\beta|\\gamma\\delta\\rangle$ Slater|\n +=============================+=============+====================================================+=====================================================+\n |$3z^2-r^2, xz, yz, xy$ | $-\\sqrt{3}B$ | $-\\sqrt{3}F_2+5\\sqrt{3}F_4$ |\n +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+\n |$3z^2-r^2, xz, xy, yz$ | $\\sqrt{3}B/2$ | $\\sqrt{3}F_2/2-5\\sqrt{3}F_4/2$ |\n +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+\n |$3z^2-r^2, yz, xz, xy$ | $-\\sqrt{3}B$ | $-\\sqrt{3}F_2+5\\sqrt{3}F_4$ |\n +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+\n |$3z^2-r^2, yz, xy, xz$ | $\\sqrt{3}B/2$ | $\\sqrt{3}F_2/2-5\\sqrt{3}F_4/2$ |\n +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+\n |$3z^2-r^2, xy, xz, yz$ | $\\sqrt{3}B/2$ | $\\sqrt{3}F_2/2-5\\sqrt{3}F_4/2$ |\n +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+\n |$3z^2-r^2, xy, yz, xz$ | $\\sqrt{3}B/2$ | $\\sqrt{3}F_2/2-5\\sqrt{3}F_4/2$ |\n +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+\n |$x^2-y^2 , xz, xy, yz$ | $-3B/2$ | $-3F_2/2+15F_4/2$ |\n +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+\n |$x^2-y^2 , yz, xy, xz$ | $3B/2$ | $3F_2/2-15F_4/2$ |\n +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+\n |$x^2-y^2 , xy, xz, yz$ | $-3B/2$ | $-3F_2/2+15F_4/2$ |\n +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+\n |$x^2-y^2 , xy, yz, xz$ | $3B/2$ | $3F_2/2-15F_4/2$ |\n +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+\n\n\n.. rubric:: Footnotes\n\n.. [1] MSugano S, Tanabe Y and Kamimura H. 1970. Multiplets of\n Transition-Metal Ions in Crystals. Academic Press, New York and London.\n\n" - ] - } - ], - "metadata": { - "kernelspec": { - "display_name": "Python 3", - "language": "python", - "name": "python3" - }, - "language_info": { - "codemirror_mode": { - "name": "ipython", - "version": 3 - }, - "file_extension": ".py", - "mimetype": "text/x-python", - "name": "python", - "nbconvert_exporter": "python", - "pygments_lexer": "ipython3", - "version": "3.10.14" - } - }, - "nbformat": 4, - "nbformat_minor": 0 -} \ No newline at end of file diff --git a/edrixs/_downloads/6cd93e346e243a1950ba57feae08db0a/example_0_ed_calculator.ipynb b/edrixs/_downloads/6cd93e346e243a1950ba57feae08db0a/example_0_ed_calculator.ipynb deleted file mode 100644 index fb3489b263..0000000000 --- a/edrixs/_downloads/6cd93e346e243a1950ba57feae08db0a/example_0_ed_calculator.ipynb +++ /dev/null @@ -1,355 +0,0 @@ -{ - "cells": [ - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n# Exact diagonalization\nHere we show how to find the eigenvalues and eigenvectors of a many-body\nHamiltonian of fermions with Coulomb interactions. We then determine their spin\nand orbital angular momentum and how this changes when we switch on spin-orbit\ncoupling.\n" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Import the necessary modules.\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "import numpy as np\nimport matplotlib.pyplot as plt\nimport scipy\nimport edrixs" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Parameters\nDefine the orbital angular momentum number $l=1$ (i.e. a `p` shell),\nthe number of spin-orbitals, the occupancy and the Slater integrals.\n$F^{k}$ with $k=0,2$:\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "l = 1\nnorb = 6\nnoccu = 2\nF0, F2 = 4.0, 1.0" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Coulomb interactions\nThe Coulomb interactions in EDRIXS are described by a tensor. Understanding this\nin full is complicated and requires careful consideration of the symmetry of the\ninteractions. See example 6 for more discussion if desired.\nEDRIXS can construct the matrix via\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "umat = edrixs.get_umat_slater('p', F0, F2)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Create basis\nNow we build the binary form of the Fock basis $|F>$ (we consider it\npreferable to use the standard $F$ and trust the reader to avoid\nconfusing it with the interaction parameters.)\nThe Fock basis is the simplest legitimate form for the basis and it consists\nof a series of 1s and 0s where 1 means occupied and\n0 means empty. These are in order up, down, up, down, up, down.\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "basis = edrixs.get_fock_bin_by_N(norb, noccu)\nprint(np.array(basis))" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We expect the number of these states to be given by the mathematical\ncombination of two electrons distributed among six states (three spin-orbitals\nwith two spins per orbital).\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "message = (\"We predict C(norb={}, noccu={})={:.0f} states and we got {:d}, \"\n \"which is reassuring!\")\nprint(message.format(norb, noccu, edrixs.combination(norb, noccu), len(basis)))" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Note that in more complicated problems with both valence and core\nelectrons, the edrixs convention is to list the valence electrons first.\n\n" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Transform interactions into Fock basis\nedrixs works by initiailly creating a Hamiltonian matrix\n$\\hat{H}$ in the single particle basis and then transforming into\nour chosen Fock basis. In the single particle basis, we have four fermion\ninteractions with this form\n\n .. math::\n \\hat{H} = \n\ngenerated as\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "n_fermion = 4\nH = edrixs.build_opers(n_fermion, umat, basis)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We needed to specify :code:`n_fermion = 4` because the\n:code:`edrixs.build_opers` function can also make two fermion terms.\n\n" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Diagonalize the matrix\nFor a small problem such as this it is convenient to use the native\n[scipy](https://scipy.org) diagonalization routine. This returns eigenvalues\n:code:`e` and eignvectors :code:`v` where eigenvalue :code:`e[i]` corresponds\nto eigenvector :code:`v[:,i]`.\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "e, v = scipy.linalg.eigh(H)\nprint(\"{} eignvalues and {} eigvenvectors {} elements long.\".format(len(e),\n v.shape[1],\n v.shape[0]))" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Computing expectation values\nTo interpret the results, it is informative to compute the expectations values\nrelated to the spin $\\mathbf{S}$, orbital $\\mathbf{L}$,\nand total $\\mathbf{J}$, angular momentum. We first load the relevant\nmatrices for these quantities for a `p` atomic shell. We need to specify\nthat we would like to include spin when loading the orbital operator.\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "orb_mom = edrixs.get_orb_momentum(l, ispin=True)\nspin_mom = edrixs.get_spin_momentum(l)\ntot_mom = orb_mom + spin_mom" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We again transform these matrices to our Fock basis to build the operators\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "n_fermion = 2\nopL, opS, opJ = edrixs.build_opers(n_fermion, [orb_mom, spin_mom, tot_mom],\n basis)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Recall that quantum mechanics forbids us from knowing all three Cartesian\ncomponents of angular momentum at once, so we want to compute the squares of\nthese operators i.e.\n\n .. math::\n \\mathbf{S}^2 = S^2_x + S^2_y + S^2_z\\\\\n \\mathbf{L}^2 = L^2_x + L^2_y + L^2_z\\\\\n \\mathbf{J}^2 = J^2_x + J^2_y + J^2_z\n\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "L2 = np.dot(opL[0], opL[0]) + np.dot(opL[1], opL[1]) + np.dot(opL[2], opL[2])\nS2 = np.dot(opS[0], opS[0]) + np.dot(opS[1], opS[1]) + np.dot(opS[2], opS[2])\nJ2 = np.dot(opJ[0], opJ[0]) + np.dot(opJ[1], opJ[1]) + np.dot(opJ[2], opJ[2])" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Remember that the eigenvalues of $\\mathbf{S}^2$ are in the form\n$S(S+1)$ etc. and that they can be obtained by calculating the\nprojection of the operators onto our eigenvectors.\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "L2_val = edrixs.cb_op(L2, v).diagonal().real\nS2_val = edrixs.cb_op(S2, v).diagonal().real\nJ2_val = edrixs.cb_op(J2, v).diagonal().real" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We can determine the degeneracy of the eigenvalues numerically and print out\nthe values as follows\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "e = np.round(e, decimals=6)\ndegeneracy = [sum(eval == e) for eval in e]\nheader = \"{:<3s}\\t{:>8s}\\t{:>8s}\\t{:>8s}\\t{:>8s}\"\nprint(header.format(\"# \", \"E \", \"S(S+1)\", \"L(L+1)\", \"Degen.\"))\nfor i, eigenvalue in enumerate(e):\n values_list = [i, eigenvalue, S2_val[i], L2_val[i], degeneracy[i]]\n print(\"{:<3d}\\t{:8.3f}\\t{:8.3f}\\t{:8.3f}\\t{:>3d}\".format(*values_list))" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We see $S=0$ and $S=1$ states coming from the\ntwo combinations of the spin 1/2 particles. $L$ can take values of\n0, 1, 2. Remember that spin states have degeneracy of $2S+1$ and the\nsame is true for orbital states.\nWe must multiply these $S$ and\n$L$ degeneracies to get the total degeneracy.\nSince these particles are fermions, the\noverall state must be antisymmetric, which dictates the allowed combinations\nof $S$ and $L$.\n\n" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Energy level diagram\nLet us show our findings graphically\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "fig, ax = plt.subplots()\nfor i, eigenvalue in enumerate(np.unique(e)):\n art = ax.plot([0, 1], [eigenvalue, eigenvalue], '-', color='C{}'.format(i))\n ind = np.where(eigenvalue == e)[0][0]\n L = (-1 + np.sqrt(1 + 4*L2_val[ind]))/2\n S = (-1 + np.sqrt(1 + 4*S2_val[ind]))/2\n message = \"L={:.0f}, S={:.0f} ({:.0f})\"\n ax.text(1, eigenvalue, message.format(L, S, degeneracy[ind]),\n horizontalalignment='right',\n verticalalignment='bottom',\n color='C{}'.format(i))\n\nax.set_ylabel('Energy')\nfor loc in ['right', 'top', 'bottom']:\n ax.spines[loc].set_visible(False)\n\nax.yaxis.set_ticks_position('left')\nax.set_xticks([])\nplt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We see Hund's rules in action! Rule 1 says that the highest spin $S=1$\nstate has the lowest energy. Of the two $S=0$ states, the state with\nlarger $L=1$ is lower energy following rule 2.\n\n" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Spin orbit coupling\nFor fun, we can see how this changes when we add spin orbit coupling (SOC).\nThis is a two-fermion operator that we create, transform into the Fock basis\nand add to the prior Hamiltonian. To make things easy, let us make the SOC\nsmall so that the LS coupling approximation is valid and we can\nstill track the states.\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "soc = edrixs.atom_hsoc('p', 0.1)\nn_fermion = 2\nH2 = H + edrixs.build_opers(n_fermion, soc, basis)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Then, we redo the diagonalization and print the results.\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "e2, v2 = scipy.linalg.eigh(H2)\ne2 = np.round(e2, decimals=6)\ndegeneracy2 = [sum(eval == e2) for eval in e2]\nprint()\nmessage = \"With SOC\\n {:<3s}\\t{:>8s}\\t{:>8s}\\t{:>8s}\\t{:>8s}\\t{:>8s}\"\nprint(message.format(\"#\", \"E\", \"S(S+1)\", \"L(L+1)\", \"J(J+1)\", \"degen.\"))\nJ2_val_soc = edrixs.cb_op(J2, v2).diagonal().real\nL2_val_soc = edrixs.cb_op(L2, v2).diagonal().real\nS2_val_soc = edrixs.cb_op(S2, v2).diagonal().real\nfor i, eigenvalue in enumerate(e2):\n values_list = [i, eigenvalue, S2_val_soc[i], L2_val_soc[i], J2_val_soc[i],\n degeneracy2[i]]\n print(\"{:<3d}\\t{:8.3f}\\t{:8.3f}\\t{:8.3f}\\t{:8.3f}\\t{:8.3f}\".format(*values_list))" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and we make an equivalent energy level diagram.\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "fig, ax = plt.subplots()\nfor i, eigenvalue in enumerate(np.unique(e2)):\n art = ax.plot([0, 1], [eigenvalue, eigenvalue], '-', color='C{}'.format(i))\n ind = np.where(eigenvalue == e2)[0][0]\n J = (-1 + np.sqrt(1+4*J2_val_soc[ind]))/2\n message = \"J={:.0f} ({:.0f})\"\n ax.text(1, eigenvalue, message.format(J, degeneracy2[ind]),\n horizontalalignment='right',\n verticalalignment='bottom',\n color='C{}'.format(i))\n\nax.set_ylabel('Energy')\nfor loc in ['right', 'top', 'bottom']:\n ax.spines[loc].set_visible(False)\n\nax.yaxis.set_ticks_position('left')\nax.set_xticks([])\nplt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "It is clear that we have split the $S=1$ state, which branches into\nthree states from $J=|L-S|, |L-S|+1, ..., |L+S|$. Since the shell is\nless than half full, Hund's third rule dictates that the smaller $J$\nstates have the lower energies.\n\n" - ] - } - ], - "metadata": { - "kernelspec": { - "display_name": "Python 3", - "language": "python", - "name": "python3" - }, - "language_info": { - "codemirror_mode": { - "name": "ipython", - "version": 3 - }, - "file_extension": ".py", - "mimetype": "text/x-python", - "name": "python", - "nbconvert_exporter": "python", - "pygments_lexer": "ipython3", - "version": "3.10.16" - } - }, - "nbformat": 4, - "nbformat_minor": 0 -} \ No newline at end of file diff --git a/edrixs/_downloads/6dffa254d364294a4b733a81a90a1732/example_9_Coulomb.zip b/edrixs/_downloads/6dffa254d364294a4b733a81a90a1732/example_9_Coulomb.zip deleted file mode 100644 index 420846e43dbf3bc8ea9f2afbe54e0aca43febae6..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 45176 zcmeHQOLH67mDV^jvtYKVn(ZA3Wr7h2@g>J*g$qkIW5s!>EIOHlErAW7LA2TEhPxXg 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zecISOLQ0~@sGRgd6TGjTTk?lV5bBDJ1z+2X=-k4#qmrf`KVgmu?b}I`J*1j^Z-O7{ zjtCi7bQd$=(i%`p*En*4I=9iBrQG$jSh?315~BJ-^)^dLCKZv9?}Am`1_%eD6as+3 z`+9q4ZP4A=>Qq;IgiQ>%@3{ItaSlU^W%G=LBeDy~R0FKej3HlV9(D#u$;zCUpSkpD7_~2f@BS zAe#(m-V!)Bc1bXmMv|Td*w*HNOUu>8F~sgB?jfM0NSnfqYpe^q$Eoqm4p{~xf>?#8 zE*{N~^gk0HJa&Bzxs*rmDLV#W-_Ah2kKVO=yd`-5uthpSzazXrHj~BpXBG))TxeVl z6UN)vq|Nl!INM;Uc*2r!$LT@~)uOKl*I#<_{@dU9=C^+7TSTK(Do=<}`_G^K`Cr|e zt5o#$n|9T|efFt-2Apl{JFS8N4Hqm_E$>K4uYdf%KRx{8->p>iRqp7=bR9cdYQ446 zvcyY7m5g_EOh+I8**E|2qk5&HFJ;e;|33S4^}V8_Ml0yBEz@DgyLwDlpZ)0H|K{H= zRx0|c#(wzu*{47IgQBZu3|5ym(E8rZ@2`yarZ@bDKl#bW|MNkmqOad5_4ez3S@gEp zx_0xst?yr((23^zrGNU{f9(FCQqfnrllrfIBTdsQt)L!F9!dZ1-~VLww<;BV{dQ?2 z|I#ixxzuvoxUU_N0OOr$zJGYQ_+Ni|rBcyXdA{%eu;{Gb>TI-7&kf~FG5LvmU^MZ0 z9NK4}zVg>aSEE?%v92^vqp_1kSJ!@8baiqh<@;~{@~^1POZ@lW->g*r`_ -# if the code is confusing. -e, v = diagonlize(scaleU=0) -e = e.round(decimals=6) -unique_e = np.unique(e) -degeneracies = [sum(evalue == e) for evalue in unique_e] - -print("E \tDegeneracy") -for evalue, degenvalue in zip(unique_e, degeneracies): - print("{:.1f}\t{:.0f}".format(evalue, degenvalue)) -print("{} distinct energies".format(len(unique_e))) - -################################################################################ -# We see 10 distinct energies, which is the number of ways one can arrange -# two holes among 4 energy levels -- which makes sense as the tetragonal field -# involves four levels :math:`zx/zy, xy, 3z^2-r^2, x^2-y^2`. To see what is going -# on in more detail, we can also calculate the expectation -# values of the occupancy number of the orbitals -# :math:`3z^2-r^2, zx, zy, x^2-y^2, xy`. -# To create the operator, first write the matrix in the real harmonics basis -# :math:`|3z^2-r^2,\uparrow>`, :math:`|3z^2-r^2,\downarrow>`, -# :math:`|zx,\uparrow>`, :math:`|zx,\downarrow>`, etc. -# In this basis, they take a simple form: only the diagonal terms have element -# 1. We therefore make a 3D empty array and assign the diagonal as 1. Check -# out the -# `numpy indexing notes `_ -# if needed. -nd_real_harmoic_basis = np.zeros((norb, norb, norb), dtype=complex) -indx = np.arange(norb) -nd_real_harmoic_basis[indx, indx, indx] = 1 - -################################################################################ -# Recalling the necessity to put everything in the same basis, we transform -# into the complex harmonic basis and then transform into our Fock basis -nd_complex_harmoic_basis = edrixs.cb_op(nd_real_harmoic_basis, - edrixs.tmat_r2c('d', True)) -nd_op = edrixs.build_opers(2, nd_complex_harmoic_basis, basis) - - -################################################################################ -# We apply the operator and print out as follows. Check the -# `numpy docs `_ -# if the details of how the spin pairs have been added up is not immediately -# transparent. -nd_expt = np.array([edrixs.cb_op(nd_vec, v).diagonal().real for nd_vec in nd_op]) - -message = "{:>3s}" + "\t{:>6s}"*5 -print(message.format(*"E 3z2-r2 zx zy x2-y2 xy".split(" "))) - -message = "{:>3.1f}" + "\t{:>6.1f}"*5 -for evalue, row in zip(e, nd_expt.T): - spin_pairs = row.reshape(-1, 2).sum(1) - print(message.format(evalue, *spin_pairs)) - -################################################################################ -# The lowest energy state involves putting both holes in the :math:`x^2-y^2` -# orbital, which makes sense. Now, let us redo the proceedure including Coulomb -# repulsion, which imposes an energy cost to putting multiple electrons in the -# same orbital. - -e, v = diagonlize(scaleU=1) - -nd_expt = np.array([edrixs.cb_op(nd_vec, v).diagonal().real for nd_vec in nd_op]) - -message = "{:>3s}" + "\t{:>6s}"*5 -print(message.format(*"E 3z2-r2 zx zy x2-y2 xy".split(" "))) - -message = "{:>3.1f}" + "\t{:>6.1f}"*5 -for evalue, row in zip(e, nd_expt.T): - spin_pairs = row.reshape(-1, 2).sum(1) - print(message.format(evalue, *spin_pairs)) - -################################################################################ -# Now the lowest energy state involves splitting the holes between the two -# highest energy :math:`x^2-y^2` and :math:`3z^2-r^2` orbitals. i.e. we have -# gone from low-spin to high spin. Working out the balance between these two -# states is often one of the first things one wants to determine upon the -# discovery of an interesting new material [1]_. - -############################################################################## -# -# .. rubric:: Footnotes -# -# .. [1] M. Rossi et al., `arXiv:2011.00595 (2021) `_. diff --git a/edrixs/_downloads/7d2302527f77719856548abe89da7a72/example_8_Hunds_interactions.py b/edrixs/_downloads/7d2302527f77719856548abe89da7a72/example_8_Hunds_interactions.py deleted file mode 100644 index f33384ac8c..0000000000 --- a/edrixs/_downloads/7d2302527f77719856548abe89da7a72/example_8_Hunds_interactions.py +++ /dev/null @@ -1,290 +0,0 @@ -#!/usr/bin/env python -""" -Hund's Interactions in charge transfer insulators -================================================= -In this exercise we will solve a toy model relevant to cubic :math:`d^8` charge transfer insulators -such as NiO or NiPS\\ :sub:`3`. We are interested in better understanding the interplay between the -Hund's interactions and the charge transfer energy in terms of the energy of the triplet-singlet -excitations of this model. These seem to act against each other in that the Hund's interactions -impose a energy cost for the triplet-singlet excitations whenever there are two holes on -the Ni :math:`d` orbitals. The charge transfer physics, on the other hand, will promote a -:math:`d^9\\underline{L}` ground state in which the Hund's interactions are not active. - -The simplest model that captures this physics requires four Ni spin-orbitals, representing the Ni -:math:`e_g` manifold. We will represent the ligand states in the same way as the Anderson impurity -model in terms of one effective ligand spin-orbital per Ni spin-orbital. We assume these effective -orbitals have been constructed so that each Ni orbital only bonds to one sister orbital. For -simplicity, we will treat all Ni and all ligand orbitals as equivalent, even though a more -realistic model would account for the different Coulomb and hopping of the :math:`d_{3z^2-r^2}` -and :math:`d_{x^2-y^2}` orbitals. We therefore simply connect Ni and ligand orbitals via a constant -hopping :math:`t`. We also include the ligand energy parameter :math:`e_L`. - -The easiest way to implement the requried Coulomb interactions is to use the so-called Kanamori -Hamiltonian, which is a simplfied form for the interactions, which treats all orbitals as -equivalent. Daniel Khomskii's book provides a great explanation of this physics [1]_. We -parameterize the interactions via Coulomb repulsion parameter :math:`U` and Hund's exchange -:math:`J_H`. EDRIXS provides this functionality via the more general -:func:`.get_umat_kanamori` function. - -It's also easiest to consider this problem in hole langauge, which means our eight spin-orbitals -are populated by two fermions. -""" - -################################################################################ -# Setup -# ------------------------------------------------------------------------------ -# We start by loading the necessary modules, and defining the total number of -# orbitals and electrons. -import edrixs -import scipy -import numpy as np -import matplotlib.pyplot as plt - -norb = 8 -noccu = 2 - -################################################################################ -# Diagonalization -# ------------------------------------------------------------------------------ -# Let's write a function to diagonalize our model in a similar way to -# the :ref:`sphx_glr_auto_examples_example_6_Hubbard_dimer.py` example. -# Within this function, we also create operators to count the number of -# :math:`d` holes and operators to calculate expectation values for -# :math:`S^2` and :math:`S_z`. For the latter to make sense, we also include a -# small effective spin interaction along :math:`z`. - - -def diagonalize(U, JH, t, eL, n=1): - # Setup Coulomb matrix - umat = np.zeros((norb, norb, norb, norb), dtype=complex) - uNi = edrixs.get_umat_kanamori(norb//2, U, JH) - umat[:norb//2, :norb//2, :norb//2, :norb//2] = uNi - - # Setup hopping matrix - emat = np.zeros((norb, norb), dtype=complex) - ind = np.arange(norb//2) - emat[ind, ind + norb//2] = t - emat[ind+norb//2, ind] = np.conj(t) # conj is not needed, but is good practise. - ind = np.arange(norb//2, norb) - emat[ind, ind] += eL - - # Spin operator - spin_mom = np.zeros((3, norb, norb), dtype=complex) - spin_mom[:, :2, :2] = edrixs.get_spin_momentum(0) - spin_mom[:, 2:4, 2:4] = edrixs.get_spin_momentum(0) - spin_mom[:, 4:6, 4:6] = edrixs.get_spin_momentum(0) - spin_mom[:, 6:8, 6:8] = edrixs.get_spin_momentum(0) - - # add small effective field along z - emat += 1e-6*spin_mom[2] - - # Diagonalize - basis = edrixs.get_fock_bin_by_N(norb, noccu) - H = edrixs.build_opers(2, emat, basis) + edrixs.build_opers(4, umat, basis) - e, v = scipy.linalg.eigh(H) - e -= e[0] # Define ground state as zero energy - - # Operator for holes on Ni - basis = np.array(basis) - num_d_electrons = basis[:, :4].sum(1) - d0 = np.sum(np.abs(v[num_d_electrons == 0, :])**2, axis=0) - d1 = np.sum(np.abs(v[num_d_electrons == 1, :])**2, axis=0) - d2 = np.sum(np.abs(v[num_d_electrons == 2, :])**2, axis=0) - - # S^2 and Sz operators - opS = edrixs.build_opers(2, spin_mom, basis) - S_squared_op = np.dot(opS[0], opS[0]) + np.dot(opS[1], opS[1]) + np.dot(opS[2], opS[2]) - S_squared_exp = edrixs.cb_op(S_squared_op, v).diagonal().real - S_z_exp = edrixs.cb_op(opS[2], v).diagonal().real - - return e[:n], d0[:n], d1[:n], d2[:n], S_squared_exp[:n], S_z_exp[:n] - - -################################################################################ -# The atomic limit -# ------------------------------------------------------------------------------ -# For simplicity, let's start in the atomic limit with :math:`e_L \gg t \gg U` -# where all holes are on nickel. In this case, there are six ways to distribute -# two holes on the four Ni spin-orbitals. Let's examine the expectation values -# of the :math:`S^2` and :math:`S_z` operators. -U = 10 -JH = 2 -t = 100 -eL = 1e10 - -e, d0, d1, d2, S_squared_exp, S_z_exp = diagonalize(U, JH, t, eL, n=6) - -print("Ground state\nE\t") -for i in range(3): - print(f"{e[i]:.2f}\t{S_squared_exp[i]:.2f}\t{S_z_exp[i]:.2f}") - -print("\nExcited state\nE\t") -for i in range(3, 6): - print(f"{e[i]:.2f}\t{S_squared_exp[i]:.2f}\t{S_z_exp[i]:.2f}") - -################################################################################ -# The ground state is a high-spin triplet. The fourth and fifth -# states (the first excited state) are low-spin singlet excitons at -# :math:`2 J_H`. These have one hole on each orbital in the antisymmetric -# combination of :math:`|\uparrow\downarrow>-|\downarrow\uparrow>`. -# The state at :math:`3 J_H` also has one hole on each orbital in the symmetric -# :math:`|\uparrow\downarrow>+|\downarrow\uparrow>` configuration. - -################################################################################ -# Where are the holes for large hopping -# ------------------------------------------------------------------------------ -# As discussed at the start, we are interested to see interplay between Hund's -# and charge-transfer physics, which will obviously depend strongly on whether -# the holes are on Ni or the ligand. Let's see what happens as :math:`e_L` is -# reduced while observing the location of the ground state and exciton holes. -U = 10 -JH = 2 -t = 100 - -eLs = np.linspace(0, 1000, 30) - -fig, axs = plt.subplots(1, 2, figsize=(8, 4)) - -for ax, ind in zip(axs.ravel(), [0, 3]): - ds = np.array([diagonalize(U, JH, t, eL, n=6) - for eL in eLs]) - - ax.plot(eLs, ds[:, 1, ind], 'o', label='$d^0$') - ax.plot(eLs, ds[:, 2, ind], 's', label='$d^1$') - ax.plot(eLs, ds[:, 3, ind], '^', label='$d^2$') - ax.set_xlabel("Energy of ligands $e_L$") - ax.set_ylabel("Number of electrons") - ax.legend() - -axs[0].set_title("Location of ground state holes") -axs[1].set_title("Location of exciton holes") - -plt.tight_layout() -plt.show() -################################################################################ -# For large :math:`e_L`, we see that both holes are on nickel as expected. In -# the opposite limit of :math:`|e_L| \ll t` and :math:`U \ll t` the holes are -# shared in the ratio 0.25:0.5:0.25 as there are two ways to have one hole on -# Ni. In the limit of large :math:`e_L`, all holes move onto Ni. Since -# :math:`t` is large, this applies equally to both the ground state and the -# exciton. - - -################################################################################ -# Connecton between atomic and charge transfer limits -# ------------------------------------------------------------------------------ -# We now examine the quantum numbers during cross over between the two limits -# with :math:`e_L`. Let's first look at the how :math:`` changes for the -# ground state and exciton and then examine how the exciton energy changes. - -U = 10 -JH = 2 -t = 100 - -eLs = np.linspace(0, 1000, 30) - -info = np.array([diagonalize(U, JH, t, eL, n=6) - for eL in eLs]) - -fig, axs = plt.subplots(1, 2, figsize=(8, 4)) - - -axs[0].plot(eLs, info[:, 4, 0], label='Ground state') -axs[0].plot(eLs, info[:, 4, 3], label='Exciton') -axs[0].set_xlabel("Energy of ligands $e_L$") -axs[0].set_ylabel('$$') -axs[0].set_title('Quantum numbers') -axs[0].legend() - -axs[1].plot(eLs, info[:, 0, 3], '+', color='C0') -axs[1].set_xlabel("Energy of ligands $e_L$") -axs[1].set_ylabel('Exciton energy', color='C0') -axr = axs[1].twinx() -axr.plot(eLs, info[:, 3, 5], 'x', color='C1') -axr.set_ylabel('$d^2$ fraction', color='C1') - -for ax, color in zip([axs[1], axr], ['C0', 'C1']): - for tick in ax.get_yticklabels(): - tick.set_color(color) - -axs[1].set_ylim(0, 2*JH) -axr.set_ylim(0, 1) - -axs[1].set_title('Exciton energy vs. $d^2$ character') -plt.tight_layout() -plt.show() -############################################################################## -# In the left panel, we see that the two limits are adiabatically connected -# as they preseve the same quantum numbers. This is because there is always -# an appreciable double occupancy under conditions where the -# :math:`d^9\underline{L}` character is maximized and this continues to favor -# the high spin ground state. Other interactions such as strong tetragonal -# crystal field would be needed to overcome the Hund's interactions and break -# this paradigm. In the right panel, we see that the exciton energy simply -# scales with the double occupancy. Overall, even though -# Hund's interactions are irrelevant for the :math:`d^9\underline{L}` -# electronic configuration, whenever :math:`t` is appreciable there is a -# strong mixing with the :math:`d^8` component is always present, which -# dominates the energy of the exciton. - -################################################################################ -# Charge transfer excitons -# ------------------------------------------------------------------------------ -# Another limiting case of the model is where :math:`t` is smaller than the -# Coulomb interactions. This, however, tends to produce -# ground state and exciton configurations that correspond to those of distinct -# atomic models. Let's look at the :math:`e_L` dependence in this case. -U = 10 -JH = 2 -t = .5 -eL = 7 - -eLs = np.linspace(0, 20, 30) - -fig, axs = plt.subplots(1, 2, figsize=(8, 4)) - -for ax, ind in zip(axs.ravel(), [0, 3]): - ds = np.array([diagonalize(U, JH, t, eL, n=6) - for eL in eLs]) - - ax.plot(eLs, ds[:, 1, ind], 'o', label='$d^0$') - ax.plot(eLs, ds[:, 2, ind], 's', label='$d^1$') - ax.plot(eLs, ds[:, 3, ind], '^', label='$d^2$') - ax.set_xlabel("Energy of ligands $e_L$") - ax.set_ylabel("Number of electrons") - ax.legend() - -axs[0].axvline(x=eL, linestyle=':', color='k') -axs[1].axvline(x=eL, linestyle=':', color='k') - -axs[0].set_title("Location of ground state holes") -axs[1].set_title("Location of exciton holes") - -plt.tight_layout() -plt.show() -################################################################################ -# Around :math:`e_L = 7` the plot shows that the excition is primairly a -# :math:`d^2 \rightarrow d^1` transition or a -# :math:`d^8 \rightarrow d^{9}\underline{L}` transition in electron language. -# Let's examine the energy and quantum numbers. - -e, d0, d1, d2, S_squared_exp, S_z_exp = diagonalize(U, JH, t, eL, n=6) - -print("Ground state\nE\t") -for i in range(3): - print(f"{e[i]:.2f}\t{S_squared_exp[i]:.2f}\t{S_z_exp[i]:.2f}") - -print("\nExcited state\nE\t") -for i in range(3, 6): - print(f"{e[i]:.2f}\t{S_squared_exp[i]:.2f}\t{S_z_exp[i]:.2f}") - -################################################################################ -# We once again see the same quantum numbers, despite the differences in mixing -# in the ground state and exciton. - - -############################################################################## -# -# .. rubric:: Footnotes -# -# .. [1] D. Khomskii, Transition Metal Compounds, Cambridge University Press (2014) diff --git a/edrixs/_downloads/86f83fa0327e3b951a4dcf9df5773fd6/example_3_AIM_XAS.py b/edrixs/_downloads/86f83fa0327e3b951a4dcf9df5773fd6/example_3_AIM_XAS.py deleted file mode 100644 index 37033b93e1..0000000000 --- a/edrixs/_downloads/86f83fa0327e3b951a4dcf9df5773fd6/example_3_AIM_XAS.py +++ /dev/null @@ -1,344 +0,0 @@ -#!/usr/bin/env python -""" -Anderson impurity model for NiO XAS -================================================================================ -Here we calculate the :math:`L`-edge XAS spectrum of an Anderson impurity model, -which is sometimes also called a charge-transfer multiplet model. This model -considers a set of correlated orbitals, often called the impurity or metal -states, that hybridize with a set of uncorrelated orbitals, often called the -ligands or bath states. Everyone's favorite test case for x-ray spectroscopic -calculations of the Anderson impurity model is NiO and we won't risk being -original! This means that our correlated states will -be the Ni :math:`3d` orbitals and the uncorrelated states come from O -:math:`2p` orbitals. The fact that we include these interactions means that our -spectrum can include processes where electrons transition from the bath to the -impurity, as such the Anderson Impurity Model is often more accurate than atomic -models, especially if the material has strong covalency. - -When defining the bath states, it is useful to use the so-called -symmetry adapted linear combinations of orbitals as the basis. These states -take into account the symmetry relationships between the different bath atom -orbitals and the fact that there are bath orbital combinations that do not -interact with the impurity by symmetry. By doing this the problem can be -represented with fewer orbitals, which makes the calculation far more efficient. -The standard EDRIXS solver that we will use assumes that the bath states are -represented by an integer number of bath sites set by :code:`nbath`, each of -which hosts the same number of spin-orbits as the impurity e.g. 10 for a -:math:`d`-electron material. - -NiO has a rocksalt structure in which all Ni atoms are surrounded by six O -atoms. This NiO cluster used to simulate the -crystal would then contain 10 Ni :math:`3d` spin-orbitals and :math:`6` -spin-orbitals per O :math:`\\times 6` O atoms :math:`=36` oxygen -spin-orbitals. As explained by, for example, Maurits Haverkort -et al. in [1]_ symmetry allows us to represent the bath with 10 symmetry -adapted linear combinations of the different O :math:`p_x, p_y, p_z` states. -The crystal field and hopping parameters for -such a calculation can be obtained by post-processing DFT calculations. We will -use values for NiO from [1]_. If you use values from a paper the relevant -references should, of course, be cited. - -""" -import edrixs -import numpy as np -import matplotlib.pyplot as plt - -################################################################################ -# Number of electrons -# ------------------------------------------------------------------------------ -# When formulating problems of this type, one usually thinks of a nominal -# valence for the impurity atom in this case :code:`nd = 8` and assume that the -# bath is full. The solver that we will -# use can simulate multiple bath sites. In our case we specify -# :code:`nbath = 1` sites. Electrons will be able to transition from O to Ni -# during our calculation, but the total number of valance electrons -# :code:`v_noccu` will be conserved. -nd = 8 -norb_d = 10 -norb_bath = 10 -nbath = 1 -v_noccu = nd + nbath*norb_d -shell_name = ('d', 'p') # valence and core shells for XAS calculation - -################################################################################ -# Coulomb interactions -# ------------------------------------------------------------------------------ -# The atomic Coulomb interactions are usually initialized based on Hartree-Fock -# calculations from, for example, -# `Cowan's code `_. -# edrixs has a database of these. -info = edrixs.utils.get_atom_data('Ni', '3d', nd, edge='L3') - -################################################################################ -# The atomic values are typically scaled to account for screening in the solid. -# Here we use 80% scaling. Let's write these out in full, so that nothing is -# hidden. Values for :math:`U_{dd}` and :math:`U_{dp}` are those of Ref. [1]_ -# obtained by comparing theory and experiment [2]_ [3]_. -scale_dd = 0.8 -F2_dd = info['slater_i'][1][1] * scale_dd -F4_dd = info['slater_i'][2][1] * scale_dd -U_dd = 7.3 -F0_dd = U_dd + edrixs.get_F0('d', F2_dd, F4_dd) - -scale_dp = 0.8 -F2_dp = info['slater_n'][4][1] * scale_dp -G1_dp = info['slater_n'][5][1] * scale_dp -G3_dp = info['slater_n'][6][1] * scale_dp -U_dp = 8.5 -F0_dp = U_dp + edrixs.get_F0('dp', G1_dp, G3_dp) - -slater = ([F0_dd, F2_dd, F4_dd], # initial - [F0_dd, F2_dd, F4_dd, F0_dp, F2_dp, G1_dp, G3_dp]) # with core hole - -################################################################################ -# Energy of the bath states -# ------------------------------------------------------------------------------ -# In the notation used in EDRIXS, :math:`\Delta` sets the energy difference -# between the bath and impurity states. :math:`\Delta` is defined in the atomic -# limit without crystal field (i.e. in terms of the centers of the impurity and -# bath states before hybridization is considered) as the energy for a -# :math:`d^{n_d} \rightarrow d^{n_d + 1} \underline{L}` transition. -# Note that as electrons are moved one has to pay energy -# costs associated with the Coulomb interactions. The -# energy splitting between the bath and impurity is consequently not simply -# :math:`\Delta`. One must therefore determine the energies by solving -# a set of linear equations. See the :ref:`edrixs.utils functions ` -# for details. We can call these functions to get the impurity energy -# :math:`E_d`, bath energy :math:`E_L`, impurity energy with a core hole -# :math:`E_{dc}`, bath energy with a core hole :math:`E_{Lc}` and the -# core hole energy :math:`E_p`. The -# :code:`if __name__ == '__main__'` code specifies that this command -# should only be executed if the file is explicitly run. -Delta = 4.7 -E_d, E_L = edrixs.CT_imp_bath(U_dd, Delta, nd) -E_dc, E_Lc, E_p = edrixs.CT_imp_bath_core_hole(U_dd, U_dp, Delta, nd) -message = ("E_d = {:.3f} eV\n" - "E_L = {:.3f} eV\n" - "E_dc = {:.3f} eV\n" - "E_Lc = {:.3f} eV\n" - "E_p = {:.3f} eV\n") -if __name__ == '__main__': - print(message.format(E_d, E_L, E_dc, E_Lc, E_p)) - - -################################################################################ -# The spin-orbit coupling for the valence electrons in the ground state, the -# valence electrons with the core hole present, and for the core hole itself -# are initialized using the atomic values. -zeta_d_i = info['v_soc_i'][0] -zeta_d_n = info['v_soc_n'][0] -c_soc = info['c_soc'] - -################################################################################ -# Build matrices describing interactions -# ------------------------------------------------------------------------------ -# edrixs uses complex spherical harmonics as its default basis set. If we want to -# use another basis set, we need to pass a matrix to the solver, which transforms -# from complex spherical harmonics into the basis we use. -# The solver will use this matrix when implementing the Coulomb interactions -# using the :code:`slater` list of Coulomb parameters. -# Here it is easiest to -# use real harmonics. We make the complex harmonics to real harmonics transformation -# matrix via -trans_c2n = edrixs.tmat_c2r('d',True) - -################################################################################ -# The crystal field and SOC needs to be passed to the solver by constructing -# the impurity matrix in the real harmonic basis. For cubic symmetry, we need -# to set the energies of the orbitals along the -# diagonal of the matrix. These need to be in pairs as there are two -# spin-orbitals for each orbital energy. Python -# `list comprehension `_ -# and -# `numpy indexing `_ -# are used here. See :ref:`sphx_glr_auto_examples_example_1_crystal_field.py` -# for more details if needed. -ten_dq = 0.56 -CF = np.zeros((norb_d, norb_d), dtype=complex) -diagonal_indices = np.arange(norb_d) - -orbital_energies = np.array([e for orbital_energy in - [+0.6 * ten_dq, # dz2 - -0.4 * ten_dq, # dzx - -0.4 * ten_dq, # dzy - +0.6 * ten_dq, # dx2-y2 - -0.4 * ten_dq] # dxy) - for e in [orbital_energy]*2]) - - -CF[diagonal_indices, diagonal_indices] = orbital_energies - -################################################################################ -# The valence band SOC is constructed in the normal way and transformed into the -# real harmonic basis. -soc = edrixs.cb_op(edrixs.atom_hsoc('d', zeta_d_i), edrixs.tmat_c2r('d', True)) - -################################################################################ -# The total impurity matrices for the ground and core-hole states are then -# the sum of crystal field and spin-orbit coupling. We further needed to apply -# an energy shift along the matrix diagonal, which we do using the -# :code:`np.eye` function which creates a diagonal matrix of ones. -E_d_mat = E_d*np.eye(norb_d) -E_dc_mat = E_dc*np.eye(norb_d) -imp_mat = CF + soc + E_d_mat -imp_mat_n = CF + soc + E_dc_mat - -################################################################################ -# The energy level of the bath(s) is described by a matrix where the row index -# denotes which bath and the column index denotes which orbital. Here we have -# only one bath, with 10 spin-orbitals. We initialize the matrix to -# :code:`norb_d` and then split the energies according to :code:`ten_dq_bath`. -ten_dq_bath = 1.44 -bath_level = np.full((nbath, norb_d), E_L, dtype=complex) -bath_level[0, :2] += ten_dq_bath*.6 # 3z2-r2 -bath_level[0, 2:6] -= ten_dq_bath*.4 # zx/yz -bath_level[0, 6:8] += ten_dq_bath*.6 # x2-y2 -bath_level[0, 8:] -= ten_dq_bath*.4 # xy -bath_level_n = np.full((nbath, norb_d), E_Lc, dtype=complex) -bath_level_n[0, :2] += ten_dq_bath*.6 # 3z2-r2 -bath_level_n[0, 2:6] -= ten_dq_bath*.4 # zx/yz -bath_level_n[0, 6:8] += ten_dq_bath*.6 # x2-y2 -bath_level_n[0, 8:] -= ten_dq_bath*.4 # xy - -################################################################################ -# The hybridization matrix describes the hopping between the bath -# and the impurity. This is called either :math:`V` or :math:`T` in the -# literature and matrix sign can either be positive or negative based. -# This is the same shape as the bath matrix. We take our -# values from Maurits Haverkort et al.'s DFT calculations [1]_. -Veg = 2.06 -Vt2g = 1.21 - -hyb = np.zeros((nbath, norb_d), dtype=complex) -hyb[0, :2] = Veg # 3z2-r2 -hyb[0, 2:6] = Vt2g # zx/yz -hyb[0, 6:8] = Veg # x2-y2 -hyb[0, 8:] = Vt2g # xy - -################################################################################ -# We now need to define the parameters describing the XAS. X-ray polarization -# can be linear, circular or isotropic (appropriate for a powder). -poltype_xas = [('isotropic', 0)] -################################################################################ -# edrixs uses the temperature in Kelvin to work out the population of the low-lying -# states via a Boltzmann distribution. -temperature = 300 -################################################################################ -# The x-ray beam is specified by the incident angle and azimuthal angle in radians -thin = 0 / 180.0 * np.pi -phi = 0.0 -################################################################################ -# these are with respect to the crystal field :math:`z` and :math:`x` axes -# written above. (That is, unless you specify the :code:`loc_axis` parameter -# described in the :code:`edrixs.xas_siam_fort` function documentation.) - -################################################################################ -# The spectrum in the raw calculation is offset by the energy involved with the -# core hole state, which is roughly :math:`5 E_p`, so we offset the spectrum by -# this and use :code:`om_shift` as an adjustable parameters for comparing -# theory to experiment. We also use this to specify :code:`ominc_xas` -# the range we want to compute the spectrum over. The core hole lifetime -# broadening also needs to be set via :code:`gamma_c_stat`. -om_shift = 857.6 -c_level = -om_shift - 5*E_p -ominc_xas = om_shift + np.linspace(-15, 25, 1000) - -################################################################################ -# The final state broadening is specified in terms of half-width at half-maximum -# You can either pass a constant value or an array the same size as -# :code:`om_shift` with varying values to simulate, for example, different state -# lifetimes for higher energy states. -gamma_c = np.full(ominc_xas.shape, 0.48/2) - -################################################################################ -# Magnetic field is a three-component vector in eV specified with respect to the -# same local axis as the x-ray beam. Since we are considering a powder here -# we create an isotropic normalized vector. :code:`on_which = 'both'` specifies to -# apply the operator to the total spin plus orbital angular momentum as is -# appropriate for a physical external magnetic field. You can use -# :code:`on_which = 'spin'` to apply the operator to spin in order to simulate -# magnetic order in the sample. The value of the Bohr Magneton can -# be useful for converting here :math:`\mu_B = 5.7883818012\times 10^{−5}`. -# For this example, we will account for magnetic order in the sample by -ext_B = np.array([0.00, 0.00, 0.12]) -on_which = 'spin' - -################################################################################ -# The number crunching uses -# `mpi4py `_. You can safely ignore -# this for most purposes, but see -# `Y. L. Wang et al., Computer Physics Communications 243, 151-165 (2019) `_ -# if you would like more details. -# The main thing to remember is that you should call this script via:: -# -# mpirun -n python example_AIM_XAS.py -# -# where :code:`` is the number of processors -# you'd like to us. Running it as normal will work, it will just be slower. -if __name__ == '__main__': - from mpi4py import MPI - comm = MPI.COMM_WORLD - rank = comm.Get_rank() - size = comm.Get_size() - -################################################################################ -# Calling the :code:`edrixs.ed_siam_fort` solver will find the ground state and -# write input files, *hopping_i.in*, *hopping_n.in*, *coulomb_i.in*, *coulomb_n.in* -# for following XAS (or RIXS) calculation. We need to specify :code:`siam_type=0` -# which says that we will pass *imp_mat*, *bath_level* and *hyb*. -# We need to specify :code:`do_ed = 1`. For this example, we cannot use -# :code:`do_ed = 0` for a ground state search as we have set the impurity and -# bath energy levels artificially, which means edrixs will have trouble to know -# which subspace to search to find the ground state. -if __name__ == '__main__': - do_ed = 1 - eval_i, denmat, noccu_gs = edrixs.ed_siam_fort( - comm, shell_name, nbath, siam_type=0, imp_mat=imp_mat, imp_mat_n=imp_mat_n, - bath_level=bath_level, bath_level_n=bath_level_n, hyb=hyb, c_level=c_level, - c_soc=c_soc, slater=slater, ext_B=ext_B, - on_which=on_which, trans_c2n=trans_c2n, v_noccu=v_noccu, do_ed=do_ed, - ed_solver=2, neval=50, nvector=3, ncv=100, idump=True) -################################################################################ -# Let's check that we have all the electrons we think we have and print how -# the electron are distributed between the Ni (impurity) and O (bath). -if __name__ == '__main__': - assert np.abs(noccu_gs - v_noccu) < 1e-6 - impurity_occupation = np.sum(denmat[0].diagonal()[0:norb_d]).real - bath_occupation = np.sum(denmat[0].diagonal()[norb_d:]).real - print('Impurity occupation = {:.6f}\n'.format(impurity_occupation)) - print('Bath occupation = {:.6f}\n'.format(bath_occupation)) -################################################################################ -# We see that 0.18 electrons move from the O to the Ni in the ground state. -# -# We can now construct the XAS spectrum edrixs by applying a transition -# operator to create the excited state. We need to be careful to specify how -# many of the low energy states are thermally populated. In this case -# :code:`num_gs=3`. This can be determined by inspecting the function output. -if __name__ == '__main__': - xas, xas_poles = edrixs.xas_siam_fort( - comm, shell_name, nbath, ominc_xas, gamma_c=gamma_c, v_noccu=v_noccu, thin=thin, - phi=phi, num_gs=3, nkryl=200, pol_type=poltype_xas, temperature=temperature - ) -################################################################################ -# Let's plot the data and save it just in case -if __name__ == '__main__': - fig, ax = plt.subplots() - - ax.plot(ominc_xas, xas) - ax.set_xlabel('Energy (eV)') - ax.set_ylabel('XAS intensity') - ax.set_title('Anderson impurity model for NiO') - plt.show() - - np.savetxt('xas.dat', np.concatenate((np.array([ominc_xas]).T, xas), axis=1)) - -############################################################################## -# -# .. rubric:: Footnotes -# -# .. [1] Maurits Haverkort et al -# `Phys. Rev. B 85, 165113 (2012) `_. -# .. [2] A. E. Bocquet et al., -# `Phys. Rev. B 53, 1161 (1996) `_. -# .. [3] Arata Tanaka, and Takeo Jo, -# `J. Phys. Soc. Jpn. 63, 2788-2807(1994) `_. diff --git a/edrixs/_downloads/9014043ee6a1e0681078cd8cc8ad3780/example_5_Hubbard_dimer.ipynb b/edrixs/_downloads/9014043ee6a1e0681078cd8cc8ad3780/example_5_Hubbard_dimer.ipynb deleted file mode 100644 index 331bf277f4..0000000000 --- a/edrixs/_downloads/9014043ee6a1e0681078cd8cc8ad3780/example_5_Hubbard_dimer.ipynb +++ /dev/null @@ -1,190 +0,0 @@ -{ - "cells": [ - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "%matplotlib inline" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n# Hubbard Dimer\nThis exercise will demonstrate how to handle hopping and multi-site problems within\nedrixs using the example of a Hubbard dimer. We want to solve the equation\n\n .. math::\n \\begin{equation}\n \\hat{H} = \\sum_{i,j} \\sum_{\\sigma} t_{i,j} \\hat{f}^{\\dagger}_{i,\\sigma} \\hat{f}_{j, \\sigma}\n + U \\sum_{i} \\hat{n}_{i,\\uparrow}\\hat{n}_{i,\\downarrow},\n \\end{equation}\n\nwhich involves two sites labeled with indices $i$ or $j$ with two\nelectrons of spin $\\sigma\\in{\\uparrow,\\downarrow}$. $t_{i,j}$\nis the hopping between sites, $\\hat{f}^{\\dagger}_{i,\\sigma}$ is the\ncreation operators, and\n$\\hat{n}^{\\dagger}_{i,\\sigma}=\\hat{f}^{\\dagger}_{i,\\sigma}\\hat{f}_{i,\\sigma}$\nis the number operator. The main task is to represent this Hamiltonian and\nthe related spin operator using the EDRIXS two-fermion and four-fermion form\nwhere $\\alpha,\\beta,\\delta,\\gamma$ are the indices of the single\nparticle basis.\n\n .. math::\n \\begin{equation}\n \\hat{H} = \\sum_{\\alpha,\\beta} t_{\\alpha,\\beta} \\hat{f}^{\\dagger}_{\\alpha} \\hat{f}_{\\beta}\n + \\sum_{\\alpha,\\beta,\\gamma,\\delta} U_{\\alpha,\\beta,\\gamma,\\delta}\n \\hat{f}^{\\dagger}_{\\alpha}\\hat{f}^{\\dagger}_{\\beta}\\hat{f}_{\\gamma}\\hat{f}_{\\delta}.\n \\end{equation}\n" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Initialize matrices\nWe start by noting that each of the two sites is like an $l=0$\n$s$-orbital with two spin-orbitals each. We will include\ntwo electron occupation and build the Fock basis.\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "import numpy as np\nimport matplotlib.pyplot as plt\nimport scipy\nimport edrixs\nnp.set_printoptions(precision=4)\n\n\nll = 0\ncase = 's'\nnorb = 4\nnoccu = 2\nbasis = edrixs.get_fock_bin_by_N(norb, noccu)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Create function to populate and diagonalize matrices\nThe Coulomb and hopping matrices :code:`umat` and :code:`emat` will be\nrepresented by $4\\times4\\times4\\times4$ and $4\\times4$ matrices,\nrespectively. Note that we needed to specify\nthat these are, in general, complex, although\nthey happen to contain only real numbers in this case. We follow the convention\nthat these are ordered first by site and then by spin:\n$|0\\uparrow>, |0\\downarrow>, |1\\uparrow>, |1\\downarrow>$.\nConsequently the $2\\times2$ and $2\\times2\\times2\\times2$ block\ndiagonal structures of the matrices will contain the on-site interactions.\nThe converse is true for the hopping between the sites.\nFrom here let us generate a function to build and diagonalize the Hamiltonian.\nWe need to generate the Coulomb matrix for the on-site interactions and\napply it to the block diagonal. The hopping connects off-site indices with\nthe same spin.\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "def diagonalize(U, t, extra_emat=None):\n \"\"\"Diagonalize 2 site Hubbard Hamiltonian\"\"\"\n umat = np.zeros((norb, norb, norb, norb), dtype=np.complex128)\n emat = np.zeros((norb, norb), dtype=np.complex128)\n U_mat_1site = edrixs.get_umat_slater('s', U)\n umat[:2, :2, :2, :2,] = umat[2:, 2:, 2:, 2:] = U_mat_1site\n emat[2, 0] = emat[3, 1] = emat[0, 2] = emat[1, 3] = t\n\n if extra_emat is not None:\n emat = emat + extra_emat\n\n H = (edrixs.build_opers(2, emat, basis)\n + edrixs.build_opers(4, umat, basis))\n\n e, v = scipy.linalg.eigh(H)\n return e, v" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## The large $U$ limit\nLet us see what happens with $U \\gg t$.\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "e, v = diagonalize(1000, 1)\nprint(\"Energies are\")\nprint(e)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "To analyze what is going on we can determine the spin expectation values\nof the cluster. Building the operators follows the same form as the\nHamiltonian and the previous example.\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "spin_mom_one_site = edrixs.get_spin_momentum(ll)\nspin_mom = np.zeros((3, norb, norb), dtype=np.complex128)\nspin_mom[:, :2, :2] = spin_mom[:, 2:, 2:] = spin_mom_one_site\n\nopS = edrixs.build_opers(2, spin_mom, basis)\nopS_squared = (np.dot(opS[0], opS[0]) + np.dot(opS[1], opS[1])\n + np.dot(opS[2], opS[2]))" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "This time let us include a tiny magnetic field along the $z$-axis, so\nthat we have a well-defined measurement axis and print out the expectation\nvalues.\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "zeeman = np.zeros((norb, norb), dtype=np.complex128)\nzeeman[:2, :2] = zeeman[2:, 2:] = 1e-8*spin_mom_one_site[2]\ne, v = diagonalize(1000, 1, extra_emat=zeeman)\n\nSsq_exp = edrixs.cb_op(opS_squared, v).diagonal().real\nSz_exp = edrixs.cb_op(opS[2], v).diagonal().real\n\nheader = \"{:<10s}\\t{:<6s}\\t{:<6s}\"\nprint(header.format(\"E\", \"S(S+1)\", \"\"))\nfor i in range(len(e)):\n print(\"{:<2f}\\t{:.1f}\\t{:.1f}\".format(e[i], Ssq_exp[i], Sz_exp[i]))" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "For $U \\gg t$ the two states with double occupancy acquire an energy of\napproximately $U$. The low energy states are a $S=0$ singlet and\nand $S=1$ triplet, which are split by $4t^2/U$, which is the\nmagnetic exchange term.\n\n" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## $U$ dependence\nLet us plot the changes in energy with $U$.\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "plt.figure()\n\nt = 1\nUs = np.linspace(0.01, 10, 50)\nEs = np.array([diagonalize(U, t, extra_emat=zeeman)[0] for U in Us])\n\nplt.plot(Us/t, Es/t)\nplt.xlabel('U/t')\nplt.ylabel('Eigenstate energies/t')\nplt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "To help interpret this, we can represent the eigenvectors in terms of a sum\nof the single particle states.\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "def get_single_particle_repesentations(v):\n reps = []\n for i in range(6):\n rep = sum([vec*weight for weight, vec\n in zip(v[:, i], np.array(basis))])\n reps.append(rep)\n\n return np.array(reps)\n\nt = 1\nfor U in [10000, 0.0001]:\n e, v = diagonalize(U, t, extra_emat=zeeman)\n repesentations = get_single_particle_repesentations(v)\n print(\"For U={} t={} states are\".format(U, t))\n print(repesentations.round(3).real)\n print(\"\\n\")" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "For $U \\gg t$ the ground state maximizes its magnetic exchange\nenergy saving. In the $U \\ll t$ condition the ground state maximizes\nits kinetic energy saving. Since both states share the same parity, the\ncross-over between them is smooth. This type of physics is at play in current\nresearch on quantum materials [1]_ [2]_.\n\n" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - ".. rubric:: Footnotes\n\n.. [1] Y. Wang et al., `Phys. Rev. Lett. 122, 106401 (2019) `_.\n.. [2] A. Revelli et al., `Science Advances 5, eaav4020 (2019) `_.\n\n" - ] - } - ], - "metadata": { - "kernelspec": { - "display_name": "Python 3", - "language": "python", - "name": "python3" - }, - "language_info": { - "codemirror_mode": { - "name": "ipython", - "version": 3 - }, - "file_extension": ".py", - "mimetype": "text/x-python", - "name": "python", - "nbconvert_exporter": "python", - "pygments_lexer": "ipython3", - "version": "3.8.12" - } - }, - "nbformat": 4, - "nbformat_minor": 0 -} \ No newline at end of file diff --git a/edrixs/_downloads/9d9d4281e4fa8bdc87c5f8d67a01fd4a/example_4_GS_analysis.zip b/edrixs/_downloads/9d9d4281e4fa8bdc87c5f8d67a01fd4a/example_4_GS_analysis.zip deleted file mode 100644 index a9a73f9aa2298e0c497dd8bd2e92276854aa8cde..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 17653 zcmds9TW{OQ6;5~CKJa-T3KRvVicm{gnX#3iNDMf@ZnxP4Nw)@)UQjckC~+jsh8nUY zWn1?8VGH!RKwtU?_74>OF@5N7DbVkn;ZW2qmL1usSAipO?sMkc=6pvwc=7QMKe4n>v+xfHgC=*Ow}O8wT^-|)+MJye+}R*6}nC_c_aK9piFmHoX= zmJCn)L6rKzIFJ3JU1mS{5B$%b|Hl90XRolOj$?-ju>M}GGZo6z43-5WDMoQ9BUkW@ zxkACQ6xlF77W9|LX0xZ#q1lgvBY7lyd5i&+7P(AEnTY!$aP~$)KHTegUJxb2KztLv zef09}+3wEy6Y;6=yspf-@bSyHkDuUq5R67#Kuuo0ee|T`q%uimCN(ImV(mK4b1lMH z=>hf)VqjIrDvHD)#xo9MFl2hj6Pfl@CTSJjv5G?KKsyapFoJ%ls_-DaLMR}}M zplR^|b~C{kL5`g!vOkWZiReW^s`^T1bGsKYwx2plik4|k^_Y(+VyMR~qDD!a=0b+4 zI?0^kNsmX#LZG{;hgE+p}7v+`29UEjSd>K`u4pGd(#p}zK(mn@rB)?AAN{Y3koA2`Vov+bYY+2q}u{< za{u#|$kRY)ey^>Y4iN%k*50`eefqSjwy7sj4HRj=I8{hdAw_B&|lq z4o0{*ViBZLgfi0$*y^i`UHudD%WTt#*NjPvR*|q&*$1Ze>&y`t(@zcgu zxCQZCl+q>-fqpu1Ph}csjRr{rBy;ZnW=n+mB$4~YvP{RrJuAU0KfDp2?u)r~I@hAQ zUsoK4>l3ttR352#oWZpOdyE3G=Mq8bkXNv6-IwBE3;8k*FspV{z{L(b+eRp>K8h<| zNSP2bjKD-RSE%5j5T6tb0@jI`JBT?*WQjehI=_@dlAk5(fu2XV1>q`0Vk)b17J}x_ z)TwkG@@@Qd*j>RtCC_XXOtaN2yBzM69AiH*rxa@adKiTCpz-gfMx!I7mg#^l09Ji< zaimESDdQ0meeBhvxFdjSQIO?}8lWRm_;XM)_AX)M-N6Z-7$U*1&;!}9Vq@O#wJU_7 zwk^;&JYj-NS9-Yf=DNuO0}0OMJK#v2Fl*pI#;MAo%)y1U7;|GbJ~UYk&<8izIAu?I zmTYmqzg?2f3=6`{I>xloU9p5XO{Q9VNsaXz7Ra`1=+~w%uBb+bY9Kc#tH9ytzEXKz66e3EujFOh!DDJ&8_jJpI zRdFr!jhO-cX`AB##W{EZNo;P{G(8t6HjJj1>*V-gs4geqR5k>2x}Q8mMS8$+bGy06 z-X6{y9hJCOICfk}!yTe#4}YIpzMp-5W2WQNz^RmrZ15*`(r`R{l0DWo#(uiJt{*22 zd&2PPFxB;(<7<$L++=uZG6o+4}6HVQQOBHkqrMOsh`ks!q$QQwcI^swVds zf`dbBwDDhd(nP$+ltON7>10|8akHhTg&xBxty1u2qH9rPpHsAmY5o-AQMgNV3^t|_ z-~j3Qi!zc#N^9f3IE3E#b-b4T>{>Xa#T9l{6ulb5`=zj0;pb+!8peIBt+d6)1`F$? zwuCkArXM60ouQqltQTCxVu0O89gJ2wimrGiS#z&c0k`)CG``gvbXA0h)Pz<7Lvu{etu}qw zVn5Y3BaMUWD8}XRof|2{9tW;egJI(t_*`s5Sgy+^znY0kUk8(Z1+xUFa)|h^2ynGx zRiy?9l2@X^II*cHwWz(dWQl@=#pV))<#809t}zsQr6C`+#ABAV-5tw=aC1urY{5~Y zTO#Sj2nTNhc?i9Zqj4rOgu_yb0&+wa#;O-Xi}n<5aI;=o7^2lUp_>`BAVw+J-Nd{) zt@4@TO?V{Xw@oQMbsb*U@iAWt%|pJ5G8;L`)7OZgZ{af6J^Q=-fHnwLx3KxQll8hCEqE?8|D}w zd=OS5Dt|I0!bhls>)7xC0{hn?i6$Bt#qm3aC@g>S9@k-EaTKJuj3Ok0)mcsI^woek z50nQG0bcDAE|E2ehfiGk+<YxVQy29T?WyC^R_Vdyu6CdTZVw>n z?D6?rb8{xu_FTdp3{Z6Ti2KkSFcSU=FQu{dYqkhb>A!E91yJRrs62=hvP8IOP3v-% zN3yX6?9P-$61ptJetG*qzG&+khD#`@4`ZLPEj1=sL~1}!*etGGKJlcj-s$-&q%fCsd#Q&nR?+Ta?HwV=ptL~bS)VtMEb3gFMQZ+oT; zrkP@BfgrRBGX`PEw}2qRFj=0Cd-)h^w{hqtD2VGE%pkJ=`0sy?|N4tNcj)h@D-l@$ zVY+(;6I|;d44KvT#3A1jsEA%%>{Fx(Rkcx&z6;}HU9+Lx8sg+Z7~}zJ&(6!z3_;c& z^)Q3+K%TxUsyYW$GpI0=6nx_8*Az{ex-nBswsRRc*3)(fI(QppH_Grbc&w*MeXay# zJzc*t7F+l992o0qPe0!r9BbBI^=Qe}zC0f5=~aNNr)vd^yy5ra@R#!q1$m}C14Owu zCT-#O0Qu)&6OCYU$vjn-DyASl1VkSsz#`GS4_rhRRa2&e_k$=hv+$d8uw|Tx5Xuxr z_H^-B!I8@{*fvtl3qW$&fL+n@X#tQaYs?%N$J1+q$(Nw+he+k!8H5?4!+!XN@L-Z? z=&UN@Vxmq6vJ|>(DMmtaTfo8aOg4rFLnt_u22q%ad6DcvMU_}EKXFRM3>IvQXR+V{ zuC-`z1rDAqs`h3;xPl0qmVkt(P^%fobfWYQ0}9U!*(#De@NR%v28Q3?WQ|qWMF1bJ zE|L6-^V#lzp$#&9Zu?M)Sxd_{odJo7(w9;kAIRFz=h0(NuaEV5`ch`c8m?-@-_zFo zSdOFc-6F{AY2#MF{8^0ml7#yI(y&*6rwzZ5r?BF60X}_Uf2iOEfcnBt%Mof%(=o!8 zZ|~fknqC528~wUeg+j-X_fik*>j2E2W_-C0M7e~P7e3T0;4&ycX%V>RPqF1XLf)p+ z;s;R5c3%LNVI(($E3>pMhn0yAHdc2LpbTLo#96$L8UT~)gbA$%ld+f&!jf;b{LkUb zkd#W`a4j6%gFA+c(-nYl)ANc*_|1v)B0BErrC4^A4eQHb<5?IR*OIXQ;rKN)!WH0a zbM2T(4XJ7c=%Q+B69HBXT;FP5w{X`AUR?sZ!^;M56Hvbjlqqc04RIqm=^Me}J~ZD8OCp()k3p{-BrVLXgjb^o2`t!IEr< zIX-z`I>Zg}3c{x!njm3}z{>oHS%UyYhOgF@E2#Q`Mcg54to*T(DffG*`W@dVOZCyLXq`vzI_(F-FPjgE; z7oaFh(n>JShO$a9l1VDL)mrWaJ;(&Q!E&tF%xO@42)Y*#A|zH?&QY zu3((t%Vysoz(AQI>vUdBzwxw5WLT&7lKh(+5(e^Hs@VnnRTq?1GjeVNWH!|~vk9Lb!53xAiwIwl(ggnJ%%!4D+di&NWFqUm zP9D<|m55vqPU)CQH{t}4+B=!T*CiB+Qjm(IkXop+Bnl?J#s;MFB-`(X#Zd#r=_p(8yZOVm-tZ$Zc;` (we consider it -# preferable to use the standard :math:`F` and trust the reader to avoid -# confusing it with the interaction parameters.) -# The Fock basis is the simplest legitimate form for the basis and it consists -# of a series of 1s and 0s where 1 means occupied and -# 0 means empty. These are in order up, down, up, down, up, down. -basis = edrixs.get_fock_bin_by_N(norb, noccu) -print(np.array(basis)) -################################################################################ -# We expect the number of these states to be given by the mathematical -# combination of two electrons distributed among six states (three spin-orbitals -# with two spins per orbital). -message = ("We predict C(norb={}, noccu={})={:.0f} states and we got {:d}, " - "which is reassuring!") -print(message.format(norb, noccu, edrixs.combination(norb, noccu), len(basis))) -################################################################################ -# Note that in more complicated problems with both valence and core -# electrons, the edrixs convention is to list the valence electrons first. - -################################################################################ -# Transform interactions into Fock basis -# ------------------------------------------------------------------------------ -# edrixs works by initiailly creating a Hamiltonian matrix -# :math:`\hat{H}` in the single particle basis and then transforming into -# our chosen Fock basis. In the single particle basis, we have four fermion -# interactions with this form -# -# .. math:: -# \hat{H} = -# -# generated as -n_fermion = 4 -H = edrixs.build_opers(n_fermion, umat, basis) - -################################################################################ -# We needed to specify :code:`n_fermion = 4` because the -# :code:`edrixs.build_opers` function can also make two fermion terms. - -################################################################################ -# Diagonalize the matrix -# ------------------------------------------------------------------------------ -# For a small problem such as this it is convenient to use the native -# `scipy `_ diagonalization routine. This returns eigenvalues -# :code:`e` and eignvectors :code:`v` where eigenvalue :code:`e[i]` corresponds -# to eigenvector :code:`v[:,i]`. -e, v = scipy.linalg.eigh(H) -print("{} eignvalues and {} eigvenvectors {} elements long.".format(len(e), - v.shape[1], - v.shape[0])) - -################################################################################ -# Computing expectation values -# ------------------------------------------------------------------------------ -# To interpret the results, it is informative to compute the expectations values -# related to the spin :math:`\mathbf{S}`, orbital :math:`\mathbf{L}`, -# and total :math:`\mathbf{J}`, angular momentum. We first load the relevant -# matrices for these quantities for a `p` atomic shell. We need to specify -# that we would like to include spin when loading the orbital operator. -orb_mom = edrixs.get_orb_momentum(l, ispin=True) -spin_mom = edrixs.get_spin_momentum(l) -tot_mom = orb_mom + spin_mom - -################################################################################ -# We again transform these matrices to our Fock basis to build the operators -n_fermion = 2 -opL, opS, opJ = edrixs.build_opers(n_fermion, [orb_mom, spin_mom, tot_mom], - basis) - -################################################################################ -# Recall that quantum mechanics forbids us from knowing all three Cartesian -# components of angular momentum at once, so we want to compute the squares of -# these operators i.e. -# -# .. math:: -# \mathbf{S}^2 = S^2_x + S^2_y + S^2_z\\ -# \mathbf{L}^2 = L^2_x + L^2_y + L^2_z\\ -# \mathbf{J}^2 = J^2_x + J^2_y + J^2_z -# -L2 = np.dot(opL[0], opL[0]) + np.dot(opL[1], opL[1]) + np.dot(opL[2], opL[2]) -S2 = np.dot(opS[0], opS[0]) + np.dot(opS[1], opS[1]) + np.dot(opS[2], opS[2]) -J2 = np.dot(opJ[0], opJ[0]) + np.dot(opJ[1], opJ[1]) + np.dot(opJ[2], opJ[2]) - -################################################################################ -# Remember that the eigenvalues of :math:`\mathbf{S}^2` are in the form -# :math:`S(S+1)` etc. and that they can be obtained by calculating the -# projection of the operators onto our eigenvectors. -L2_val = edrixs.cb_op(L2, v).diagonal().real -S2_val = edrixs.cb_op(S2, v).diagonal().real -J2_val = edrixs.cb_op(J2, v).diagonal().real -################################################################################ -# We can determine the degeneracy of the eigenvalues numerically and print out -# the values as follows -e = np.round(e, decimals=6) -degeneracy = [sum(eval == e) for eval in e] -header = "{:<3s}\t{:>8s}\t{:>8s}\t{:>8s}\t{:>8s}" -print(header.format("# ", "E ", "S(S+1)", "L(L+1)", "Degen.")) -for i, eigenvalue in enumerate(e): - values_list = [i, eigenvalue, S2_val[i], L2_val[i], degeneracy[i]] - print("{:<3d}\t{:8.3f}\t{:8.3f}\t{:8.3f}\t{:>3d}".format(*values_list)) - -################################################################################ -# We see :math:`S=0` and :math:`S=1` states coming from the -# two combinations of the spin 1/2 particles. :math:`L` can take values of -# 0, 1, 2. Remember that spin states have degeneracy of :math:`2S+1` and the -# same is true for orbital states. -# We must multiply these :math:`S` and -# :math:`L` degeneracies to get the total degeneracy. -# Since these particles are fermions, the -# overall state must be antisymmetric, which dictates the allowed combinations -# of :math:`S` and :math:`L`. - -################################################################################ -# Energy level diagram -# ------------------------------------------------------------------------------ -# Let us show our findings graphically -fig, ax = plt.subplots() -for i, eigenvalue in enumerate(np.unique(e)): - art = ax.plot([0, 1], [eigenvalue, eigenvalue], '-', color='C{}'.format(i)) - ind = np.where(eigenvalue == e)[0][0] - L = (-1 + np.sqrt(1 + 4*L2_val[ind]))/2 - S = (-1 + np.sqrt(1 + 4*S2_val[ind]))/2 - message = "L={:.0f}, S={:.0f} ({:.0f})" - ax.text(1, eigenvalue, message.format(L, S, degeneracy[ind]), - horizontalalignment='right', - verticalalignment='bottom', - color='C{}'.format(i)) - -ax.set_ylabel('Energy') -for loc in ['right', 'top', 'bottom']: - ax.spines[loc].set_visible(False) - -ax.yaxis.set_ticks_position('left') -ax.set_xticks([]) -plt.show() - -################################################################################ -# We see Hund's rules in action! Rule 1 says that the highest spin :math:`S=1` -# state has the lowest energy. Of the two :math:`S=0` states, the state with -# larger :math:`L=1` is lower energy following rule 2. - -################################################################################ -# Spin orbit coupling -# ------------------------------------------------------------------------------ -# For fun, we can see how this changes when we add spin orbit coupling (SOC). -# This is a two-fermion operator that we create, transform into the Fock basis -# and add to the prior Hamiltonian. To make things easy, let us make the SOC -# small so that the LS coupling approximation is valid and we can -# still track the states. -soc = edrixs.atom_hsoc('p', 0.1) -n_fermion = 2 -H2 = H + edrixs.build_opers(n_fermion, soc, basis) - -################################################################################ -# Then, we redo the diagonalization and print the results. -e2, v2 = scipy.linalg.eigh(H2) -e2 = np.round(e2, decimals=6) -degeneracy2 = [sum(eval == e2) for eval in e2] -print() -message = "With SOC\n {:<3s}\t{:>8s}\t{:>8s}\t{:>8s}\t{:>8s}\t{:>8s}" -print(message.format("#", "E", "S(S+1)", "L(L+1)", "J(J+1)", "degen.")) -J2_val_soc = edrixs.cb_op(J2, v2).diagonal().real -L2_val_soc = edrixs.cb_op(L2, v2).diagonal().real -S2_val_soc = edrixs.cb_op(S2, v2).diagonal().real -for i, eigenvalue in enumerate(e2): - values_list = [i, eigenvalue, S2_val_soc[i], L2_val_soc[i], J2_val_soc[i], - degeneracy2[i]] - print("{:<3d}\t{:8.3f}\t{:8.3f}\t{:8.3f}\t{:8.3f}\t{:8.3f}".format(*values_list)) - -################################################################################ -# and we make an equivalent energy level diagram. - -fig, ax = plt.subplots() -for i, eigenvalue in enumerate(np.unique(e2)): - art = ax.plot([0, 1], [eigenvalue, eigenvalue], '-', color='C{}'.format(i)) - ind = np.where(eigenvalue == e2)[0][0] - J = (-1 + np.sqrt(1+4*J2_val_soc[ind]))/2 - message = "J={:.0f} ({:.0f})" - ax.text(1, eigenvalue, message.format(J, degeneracy2[ind]), - horizontalalignment='right', - verticalalignment='bottom', - color='C{}'.format(i)) - -ax.set_ylabel('Energy') -for loc in ['right', 'top', 'bottom']: - ax.spines[loc].set_visible(False) - -ax.yaxis.set_ticks_position('left') -ax.set_xticks([]) -plt.show() - -################################################################################ -# It is clear that we have split the :math:`S=1` state, which branches into -# three states from :math:`J=|L-S|, |L-S|+1, ..., |L+S|`. Since the shell is -# less than half full, Hund's third rule dictates that the smaller :math:`J` -# states have the lower energies. diff --git a/edrixs/_downloads/a3ecf627cca43801e771a480c3e554b3/example_7_Hunds_interactions.py b/edrixs/_downloads/a3ecf627cca43801e771a480c3e554b3/example_7_Hunds_interactions.py deleted file mode 100644 index f33384ac8c..0000000000 --- a/edrixs/_downloads/a3ecf627cca43801e771a480c3e554b3/example_7_Hunds_interactions.py +++ /dev/null @@ -1,290 +0,0 @@ -#!/usr/bin/env python -""" -Hund's Interactions in charge transfer insulators -================================================= -In this exercise we will solve a toy model relevant to cubic :math:`d^8` charge transfer insulators -such as NiO or NiPS\\ :sub:`3`. We are interested in better understanding the interplay between the -Hund's interactions and the charge transfer energy in terms of the energy of the triplet-singlet -excitations of this model. These seem to act against each other in that the Hund's interactions -impose a energy cost for the triplet-singlet excitations whenever there are two holes on -the Ni :math:`d` orbitals. The charge transfer physics, on the other hand, will promote a -:math:`d^9\\underline{L}` ground state in which the Hund's interactions are not active. - -The simplest model that captures this physics requires four Ni spin-orbitals, representing the Ni -:math:`e_g` manifold. We will represent the ligand states in the same way as the Anderson impurity -model in terms of one effective ligand spin-orbital per Ni spin-orbital. We assume these effective -orbitals have been constructed so that each Ni orbital only bonds to one sister orbital. For -simplicity, we will treat all Ni and all ligand orbitals as equivalent, even though a more -realistic model would account for the different Coulomb and hopping of the :math:`d_{3z^2-r^2}` -and :math:`d_{x^2-y^2}` orbitals. We therefore simply connect Ni and ligand orbitals via a constant -hopping :math:`t`. We also include the ligand energy parameter :math:`e_L`. - -The easiest way to implement the requried Coulomb interactions is to use the so-called Kanamori -Hamiltonian, which is a simplfied form for the interactions, which treats all orbitals as -equivalent. Daniel Khomskii's book provides a great explanation of this physics [1]_. We -parameterize the interactions via Coulomb repulsion parameter :math:`U` and Hund's exchange -:math:`J_H`. EDRIXS provides this functionality via the more general -:func:`.get_umat_kanamori` function. - -It's also easiest to consider this problem in hole langauge, which means our eight spin-orbitals -are populated by two fermions. -""" - -################################################################################ -# Setup -# ------------------------------------------------------------------------------ -# We start by loading the necessary modules, and defining the total number of -# orbitals and electrons. -import edrixs -import scipy -import numpy as np -import matplotlib.pyplot as plt - -norb = 8 -noccu = 2 - -################################################################################ -# Diagonalization -# ------------------------------------------------------------------------------ -# Let's write a function to diagonalize our model in a similar way to -# the :ref:`sphx_glr_auto_examples_example_6_Hubbard_dimer.py` example. -# Within this function, we also create operators to count the number of -# :math:`d` holes and operators to calculate expectation values for -# :math:`S^2` and :math:`S_z`. For the latter to make sense, we also include a -# small effective spin interaction along :math:`z`. - - -def diagonalize(U, JH, t, eL, n=1): - # Setup Coulomb matrix - umat = np.zeros((norb, norb, norb, norb), dtype=complex) - uNi = edrixs.get_umat_kanamori(norb//2, U, JH) - umat[:norb//2, :norb//2, :norb//2, :norb//2] = uNi - - # Setup hopping matrix - emat = np.zeros((norb, norb), dtype=complex) - ind = np.arange(norb//2) - emat[ind, ind + norb//2] = t - emat[ind+norb//2, ind] = np.conj(t) # conj is not needed, but is good practise. - ind = np.arange(norb//2, norb) - emat[ind, ind] += eL - - # Spin operator - spin_mom = np.zeros((3, norb, norb), dtype=complex) - spin_mom[:, :2, :2] = edrixs.get_spin_momentum(0) - spin_mom[:, 2:4, 2:4] = edrixs.get_spin_momentum(0) - spin_mom[:, 4:6, 4:6] = edrixs.get_spin_momentum(0) - spin_mom[:, 6:8, 6:8] = edrixs.get_spin_momentum(0) - - # add small effective field along z - emat += 1e-6*spin_mom[2] - - # Diagonalize - basis = edrixs.get_fock_bin_by_N(norb, noccu) - H = edrixs.build_opers(2, emat, basis) + edrixs.build_opers(4, umat, basis) - e, v = scipy.linalg.eigh(H) - e -= e[0] # Define ground state as zero energy - - # Operator for holes on Ni - basis = np.array(basis) - num_d_electrons = basis[:, :4].sum(1) - d0 = np.sum(np.abs(v[num_d_electrons == 0, :])**2, axis=0) - d1 = np.sum(np.abs(v[num_d_electrons == 1, :])**2, axis=0) - d2 = np.sum(np.abs(v[num_d_electrons == 2, :])**2, axis=0) - - # S^2 and Sz operators - opS = edrixs.build_opers(2, spin_mom, basis) - S_squared_op = np.dot(opS[0], opS[0]) + np.dot(opS[1], opS[1]) + np.dot(opS[2], opS[2]) - S_squared_exp = edrixs.cb_op(S_squared_op, v).diagonal().real - S_z_exp = edrixs.cb_op(opS[2], v).diagonal().real - - return e[:n], d0[:n], d1[:n], d2[:n], S_squared_exp[:n], S_z_exp[:n] - - -################################################################################ -# The atomic limit -# ------------------------------------------------------------------------------ -# For simplicity, let's start in the atomic limit with :math:`e_L \gg t \gg U` -# where all holes are on nickel. In this case, there are six ways to distribute -# two holes on the four Ni spin-orbitals. Let's examine the expectation values -# of the :math:`S^2` and :math:`S_z` operators. -U = 10 -JH = 2 -t = 100 -eL = 1e10 - -e, d0, d1, d2, S_squared_exp, S_z_exp = diagonalize(U, JH, t, eL, n=6) - -print("Ground state\nE\t") -for i in range(3): - print(f"{e[i]:.2f}\t{S_squared_exp[i]:.2f}\t{S_z_exp[i]:.2f}") - -print("\nExcited state\nE\t") -for i in range(3, 6): - print(f"{e[i]:.2f}\t{S_squared_exp[i]:.2f}\t{S_z_exp[i]:.2f}") - -################################################################################ -# The ground state is a high-spin triplet. The fourth and fifth -# states (the first excited state) are low-spin singlet excitons at -# :math:`2 J_H`. These have one hole on each orbital in the antisymmetric -# combination of :math:`|\uparrow\downarrow>-|\downarrow\uparrow>`. -# The state at :math:`3 J_H` also has one hole on each orbital in the symmetric -# :math:`|\uparrow\downarrow>+|\downarrow\uparrow>` configuration. - -################################################################################ -# Where are the holes for large hopping -# ------------------------------------------------------------------------------ -# As discussed at the start, we are interested to see interplay between Hund's -# and charge-transfer physics, which will obviously depend strongly on whether -# the holes are on Ni or the ligand. Let's see what happens as :math:`e_L` is -# reduced while observing the location of the ground state and exciton holes. -U = 10 -JH = 2 -t = 100 - -eLs = np.linspace(0, 1000, 30) - -fig, axs = plt.subplots(1, 2, figsize=(8, 4)) - -for ax, ind in zip(axs.ravel(), [0, 3]): - ds = np.array([diagonalize(U, JH, t, eL, n=6) - for eL in eLs]) - - ax.plot(eLs, ds[:, 1, ind], 'o', label='$d^0$') - ax.plot(eLs, ds[:, 2, ind], 's', label='$d^1$') - ax.plot(eLs, ds[:, 3, ind], '^', label='$d^2$') - ax.set_xlabel("Energy of ligands $e_L$") - ax.set_ylabel("Number of electrons") - ax.legend() - -axs[0].set_title("Location of ground state holes") -axs[1].set_title("Location of exciton holes") - -plt.tight_layout() -plt.show() -################################################################################ -# For large :math:`e_L`, we see that both holes are on nickel as expected. In -# the opposite limit of :math:`|e_L| \ll t` and :math:`U \ll t` the holes are -# shared in the ratio 0.25:0.5:0.25 as there are two ways to have one hole on -# Ni. In the limit of large :math:`e_L`, all holes move onto Ni. Since -# :math:`t` is large, this applies equally to both the ground state and the -# exciton. - - -################################################################################ -# Connecton between atomic and charge transfer limits -# ------------------------------------------------------------------------------ -# We now examine the quantum numbers during cross over between the two limits -# with :math:`e_L`. Let's first look at the how :math:`` changes for the -# ground state and exciton and then examine how the exciton energy changes. - -U = 10 -JH = 2 -t = 100 - -eLs = np.linspace(0, 1000, 30) - -info = np.array([diagonalize(U, JH, t, eL, n=6) - for eL in eLs]) - -fig, axs = plt.subplots(1, 2, figsize=(8, 4)) - - -axs[0].plot(eLs, info[:, 4, 0], label='Ground state') -axs[0].plot(eLs, info[:, 4, 3], label='Exciton') -axs[0].set_xlabel("Energy of ligands $e_L$") -axs[0].set_ylabel('$$') -axs[0].set_title('Quantum numbers') -axs[0].legend() - -axs[1].plot(eLs, info[:, 0, 3], '+', color='C0') -axs[1].set_xlabel("Energy of ligands $e_L$") -axs[1].set_ylabel('Exciton energy', color='C0') -axr = axs[1].twinx() -axr.plot(eLs, info[:, 3, 5], 'x', color='C1') -axr.set_ylabel('$d^2$ fraction', color='C1') - -for ax, color in zip([axs[1], axr], ['C0', 'C1']): - for tick in ax.get_yticklabels(): - tick.set_color(color) - -axs[1].set_ylim(0, 2*JH) -axr.set_ylim(0, 1) - -axs[1].set_title('Exciton energy vs. $d^2$ character') -plt.tight_layout() -plt.show() -############################################################################## -# In the left panel, we see that the two limits are adiabatically connected -# as they preseve the same quantum numbers. This is because there is always -# an appreciable double occupancy under conditions where the -# :math:`d^9\underline{L}` character is maximized and this continues to favor -# the high spin ground state. Other interactions such as strong tetragonal -# crystal field would be needed to overcome the Hund's interactions and break -# this paradigm. In the right panel, we see that the exciton energy simply -# scales with the double occupancy. Overall, even though -# Hund's interactions are irrelevant for the :math:`d^9\underline{L}` -# electronic configuration, whenever :math:`t` is appreciable there is a -# strong mixing with the :math:`d^8` component is always present, which -# dominates the energy of the exciton. - -################################################################################ -# Charge transfer excitons -# ------------------------------------------------------------------------------ -# Another limiting case of the model is where :math:`t` is smaller than the -# Coulomb interactions. This, however, tends to produce -# ground state and exciton configurations that correspond to those of distinct -# atomic models. Let's look at the :math:`e_L` dependence in this case. -U = 10 -JH = 2 -t = .5 -eL = 7 - -eLs = np.linspace(0, 20, 30) - -fig, axs = plt.subplots(1, 2, figsize=(8, 4)) - -for ax, ind in zip(axs.ravel(), [0, 3]): - ds = np.array([diagonalize(U, JH, t, eL, n=6) - for eL in eLs]) - - ax.plot(eLs, ds[:, 1, ind], 'o', label='$d^0$') - ax.plot(eLs, ds[:, 2, ind], 's', label='$d^1$') - ax.plot(eLs, ds[:, 3, ind], '^', label='$d^2$') - ax.set_xlabel("Energy of ligands $e_L$") - ax.set_ylabel("Number of electrons") - ax.legend() - -axs[0].axvline(x=eL, linestyle=':', color='k') -axs[1].axvline(x=eL, linestyle=':', color='k') - -axs[0].set_title("Location of ground state holes") -axs[1].set_title("Location of exciton holes") - -plt.tight_layout() -plt.show() -################################################################################ -# Around :math:`e_L = 7` the plot shows that the excition is primairly a -# :math:`d^2 \rightarrow d^1` transition or a -# :math:`d^8 \rightarrow d^{9}\underline{L}` transition in electron language. -# Let's examine the energy and quantum numbers. - -e, d0, d1, d2, S_squared_exp, S_z_exp = diagonalize(U, JH, t, eL, n=6) - -print("Ground state\nE\t") -for i in range(3): - print(f"{e[i]:.2f}\t{S_squared_exp[i]:.2f}\t{S_z_exp[i]:.2f}") - -print("\nExcited state\nE\t") -for i in range(3, 6): - print(f"{e[i]:.2f}\t{S_squared_exp[i]:.2f}\t{S_z_exp[i]:.2f}") - -################################################################################ -# We once again see the same quantum numbers, despite the differences in mixing -# in the ground state and exciton. - - -############################################################################## -# -# .. rubric:: Footnotes -# -# .. [1] D. 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z^#AM9tC|!+3sGqip1eLU!js!bjG*CY_{G!+T{@Q=p^M&<9HATO?1=ZIgq#W-F<~SUYGP?$E)UFVD%eiedxZ{%aQ+BnfKupZq6irRQp|<`B80% zI~6dX_P)+-E4<s9tI|S_H%JRPY1}+5r0MD*SLcKrZ;%sGj(UAg2#vIKfdjZop;B{c z8w$g!z2PYUe8MFtmBZj#wNN1K)1UPuQs+@#d#S#d&mNUHT7+zxsXtP+`fK}To$aN| zA89vfu(F!%*b|79t7B5gG7MZ(fC#;Pl->9t&-V@ z63DY2zH_Hrm<3cgS3!lLHucT+w2LJpvbtf*!sw5!1kJ=j!Fx zKl|vDkJtYZ2KwU+^z+k?^m|=iP%>MjvzSptz3JztA1{$$w2qF-FM9d)FMdf)Kf=Gi L\")\nfor i in range(3):\n print(f\"{e[i]:.2f}\\t{S_squared_exp[i]:.2f}\\t{S_z_exp[i]:.2f}\")\n\nprint(\"\\nExcited state\\nE\\t\")\nfor i in range(3, 6):\n print(f\"{e[i]:.2f}\\t{S_squared_exp[i]:.2f}\\t{S_z_exp[i]:.2f}\")" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The ground state is a high-spin triplet. The fourth and fifth\nstates (the first excited state) are low-spin singlet excitons at\n$2 J_H$. These have one hole on each orbital in the antisymmetric\ncombination of $|\\uparrow\\downarrow>-|\\downarrow\\uparrow>$.\nThe state at $3 J_H$ also has one hole on each orbital in the symmetric\n$|\\uparrow\\downarrow>+|\\downarrow\\uparrow>$ configuration.\n\n" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Where are the holes for large hopping\nAs discussed at the start, we are interested to see interplay between Hund's\nand charge-transfer physics, which will obviously depend strongly on whether\nthe holes are on Ni or the ligand. Let's see what happens as $e_L$ is\nreduced while observing the location of the ground state and exciton holes.\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "U = 10\nJH = 2\nt = 100\n\neLs = np.linspace(0, 1000, 30)\n\nfig, axs = plt.subplots(1, 2, figsize=(8, 4))\n\nfor ax, ind in zip(axs.ravel(), [0, 3]):\n ds = np.array([diagonalize(U, JH, t, eL, n=6)\n for eL in eLs])\n\n ax.plot(eLs, ds[:, 1, ind], 'o', label='$d^0$')\n ax.plot(eLs, ds[:, 2, ind], 's', label='$d^1$')\n ax.plot(eLs, ds[:, 3, ind], '^', label='$d^2$')\n ax.set_xlabel(\"Energy of ligands $e_L$\")\n ax.set_ylabel(\"Number of electrons\")\n ax.legend()\n\naxs[0].set_title(\"Location of ground state holes\")\naxs[1].set_title(\"Location of exciton holes\")\n\nplt.tight_layout()\nplt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "For large $e_L$, we see that both holes are on nickel as expected. In\nthe opposite limit of $|e_L| \\ll t$ and $U \\ll t$ the holes are\nshared in the ratio 0.25:0.5:0.25 as there are two ways to have one hole on\nNi. In the limit of large $e_L$, all holes move onto Ni. Since\n$t$ is large, this applies equally to both the ground state and the\nexciton.\n\n" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Connecton between atomic and charge transfer limits\nWe now examine the quantum numbers during cross over between the two limits\nwith $e_L$. Let's first look at the how $$ changes for the\nground state and exciton and then examine how the exciton energy changes.\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "U = 10\nJH = 2\nt = 100\n\neLs = np.linspace(0, 1000, 30)\n\ninfo = np.array([diagonalize(U, JH, t, eL, n=6)\n for eL in eLs])\n\nfig, axs = plt.subplots(1, 2, figsize=(8, 4))\n\n\naxs[0].plot(eLs, info[:, 4, 0], label='Ground state')\naxs[0].plot(eLs, info[:, 4, 3], label='Exciton')\naxs[0].set_xlabel(\"Energy of ligands $e_L$\")\naxs[0].set_ylabel('$$')\naxs[0].set_title('Quantum numbers')\naxs[0].legend()\n\naxs[1].plot(eLs, info[:, 0, 3], '+', color='C0')\naxs[1].set_xlabel(\"Energy of ligands $e_L$\")\naxs[1].set_ylabel('Exciton energy', color='C0')\naxr = axs[1].twinx()\naxr.plot(eLs, info[:, 3, 5], 'x', color='C1')\naxr.set_ylabel('$d^2$ fraction', color='C1')\n\nfor ax, color in zip([axs[1], axr], ['C0', 'C1']):\n for tick in ax.get_yticklabels():\n tick.set_color(color)\n\naxs[1].set_ylim(0, 2*JH)\naxr.set_ylim(0, 1)\n\naxs[1].set_title('Exciton energy vs. $d^2$ character')\nplt.tight_layout()\nplt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "In the left panel, we see that the two limits are adiabatically connected\nas they preseve the same quantum numbers. This is because there is always\nan appreciable double occupancy under conditions where the\n$d^9\\underline{L}$ character is maximized and this continues to favor\nthe high spin ground state. Other interactions such as strong tetragonal\ncrystal field would be needed to overcome the Hund's interactions and break\nthis paradigm. In the right panel, we see that the exciton energy simply\nscales with the double occupancy. Overall, even though\nHund's interactions are irrelevant for the $d^9\\underline{L}$\nelectronic configuration, whenever $t$ is appreciable there is a\nstrong mixing with the $d^8$ component is always present, which\ndominates the energy of the exciton.\n\n" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Charge transfer excitons\nAnother limiting case of the model is where $t$ is smaller than the\nCoulomb interactions. This, however, tends to produce\nground state and exciton configurations that correspond to those of distinct\natomic models. Let's look at the $e_L$ dependence in this case.\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "U = 10\nJH = 2\nt = .5\neL = 7\n\neLs = np.linspace(0, 20, 30)\n\nfig, axs = plt.subplots(1, 2, figsize=(8, 4))\n\nfor ax, ind in zip(axs.ravel(), [0, 3]):\n ds = np.array([diagonalize(U, JH, t, eL, n=6)\n for eL in eLs])\n\n ax.plot(eLs, ds[:, 1, ind], 'o', label='$d^0$')\n ax.plot(eLs, ds[:, 2, ind], 's', label='$d^1$')\n ax.plot(eLs, ds[:, 3, ind], '^', label='$d^2$')\n ax.set_xlabel(\"Energy of ligands $e_L$\")\n ax.set_ylabel(\"Number of electrons\")\n ax.legend()\n\naxs[0].axvline(x=eL, linestyle=':', color='k')\naxs[1].axvline(x=eL, linestyle=':', color='k')\n\naxs[0].set_title(\"Location of ground state holes\")\naxs[1].set_title(\"Location of exciton holes\")\n\nplt.tight_layout()\nplt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Around $e_L = 7$ the plot shows that the excition is primairly a\n$d^2 \\rightarrow d^1$ transition or a\n$d^8 \\rightarrow d^{9}\\underline{L}$ transition in electron language.\nLet's examine the energy and quantum numbers.\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "e, d0, d1, d2, S_squared_exp, S_z_exp = diagonalize(U, JH, t, eL, n=6)\n\nprint(\"Ground state\\nE\\t\")\nfor i in range(3):\n print(f\"{e[i]:.2f}\\t{S_squared_exp[i]:.2f}\\t{S_z_exp[i]:.2f}\")\n\nprint(\"\\nExcited state\\nE\\t\")\nfor i in range(3, 6):\n print(f\"{e[i]:.2f}\\t{S_squared_exp[i]:.2f}\\t{S_z_exp[i]:.2f}\")" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We once again see the same quantum numbers, despite the differences in mixing\nin the ground state and exciton.\n\n" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - ".. rubric:: Footnotes\n\n.. [1] D. Khomskii, Transition Metal Compounds, Cambridge University Press (2014)\n\n" - ] - } - ], - "metadata": { - "kernelspec": { - "display_name": "Python 3", - "language": "python", - "name": "python3" - }, - "language_info": { - "codemirror_mode": { - "name": "ipython", - "version": 3 - }, - "file_extension": ".py", - "mimetype": "text/x-python", - "name": "python", - "nbconvert_exporter": "python", - "pygments_lexer": "ipython3", - "version": "3.10.14" - } - }, - "nbformat": 4, - "nbformat_minor": 0 -} \ No newline at end of file diff --git a/edrixs/_downloads/b4dc84131d87174f7c09b58fc61829a8/example_7_transitions.ipynb b/edrixs/_downloads/b4dc84131d87174f7c09b58fc61829a8/example_7_transitions.ipynb deleted file mode 100644 index 7b0469a93a..0000000000 --- a/edrixs/_downloads/b4dc84131d87174f7c09b58fc61829a8/example_7_transitions.ipynb +++ /dev/null @@ -1,176 +0,0 @@ -{ - "cells": [ - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n# X-ray transitions\nThis example explains how to calculate x-ray transition amplitudes between\nspecific orbital and spin states. We take the case of a cuprate which has one\nhole in the $d_{x^2-y^2}$ orbital and a spin ordering direction along the\nin-plane diagaonal direction and compute the angular dependence of spin-flip\nand non-spin-flip processes.\n\nThis case was chosen because the eigenvectors in question are simple enough\nfor us to write them out more-or-less by hand, so this example helps the reader\nto understand what happens under the hood in more complex cases.\n\nSome of the code here is credited to Yao Shen who used this approach for the\nanalysis of a low valence nickelate material [1]_. The task performed repeats\nanalysis done by many researchers e.g. Luuk Ament et al [2]_ as well as\nseveral other groups.\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "import edrixs\nimport numpy as np\nimport matplotlib.pyplot as plt\nimport scipy" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Eigenvectors\nLet us start by determining the eigenvectors involved in the transitions.\nThe spin direction can be set using a vector\n$\\vec{B}$ to represent a magnetic field in terms of generalized spin\noperator $\\tilde{\\sigma}=\\vec{B}\\cdot\\sigma$ based on the Pauli matrices\n$\\sigma$. Let's put the spin along the $[1, 1, 0]$ direction\nand formuate the problem in the hole basis.\nFor one particle, we know that the Hamiltonian will be diagonal in the real\nharmonic basis.\nWe can generate the required eigenvectors by making a diagonal\nmatrix, transforming it to the required\ncomplex harmonic basis (as is standard for EDRIXS) and diagonalizing it.\nAs long as the crystal field splitting is much larger than the magnetic\nfield, the eigenvectors will be independent of the exact values of both\nthese parameters.\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "B = 1e-3*np.array([1, 1, 0])\ncf_splitting = 1e3\nzeeman = sum(s*b for s, b in zip(edrixs.get_spin_momentum(2), B))\ndd_levels = np.array([energy for dd_level in cf_splitting*np.arange(5)\n for energy in [dd_level]*2], dtype=complex)\nemat_rhb = np.diag(dd_levels)\nemat = edrixs.cb_op(emat_rhb, edrixs.tmat_r2c('d', ispin=True)) + zeeman\n_, eigenvectors = np.linalg.eigh(emat)\n\ndef get_eigenvector(orbital_index, spin_index):\n return eigenvectors[:, 2*orbital_index + spin_index]" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Let's examine the $d_{x^2-y^2}$ orbital first. Recall from the\n`sphx_glr_auto_examples_example_1_crystal_field.py`\nexample that edrixs uses the standard orbital order of\n$d_{3z^2-r^2}, d_{xz}, d_{yz}, d_{x^2-y^2}, d_{xy}$. So we want\n:code:`orbital_index = 3` element. Using this, we can build spin-up and -down\neigenvectors.\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "orbital_index = 3\n\ngroundstate_vector = get_eigenvector(orbital_index, 0)\nexcitedstate_vector = get_eigenvector(orbital_index, 1)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Transition operators and scattering matrix\nHere we are considering the $L_3$-edge. This means\na $2p_{3/2} \\rightarrow 3d$\nabsoprtion transition and a $2p_{3/2} \\rightarrow 3d$\nemission transition. We can read the relevant matrix from the edrixs database,\nkeeping in mind that there are in fact three operations for\n$x, y,$ & $z$ directions. Note that edrixs provides operators\nin electron notation. If we define $D$ as the transition operator in\nelectron language, $D^\\dagger$ is the operator in the hole language\nwe are using for this example.\nThe angular dependence of a RIXS transition can be conveniently described\nusing the scattering matrix, which is a $3\\times3$ element object that\nspecifies the transition amplitude for each incoming and outgoing x-ray\npolarization. Correspondingly, we have\n\n .. math::\n \\begin{equation}\n \\mathcal{F}=\\sum_n\\langle f|D|n\\rangle\\langle n|D^{\\dagger}|g\\rangle\n \\end{equation}.\n\nIn matrix form this is\n\n .. math::\n \\begin{equation}\n \\mathcal{F}(m,n)=\\{f^{\\dagger} \\cdot D(m)\\} \\cdot \\{D^{\\dagger}(n) \\cdot g\\}\n \\end{equation}.\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "D_Tmat = edrixs.get_trans_oper('dp32')\n\ndef get_F(vector_i, vector_f):\n F = np.zeros((3, 3), dtype=complex)\n for i in range(3):\n for j in range(3):\n F[i, j] = np.dot(np.dot(np.conj(vector_f.T), D_Tmat[i]),\n np.dot(np.conj(D_Tmat[j].T), vector_i))\n return F" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Using this function, we can obtain non-spin-flip (NSF) and spin-flip (SF)\nscattering matrices by choosing whether we return to the ground state or\nwhether we access the excited state with the spin flipped.\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "F_NSF = get_F(groundstate_vector, groundstate_vector)\nF_SF = get_F(groundstate_vector, excitedstate_vector)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Angular dependence\nLet's consider the common case of fixing the total scattering angle at\n:code:`two_theta = 90` and choosing a series of incident angles :code:`thins`.\nSince the detector does not resolve polarization, we need to add both outgoing\npolarizations. It is then convenient to use function :func:`.dipole_polvec_rixs`\nto obtain the incoming and outgoing polarization vectors.\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "thins = np.linspace(0, 90)\ntwo_theta = 90\nphi = 0\n\n\ndef get_I(thin, alpha, F):\n intensity = 0\n for beta in [0, np.pi/2]:\n thout = two_theta - thin\n ei, ef = edrixs.dipole_polvec_rixs(thin*np.pi/180, thout*np.pi/180,\n phi*np.pi/180, alpha, beta)\n intensity += np.abs(np.dot(ef, np.dot(F, ei)))**2\n return intensity" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Plot\nWe now run through a few configurations specified in terms of incoming\npolarization angle $\\alpha$ (defined in radians w.r.t. the scattering\nplane), $F$, plotting label, and plotting color.\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "fig, ax = plt.subplots()\n\nconfig = [[0, F_NSF, r'$\\pi$ NSF', 'C0'],\n [np.pi/2, F_NSF, r'$\\sigma$ NSF', 'C1'],\n [0, F_SF, r'$\\pi$ SF', 'C2'],\n [np.pi/2, F_SF, r'$\\sigma$ SF', 'C3']]\n\nfor alpha, F, label, color in config:\n Is = np.array([get_I(thin, alpha, F) for thin in thins])\n ax.plot(thins, Is, label=label, color=color)\n\nax.legend()\nax.set_xlabel(r'Theta ($^\\circ$)')\nax.set_ylabel('Relative intensity')\nplt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Run through orbitals\nFor completeness, let's look at transitions from $x^2-y^2$ to all other\norbitals.\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "fig, axs = plt.subplots(5, 1, figsize=(7, 7),\n sharex=True, sharey=True)\n\norbitals = ['$d_{3z^2-r^2}$', '$d_{xz}$', '$d_{yz}$',\n '$d_{x^2-y^2}$', '$d_{xy}$']\norbital_order = [4, 1, 2, 0, 3]\n\nplot_index = 0\nfor ax, orbital_index in zip(axs, orbital_order):\n for spin_index, spin_label in zip([0, 1], ['NSF', 'SF']):\n excitedstate_vector = get_eigenvector(orbital_index, spin_index)\n F = get_F(groundstate_vector, excitedstate_vector)\n for alpha, pol_label in zip([0, np.pi/2], [r'$\\pi$', r'$\\sigma$']):\n Is = np.array([get_I(thin, alpha, F) for thin in thins])\n ax.plot(thins, Is*10, label=f'{pol_label} {spin_label}',\n color=f'C{plot_index%4}')\n plot_index += 1\n ax.legend(title=orbitals[orbital_index], bbox_to_anchor=(1.1, 1),\n loc=\"upper left\", fontsize=8)\n\n\naxs[-1].set_xlabel(r'$\\theta$ ($^\\circ$)')\naxs[2].set_ylabel('Scattering intensity')\n\nfig.subplots_adjust(hspace=0, left=.3, right=.6)\nplt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - ".. rubric:: Footnotes\n\n.. [1] Yao Shen et al.,\n [arXiv:2110.08937 (2022)](https://arxiv.org/abs/2110.08937).\n.. [2] Luuk J. P. Ament et al.,\n [Phys. Rev. Lett. 103, 117003 (2009)](https://doi.org/10.1103/PhysRevLett.103.117003)\n\n" - ] - } - ], - "metadata": { - "kernelspec": { - "display_name": "Python 3", - "language": "python", - "name": "python3" - }, - "language_info": { - "codemirror_mode": { - "name": "ipython", - "version": 3 - }, - "file_extension": ".py", - "mimetype": "text/x-python", - "name": "python", - "nbconvert_exporter": "python", - "pygments_lexer": "ipython3", - "version": "3.10.16" - } - }, - "nbformat": 4, - "nbformat_minor": 0 -} \ No newline at end of file diff --git a/edrixs/_downloads/b91e28daa42d542643c2e6a87c5adde5/example_5_charge_transfer.py b/edrixs/_downloads/b91e28daa42d542643c2e6a87c5adde5/example_5_charge_transfer.py deleted file mode 100644 index ca25920658..0000000000 --- a/edrixs/_downloads/b91e28daa42d542643c2e6a87c5adde5/example_5_charge_transfer.py +++ /dev/null @@ -1,140 +0,0 @@ -#!/usr/bin/env python -""" -Charge-transfer energy for NiO -================================================================================ -This example follows the :ref:`sphx_glr_auto_examples_example_3_AIM_XAS.py` -example and considers the same model. This time we outline how to determine -the charge transfer energy in the sense defined by Zaanen, Sawatzky, and Allen -[1]_. That is, a :math:`d^{n_d} \\rightarrow d^{n_d + 1} \\underline{L}` transition -in the atomic limit, after considering Coulomb interactions and crystal field. Although -this can be determined analytically in some cases, the easiest way is often just to -calculate it, as we will do here. -""" -import edrixs -import numpy as np -import matplotlib.pyplot as plt -import scipy -import example_3_AIM_XAS -import importlib -_ = importlib.reload(example_3_AIM_XAS) - -################################################################################ -# Determine eigenvectors and occupations -# ------------------------------------------------------------------------------ -# The first step repeats what was done in -# :ref:`sphx_glr_auto_examples_example_4_GS_analysis.py` but it does not apply -# the hybridization between the impurity and both states. - -from example_3_AIM_XAS import (F0_dd, F2_dd, F4_dd, - nd, norb_d, norb_bath, v_noccu, - imp_mat, bath_level, - hyb, ext_B, trans_c2n) -ntot = 20 -umat_delectrons = edrixs.get_umat_slater('d', F0_dd, F2_dd, F4_dd) -umat = np.zeros((ntot, ntot, ntot, ntot), dtype=complex) -umat[:norb_d, :norb_d, :norb_d, :norb_d] += umat_delectrons -emat_rhb = np.zeros((ntot, ntot), dtype='complex') -emat_rhb[0:norb_d, 0:norb_d] += imp_mat -indx = np.arange(norb_d, norb_d*2) -emat_rhb[indx, indx] += bath_level[0] -tmat = np.eye(ntot, dtype=complex) -for i in range(2): - off = i * norb_d - tmat[off:off+norb_d, off:off+norb_d] = np.conj(np.transpose(trans_c2n)) - -emat_chb = edrixs.cb_op(emat_rhb, tmat) -v_orbl = 2 -sx = edrixs.get_sx(v_orbl) -sy = edrixs.get_sy(v_orbl) -sz = edrixs.get_sz(v_orbl) -zeeman = ext_B[0] * (2 * sx) + ext_B[1] * (2 * sy) + ext_B[2] * (2 * sz) -emat_chb[0:norb_d, 0:norb_d] += zeeman -basis = np.array(edrixs.get_fock_bin_by_N(ntot, v_noccu)) -H = (edrixs.build_opers(2, emat_chb, basis) - + edrixs.build_opers(4, umat, basis)) - -e, v = scipy.linalg.eigh(H) -e -= e[0] - -num_d_electrons = basis[:, :norb_d].sum(1) -alphas = np.sum(np.abs(v[num_d_electrons == 8, :])**2, axis=0) -betas = np.sum(np.abs(v[num_d_electrons == 9, :])**2, axis=0) - -################################################################################ -# Energy to lowest energy ligand orbital -# ------------------------------------------------------------------------------ -# Let's vizualize :math:`\alpha` and :math:`\beta`. - -fig, ax = plt.subplots() - -ax.plot(e, alphas, '.-', label=r'$\alpha$ $d^8L^{10}$') -ax.plot(e, betas, '.-', label=r'$\beta$ $d^9L^{9}$') - -ax.set_xlabel('Energy (eV)') -ax.set_ylabel('Population') -ax.set_title('NiO') -ax.legend() -plt.show() - -################################################################################ -# One can see that the mixing between impurity and bath states has disappered -# because we have turned off the hybridization. The energy required to - -GS_energy = min(e[np.isclose(alphas, 1)]) -lowest_energy_to_transfer_electron = min(e[np.isclose(betas, 1)]) -E_to_ligand = lowest_energy_to_transfer_electron - GS_energy -print(f"Energy to lowest energy ligand state is {E_to_ligand:.3f} eV") - -################################################################################ -# where we have used :code:`np.isclose` to avoid errors from finite numerical -# precision. - -################################################################################ -# Diagonalizing by blocks -# ------------------------------------------------------------------------------ -# When working on a problem with a large basis, one can take advantage of the -# lack of hybridization and separately diagonalize the impurity and bath -# states - -energies = [] - -for n_ligand_holes in [0, 1]: - basis_d = edrixs.get_fock_bin_by_N(10, nd + n_ligand_holes) - Hd = (edrixs.build_opers(2, emat_chb[:10, :10], basis_d) - + edrixs.build_opers(4, umat[:10, :10, :10, :10], basis_d)) - ed = scipy.linalg.eigh(Hd, eigvals_only=True, subset_by_index=[0, 0])[0] - - basis_L = edrixs.get_fock_bin_by_N(10, 10 - n_ligand_holes) - HL = (edrixs.build_opers(2, emat_chb[10:, 10:], basis_L) - + edrixs.build_opers(4, umat[10:, 10:, 10:, 10:], basis_L)) - eL = scipy.linalg.eigh(HL, eigvals_only=True, subset_by_index=[0, 0])[0] - - energies.append(ed + eL) - -print(f"Energy to lowest energy ligand state is {energies[1] - energies[0]:.3f} eV") - -################################################################################ -# which yields the same result. - -################################################################################ -# Energy splitting in ligand states -# ------------------------------------------------------------------------------ -# The last thing to consider is that our definition of the charge transfer -# energy refers to the atomic limit with all hopping terms switched off, whereas -# the ligand states in the model are already split by the oxygen-oxygen hopping -# term :math:`T_{pp}` as illustrated below. So the final charge transer energy -# needs to account for this. -# -# .. image:: /_static/energy_level.png -# - -T_pp = 1 -print(f"Charge transfer is {energies[1] - energies[0] + T_pp:.3f} eV") - - -############################################################################## -# -# .. rubric:: Footnotes -# -# .. [1] J. Zaanen, G. A. Sawatzky, and J. W. Allen, -# `Phys. Rev. Lett. 55, 418 (1985) `_. diff --git a/edrixs/_downloads/c80f6188449857a8752ca41e52553006/example_4_GS_analysis.ipynb b/edrixs/_downloads/c80f6188449857a8752ca41e52553006/example_4_GS_analysis.ipynb deleted file mode 100644 index 572f8e7e79..0000000000 --- a/edrixs/_downloads/c80f6188449857a8752ca41e52553006/example_4_GS_analysis.ipynb +++ /dev/null @@ -1,255 +0,0 @@ -{ - "cells": [ - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n# Ground state analysis for NiO\nThis example follows the `sphx_glr_auto_examples_example_3_AIM_XAS.py`\nexample and considers the same model. This time we show how to analyze\nthe wavevectors in terms of a\n$\\alpha |d^8L^{10}> + \\beta |d^9L^9> \\gamma |d^{10}L^8>$\nrepresentation.\n\nIn doing this we will go through the exercise of building and diagonalizing\nthe Hamiltonian in a way that hopefully clarifies how to analyze other\nproperties of the model.\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "import edrixs\nimport numpy as np\nimport matplotlib.pyplot as plt\nimport scipy\nimport example_3_AIM_XAS\nimport importlib\n_ = importlib.reload(example_3_AIM_XAS)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Hamiltonian\nedrixs builds model Hamiltonians based on two fermion and four fermion terms.\nThe four fermion terms come from Coulomb interactions and will be\nassigned to :code:`umat`. All other interactions contribute to two fermion\nterms in :code:`emat`.\n\n .. math::\n \\begin{equation}\n \\hat{H}_{i} = \\sum_{\\alpha,\\beta} t_{\\alpha,\\beta}\n \\hat{f}^{\\dagger}_{\\alpha} \\hat{f}_{\\beta}\n + \\sum_{\\alpha,\\beta,\\gamma,\\delta} U_{\\alpha,\\beta,\\gamma,\\delta}\n \\hat{f}^{\\dagger}_{\\alpha}\\hat{f}^{\\dagger}_{\\beta}\n \\hat{f}_{\\gamma}\\hat{f}_{\\delta},\n \\end{equation}\n\n\n" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Import parameters\nLet's get the parammeters we need from the\n`sphx_glr_auto_examples_example_3_AIM_XAS.py` example. We need to\nconsider :code:`ntot=20` spin-orbitals\nfor this problem.\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "from example_3_AIM_XAS import (F0_dd, F2_dd, F4_dd,\n nd, norb_d, norb_bath, v_noccu,\n imp_mat, bath_level,\n hyb, ext_B, trans_c2n)\nntot = 20" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Four fermion matrix\nThe Coulomb interactions in the $d$ shell of this problem are described\nby a $10\\times10\\times10\\times10$ matrix. We\nneed to specify a $20\\times20\\times20\\times 20$ matrix since we need to\ninclude the full problem with :code:`ntot=20` spin-orbitals. The edrixs\nconvention is to put the $d$ orbitals first, so we assign them to the\nfirst $10\\times10\\times10\\times 10$ indices of the matrix. edrixs\ncreates this matrix in the complex harmmonic basis by default.\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "umat_delectrons = edrixs.get_umat_slater('d', F0_dd, F2_dd, F4_dd)\numat = np.zeros((ntot, ntot, ntot, ntot), dtype=complex)\numat[:norb_d, :norb_d, :norb_d, :norb_d] += umat_delectrons" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Two fermion matrix\nPreviously we made a $10\\times10$ two-fermion matrix describing the\n$d$-shell interactions. Keep in mind we did this in the real harmonic\nbasis. We need to specify the two-fermion matrix for\nthe full problem $20\\times20$ spin-orbitals in size.\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "emat_rhb = np.zeros((ntot, ntot), dtype='complex')\nemat_rhb[0:norb_d, 0:norb_d] += imp_mat" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The :code:`bath_level` energies need to be applied to the diagonal of the\nlast $10\\times10$ region of the matrix.\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "indx = np.arange(norb_d, norb_d*2)\nemat_rhb[indx, indx] += bath_level[0]" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The :code:`hyb` terms mix the impurity and bath states and are therefore\napplied to the off-diagonal terms of :code:`emat`.\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "indx1 = np.arange(norb_d)\nindx2 = np.arange(norb_d, norb_d*2)\nemat_rhb[indx1, indx2] += hyb[0]\nemat_rhb[indx2, indx1] += np.conj(hyb[0])" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We now need to transform into the complex harmonic basis. We assign\nthe two diagonal blocks of a $20\\times20$ matrix to the\nconjugate transpose of the transition matrix.\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "tmat = np.eye(ntot, dtype=complex)\nfor i in range(2):\n off = i * norb_d\n tmat[off:off+norb_d, off:off+norb_d] = np.conj(np.transpose(trans_c2n))\n\nemat_chb = edrixs.cb_op(emat_rhb, tmat)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The spin exchange is built from the spin operators and the effective field\nis applied to the $d$-shell region of the matrix.\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "v_orbl = 2\nsx = edrixs.get_sx(v_orbl)\nsy = edrixs.get_sy(v_orbl)\nsz = edrixs.get_sz(v_orbl)\nzeeman = ext_B[0] * (2 * sx) + ext_B[1] * (2 * sy) + ext_B[2] * (2 * sz)\nemat_chb[0:norb_d, 0:norb_d] += zeeman" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Build the Fock basis and Hamiltonain and Diagonalize\nWe create the fock basis and build the Hamiltonian using the full set of\n$20$ spin orbitals, specifying that they are occuplied by $18$\nelectrons. See the `sphx_glr_auto_examples_example_0_ed_calculator.py`\nexample for more details if needed. We also set the ground state to zero\nenergy.\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "basis = np.array(edrixs.get_fock_bin_by_N(ntot, v_noccu))\nH = (edrixs.build_opers(2, emat_chb, basis)\n + edrixs.build_opers(4, umat, basis))\n\ne, v = scipy.linalg.eigh(H)\ne -= e[0]" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## State analysis\nNow we have all the eigenvectors in the Fock basis, we need to pick out the\nstates that have 8, 9 and 10 $d$-electrons, respectively.\nThe modulus squared of these coeffcients need to be added up to get\n$\\alpha$, $\\beta$, and $\\gamma$.\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "num_d_electrons = basis[:, :norb_d].sum(1)\n\nalphas = np.sum(np.abs(v[num_d_electrons==8, :])**2, axis=0)\nbetas = np.sum(np.abs(v[num_d_electrons==9, :])**2, axis=0)\ngammas = np.sum(np.abs(v[num_d_electrons==10, :])**2, axis=0)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The ground state is the first entry.\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "message = \"Ground state\\nalpha={:.3f}\\tbeta={:.3f}\\tgamma={:.3f}\"\nprint(message.format(alphas[0], betas[0], gammas[0]))" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Plot\nLet's look how $\\alpha$, $\\beta$, and $\\gamma$ vary with\nenergy.\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "fig, ax = plt.subplots()\n\nax.plot(e, alphas, label=r'$\\alpha$ $d^8L^{10}$')\nax.plot(e, betas, label=r'$\\beta$ $d^9L^{9}$')\nax.plot(e, gammas, label=r'$\\gamma$ $d^{10}L^{8}$')\n\nax.set_xlabel('Energy (eV)')\nax.set_ylabel('Population')\nax.set_title('NiO')\nax.legend()\nplt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We see that the ligand states are mixed into the ground state, but the\nmajority of the weight for the $L^9$ and $L^8$ states\nlive at $\\Delta$ and $2\\Delta$. With a lot of additional\nstructure from the other interactions.\n\n" - ] - } - ], - "metadata": { - "kernelspec": { - "display_name": "Python 3", - "language": "python", - "name": "python3" - }, - "language_info": { - "codemirror_mode": { - "name": "ipython", - "version": 3 - }, - "file_extension": ".py", - "mimetype": "text/x-python", - "name": "python", - "nbconvert_exporter": "python", - "pygments_lexer": "ipython3", - "version": "3.10.16" - } - }, - "nbformat": 4, - "nbformat_minor": 0 -} \ No newline at end of file diff --git a/edrixs/_downloads/ceccd240871166cf69609184775c524d/example_6_Hubbard_dimer.ipynb b/edrixs/_downloads/ceccd240871166cf69609184775c524d/example_6_Hubbard_dimer.ipynb deleted file mode 100644 index 01185a72cb..0000000000 --- a/edrixs/_downloads/ceccd240871166cf69609184775c524d/example_6_Hubbard_dimer.ipynb +++ /dev/null @@ -1,179 +0,0 @@ -{ - "cells": [ - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n# Hubbard Dimer\nThis exercise will demonstrate how to handle hopping and multi-site problems within\nedrixs using the example of a Hubbard dimer. We want to solve the equation\n\n .. math::\n \\begin{equation}\n \\hat{H} = \\sum_{i,j} \\sum_{\\sigma} t_{i,j} \\hat{f}^{\\dagger}_{i,\\sigma} \\hat{f}_{j, \\sigma}\n + U \\sum_{i} \\hat{n}_{i,\\uparrow}\\hat{n}_{i,\\downarrow},\n \\end{equation}\n\nwhich involves two sites labeled with indices $i$ or $j$ with two\nelectrons of spin $\\sigma\\in{\\uparrow,\\downarrow}$. $t_{i,j}$\nis the hopping between sites, $\\hat{f}^{\\dagger}_{i,\\sigma}$ is the\ncreation operators, and\n$\\hat{n}^{\\dagger}_{i,\\sigma}=\\hat{f}^{\\dagger}_{i,\\sigma}\\hat{f}_{i,\\sigma}$\nis the number operator. The main task is to represent this Hamiltonian and\nthe related spin operator using the EDRIXS two-fermion and four-fermion form\nwhere $\\alpha,\\beta,\\delta,\\gamma$ are the indices of the single\nparticle basis.\n\n .. math::\n \\begin{equation}\n \\hat{H} = \\sum_{\\alpha,\\beta} t_{\\alpha,\\beta} \\hat{f}^{\\dagger}_{\\alpha} \\hat{f}_{\\beta}\n + \\sum_{\\alpha,\\beta,\\gamma,\\delta} U_{\\alpha,\\beta,\\gamma,\\delta}\n \\hat{f}^{\\dagger}_{\\alpha}\\hat{f}^{\\dagger}_{\\beta}\\hat{f}_{\\gamma}\\hat{f}_{\\delta}.\n \\end{equation}\n" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Initialize matrices\nWe start by noting that each of the two sites is like an $l=0$\n$s$-orbital with two spin-orbitals each. We will include\ntwo electron occupation and build the Fock basis.\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "import numpy as np\nimport matplotlib.pyplot as plt\nimport scipy\nimport edrixs\nnp.set_printoptions(precision=4)\n\n\nll = 0\ncase = 's'\nnorb = 4\nnoccu = 2\nbasis = edrixs.get_fock_bin_by_N(norb, noccu)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Create function to populate and diagonalize matrices\nThe Coulomb and hopping matrices :code:`umat` and :code:`emat` will be\nrepresented by $4\\times4\\times4\\times4$ and $4\\times4$ matrices,\nrespectively. Note that we needed to specify\nthat these are, in general, complex, although\nthey happen to contain only real numbers in this case. We follow the convention\nthat these are ordered first by site and then by spin:\n$|0\\uparrow>, |0\\downarrow>, |1\\uparrow>, |1\\downarrow>$.\nConsequently the $2\\times2$ and $2\\times2\\times2\\times2$ block\ndiagonal structures of the matrices will contain the on-site interactions.\nThe converse is true for the hopping between the sites.\nFrom here let us generate a function to build and diagonalize the Hamiltonian.\nWe need to generate the Coulomb matrix for the on-site interactions and\napply it to the block diagonal. The hopping connects off-site indices with\nthe same spin.\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "def diagonalize(U, t, extra_emat=None):\n \"\"\"Diagonalize 2 site Hubbard Hamiltonian\"\"\"\n umat = np.zeros((norb, norb, norb, norb), dtype=np.complex128)\n emat = np.zeros((norb, norb), dtype=np.complex128)\n U_mat_1site = edrixs.get_umat_slater('s', U)\n umat[:2, :2, :2, :2,] = umat[2:, 2:, 2:, 2:] = U_mat_1site\n emat[2, 0] = emat[3, 1] = emat[0, 2] = emat[1, 3] = t\n\n if extra_emat is not None:\n emat = emat + extra_emat\n\n H = (edrixs.build_opers(2, emat, basis)\n + edrixs.build_opers(4, umat, basis))\n\n e, v = scipy.linalg.eigh(H)\n return e, v" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## The large $U$ limit\nLet us see what happens with $U \\gg t$.\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "e, v = diagonalize(1000, 1)\nprint(\"Energies are\")\nprint(e)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "To analyze what is going on we can determine the spin expectation values\nof the cluster. Building the operators follows the same form as the\nHamiltonian and the previous example.\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "spin_mom_one_site = edrixs.get_spin_momentum(ll)\nspin_mom = np.zeros((3, norb, norb), dtype=np.complex128)\nspin_mom[:, :2, :2] = spin_mom[:, 2:, 2:] = spin_mom_one_site\n\nopS = edrixs.build_opers(2, spin_mom, basis)\nopS_squared = (np.dot(opS[0], opS[0]) + np.dot(opS[1], opS[1])\n + np.dot(opS[2], opS[2]))" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "This time let us include a tiny magnetic field along the $z$-axis, so\nthat we have a well-defined measurement axis and print out the expectation\nvalues.\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "zeeman = np.zeros((norb, norb), dtype=np.complex128)\nzeeman[:2, :2] = zeeman[2:, 2:] = 1e-8*spin_mom_one_site[2]\ne, v = diagonalize(1000, 1, extra_emat=zeeman)\n\nSsq_exp = edrixs.cb_op(opS_squared, v).diagonal().real\nSz_exp = edrixs.cb_op(opS[2], v).diagonal().real\n\nheader = \"{:<10s}\\t{:<6s}\\t{:<6s}\"\nprint(header.format(\"E\", \"S(S+1)\", \"\"))\nfor i in range(len(e)):\n print(\"{:<2f}\\t{:.1f}\\t{:.1f}\".format(e[i], Ssq_exp[i], Sz_exp[i]))" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "For $U \\gg t$ the two states with double occupancy acquire an energy of\napproximately $U$. The low energy states are a $S=0$ singlet and\nand $S=1$ triplet, which are split by $4t^2/U$, which is the\nmagnetic exchange term.\n\n" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## $U$ dependence\nLet us plot the changes in energy with $U$.\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "plt.figure()\n\nt = 1\nUs = np.linspace(0.01, 10, 50)\nEs = np.array([diagonalize(U, t, extra_emat=zeeman)[0] for U in Us])\n\nplt.plot(Us/t, Es/t)\nplt.xlabel('U/t')\nplt.ylabel('Eigenstate energies/t')\nplt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "To help interpret this, we can represent the eigenvectors in terms of a sum\nof the single particle states.\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "def get_single_particle_repesentations(v):\n reps = []\n for i in range(6):\n rep = sum([vec*weight for weight, vec\n in zip(v[:, i], np.array(basis))])\n reps.append(rep)\n\n return np.array(reps)\n\nt = 1\nfor U in [10000, 0.0001]:\n e, v = diagonalize(U, t, extra_emat=zeeman)\n repesentations = get_single_particle_repesentations(v)\n print(\"For U={} t={} states are\".format(U, t))\n print(repesentations.round(3).real)\n print(\"\\n\")" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "For $U \\gg t$ the ground state maximizes its magnetic exchange\nenergy saving. In the $U \\ll t$ condition the ground state maximizes\nits kinetic energy saving. Since both states share the same parity, the\ncross-over between them is smooth. This type of physics is at play in current\nresearch on quantum materials [1]_ [2]_.\n\n" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - ".. rubric:: Footnotes\n\n.. [1] Y. Wang et al., [Phys. Rev. Lett. 122, 106401 (2019)](https://www.doi.org/10.1103/PhysRevLett.122.106401).\n.. [2] A. Revelli et al., [Science Advances 5, eaav4020 (2019)](https://doi.org/10.1126/sciadv.aav4020).\n\n" - ] - } - ], - "metadata": { - "kernelspec": { - "display_name": "Python 3", - "language": "python", - "name": "python3" - }, - "language_info": { - "codemirror_mode": { - "name": "ipython", - "version": 3 - }, - "file_extension": ".py", - "mimetype": "text/x-python", - "name": "python", - "nbconvert_exporter": "python", - "pygments_lexer": "ipython3", - "version": "3.10.16" - } - }, - "nbformat": 4, - "nbformat_minor": 0 -} \ No newline at end of file diff --git a/edrixs/_downloads/d7c6df44b25b2a3a0133d7014a6a992c/example_8_Hunds_interactions.zip b/edrixs/_downloads/d7c6df44b25b2a3a0133d7014a6a992c/example_8_Hunds_interactions.zip deleted file mode 100644 index 8d037446b3950ca5c38305424fcf86f73a0a648c..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 26567 zcmeHQU2_{pc6Gd)JmA^y=|HFmv`B~qC9RBFs>&tVE7^*ygjP8f4h0T?0XWfM20Jq# z3CVic{ExitTYgafO)BTy?w$`|KmsJ_SSiyM0nAK)-M)SA>D%{!ufPA@@4xtrKOaBe z`nTV=|1o{I``Ksw_xHHwUAW^(;B_B&pHIU<()GjCi`{^4@A=_< zFFaF|S$Z0UwR*i?b>oqz(%21?p%>#)G7a1`ij&&Gx<9oSp-NBv zM0pop-1ieto#TH$2viaUXP$CZ8qL%=8hC+n#iT4emwOcxWx0u6b)upxg!^|k}A)|$RqTRvxG)m2oFOr6I<+T z%^y!9)Z!}BjedmkLx_t7<~oDf$~$v@iV|mNfz4=1rst75jRFrt3u`R@%FpKN1XIw% zNPtxsCf1-1WlLYS1g+cHQfaPSn^9{b3XgL=i>zvl7zfz#mbQWH5=I zM|95}W1&C^q1PPO*P^G88M-}<{d88-i>9b#QRpdeIP_$g%2r$GteRjN=LVpcN0Lm@ z9vXs*WL4Dc7}cqZwc6ua=)gxBQHM!Cce8?(B9qK9WN8_Hfcp6P0v5{T0RtLKu_TRs4@5FE0 zA5UuRS$^XJ*Jr$L#|O=a$$>Cq4-3qw+Ki$LF*kN-^TRsxUG!h%3nN&w{V?sM#*PIL zx*zs~=|Ebr!ZC|_;>OS$irN(Ox-UMx8+Ix)8Wud+v_?AdN!b4KbOr7*WR6i)_exA&7BiZi9|L8HbiZ?vcf0;!SvIZxJ&#k+VBRkf05WvN#CE$ z><82`5&Ira>?KU{B#6?$?*X{+9nJqFNNcqa?WhCw7;pOhDc-efw;}XrzB`ISDCK)u zKDQzMmmU@N96pR_#40ik%phyr6J^XioEQ>6aARY<&@HZ(4(w~^B$=FEbVotlb*E|6 zHDEZgZ@Q0wihDgb9&`u(*oz_86Br0V%q=Q|j}=1yfn{R*BK}BLRG;<)oq^vJ0L)%$ zC&KoNYGB>Li2+7*5#?fT(3d4d^8_u?<_S)0D##*6UrZf`Z`<0c*w5Yfup)4oGH9$q z+IMth?7o90dtt(L3|SUkgNy2tF^y^F&1F@?1faCu9H2x5M`v0=8?{;u%3hp?=9_Kx z-ScflH~Vs1g$H|EotnarS)Q4RgyKTa2>YhjfU1eud>4|;dUN#6k$lH)14Eet|(=AGzZ|RB& zWkb&!c&M>AO?hb)MFUuQ&T9gI%NbhIB|Q-p9XnR{Ao!P=Yyg5X1(phldb{Ijtovo= zcE32W)v~eWjyl__gQ@Gt$hBRRDYg#)i%iGOUEMc(k)_(52l9&|S*P5C&ZBLGU+b28 z)OozE@aw8_MvdG7pzz#U1Lh&(lm+%an`DZ3&)a$QWj4C@u@!i3c|Cc)2WOs?2Rw}W z@4DE>cYCw$t87`p9cYm~&!6s1{b10g{Yjb-ITZ}L!Wy@rfJTPJhah0CTl@2WHHgwCs=>5j3ofrz2}sG_*fTfwj^}Q)&5ic4mU3PE0sw_EcAkMh_ETKiD0H={|AJsl zHiiT2wPPQy$U}X_*pL90%BV?8X&6W(M!P{cwTO#Qor#uNC{zKB@+H!$y@49;?bg1d zDQwpWL-uxeYu-z~@o=MtF$}Qk2UxlTteKLI@T1j7tF71@E?}TXTeaFG29Rpj|5S)W zIpH@>`p3iO;k~^rCp~=sq`pccIqo^oW5d-_wjoE;iB8>Q57MXD7Pn>c&u57SfV ziiw~#sZ2v3QC+B1K|WiexIuKTYcqjY{S3BjSh%ReC|Xi5 z@8KB2z;g&= z%5I%A4twfOP#+XIvAV+WANB{R4G>`3$1u?k1c0Jm;>Bk+s1Zc{OikHpX$^8zvW(C< znVV%+7G{=s6-WYDF`2l1uL(;F`-|WEKvp&A5pfnDLP!JPtj7pP(u6IBDaPFd*7l$Y z?EPSik3$9A3%zN=5_|7Ynkd|g5sD6)*lHZH-eY6U5NbC2FM`ISE6g|dM{_@H4SoU* zVHf~70yNx<7Q5KQCHRYkNE4mq=4@NtjqYwkMSEUwaJMmdyW6;{QBg%1jLwuvN@eyg zDT9rbDf6~erd=qLATD?zU1-+7(PvI@ms&Xz)xhL8;4L^%bczqp%)_s2{3;JvMtTaUNaFp{RG)ol61yRCSmHDTs z_0P%yuu4neoHzDbX@R4*1m~_;p~jtv#Z|kl_QTF@i~rgW%~_B^U<6tgfz3N8)b`3Z z;H}UzZBa`j&w<}K!WmB#!p^biAr2V`=gh+x0(PTPo+uFSVQT`MC-5i+b;yx|NUqX= z;3DcW)6Y25ElllGeY^rQWzTI563^{&ex@yxMZ36hr-JBm7@cFRIk1I<7}_xLLO2~9 zwIJ};M|2GCn@6j*#}pptHS69Sct;#i<7N$S0v>-D$Au+hC+m;cKR%HYbHt)}Y$=gL z47Rjni-w)=(Kd{q={9R)XY7hK=S%u{1JUgzNc&13;D^KLMwWWPPOs@ztWnGDI=e*6 zjt~g<#KxrmsetTZ*^MczEVZAP`i6Kj3iAT1ndpL{H75FQLne{NzHn1p;k*A_)~EcL zvgw8$UhJ@>$~RMnQ$C#C!o(` zU6`T?D(a&#AlgARTkmkW78~Dkgdxks)gVcdy^a)GR$hOoEXpqS{*{KUcH6~q2!+zulVYeWM zZm6ms&#(gtfQA`A)_Ds(coYfuvrW1S>@2nHD)M}!t35>6-bu&MWu93N{Lwgr12F;D zvbmfy_!4NEp{fM4iBXF`;2*FBj+;C^1Y0MS~@GJ#YTM*SS#&F zp^P>?sf1LDs=*o1y2EkEUQ3Pr3%cNJIGI8)(+mev*i<7kBGWgmvCKA+eH`i;AjpT! 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[2] Luuk J. P. Ament et al., -# `Phys. Rev. Lett. 103, 117003 (2009) `_ diff --git a/edrixs/_downloads/db7d7888e6c3547d28a97a5460ad1cae/example_7_Coulomb.py b/edrixs/_downloads/db7d7888e6c3547d28a97a5460ad1cae/example_7_Coulomb.py deleted file mode 100644 index 65b3c07794..0000000000 --- a/edrixs/_downloads/db7d7888e6c3547d28a97a5460ad1cae/example_7_Coulomb.py +++ /dev/null @@ -1,381 +0,0 @@ -#!/usr/bin/env python -""" -Coulomb interactions -===================================== -In this example we provide more details on how Coulomb interactions are -implemented in multiplet calculations and EDRIXS in particular. We aim -to clarify the form of the matrices, how they are parametrized, -and how the breaking of spherical symmetry can switch on additional elements -that one might not anticipate. Our example is based on a :math:`d` atomic shell. -""" - -################################################################################ -# Create matrix -# ------------------------------------------------------------------------------ -# The Coulomb interaction between two particles can be written as -# -# .. math:: -# \begin{equation} -# \hat{H} = \frac{1}{2} -# \int d\mathbf{r} \int d\mathbf{r}^\prime -# \Sigma_{\sigma, \sigma^\prime} -# |\hat{\psi}^\sigma(\mathbf{r})|^2 \frac{e^2}{R} -# |\hat{\psi}^{\sigma^\prime}(\mathbf{r})|^2, -# \end{equation} -# -# where :math:`\hat{\psi}^\sigma(\mathbf{r})` is the electron wavefunction, with -# spin :math:`\sigma`, and :math:`R=|r-r^\prime|` is the electron separation. -# Solving our problem in this form is difficult due to the need to symmeterize -# the wavefunction to follow fermionic statistics. -# Using second quantization, we can use operators to impose the required -# particle exchange statistics and write the equation in terms of -# a tensor :math:`U` -# -# .. math:: -# \begin{equation} -# \hat{H} = \sum_{\alpha,\beta,\gamma,\delta,\sigma,\sigma^\prime} -# U_{\alpha\sigma,\beta\sigma^\prime,\gamma\sigma^\prime,\delta\sigma} -# \hat{f}^{\dagger}_{\alpha\sigma} -# \hat{f}^{\dagger}_{\beta\sigma^\prime} -# \hat{f}_{t\sigma^\prime}\hat{f}_{\delta\sigma}, -# \end{equation} -# -# where :math:`\alpha`, :math:`\beta`, :math:`\gamma`, :math:`\delta` are -# orbital indices and :math:`\hat{f}^{\dagger}` -# (:math:`\hat{f}`) are the creation (anihilation) operators. -# For a :math:`d`-electron system, we have :math:`10` distinct spin-orbitals -# (:math:`5` orbitals each with :math:`2` spins), which makes matrix the -# :math:`10\times10\times10\times10` in total size. -# In EDRIXS the matrix can be created as follows: -import edrixs -import numpy as np -import scipy -import matplotlib.pyplot as plt -import itertools - -F0, F2, F4 = 6.94, 14.7, 4.41 -umat_chb = edrixs.get_umat_slater('d', F0, F2, F4) -################################################################################ -# We stored this under variable :code:`umat_chb` where "cbh" stands for -# complex harmonic basis, which is the default basis in EDRIXS. - -################################################################################ -# Parameterizing interactions -# ------------------------------------------------------------------------------ -# EDRIXS parameterizes the interactions in :math:`U` via Slater integral -# parameters :math:`F^{k}`. These relate to integrals of various spherical -# Harmonics as well as Clebsch-Gordon coefficients, Gaunt coefficients, -# and Wigner 3J symbols. Textbooks such as [1]_ can be used for further -# reference. If you are interested in the details of how -# EDRIXS does this (and you probably aren't) function :func:`.umat_slater`, -# constructs the required matrix via Gaunt coeficents from -# :func:`.get_gaunt`. Two alternative parameterizations are common. -# The first are the Racah parameters, which are -# -# .. math:: -# \begin{eqnarray} -# A &=& F^0 - \frac{49}{441} F^4 \\ -# B &=& \frac{1}{49}F^2 - \frac{5}{441}F^4 \\ -# C &=& \frac{35}{441}F^4. -# \end{eqnarray} -# -# or an alternative form for the Slater integrals -# -# .. math:: -# \begin{eqnarray} -# F_0 &=& F^0 \\ -# F_2 &=& \frac{1}{49}F^2 \\ -# F_4 &=& \frac{1}{441}F^4, -# \end{eqnarray} -# -# which involves different normalization parameters. - -################################################################################ -# Basis transform -# ------------------------------------------------------------------------------ -# If we want to use the real harmonic basis, we can use a tensor -# transformation, which imposes the following orbital order -# :math:`3z^2-r^2, xz, yz, x^2-y^2, xy`, each of which involves -# :math:`\uparrow, \downarrow` spin pairs. Let's perform this transformation and -# store a list of these orbitals. -umat = edrixs.transform_utensor(umat_chb, edrixs.tmat_c2r('d', True)) -orbitals = ['3z^2-r^2', 'xz', 'yz', 'x^2-y^2', 'xy'] - -################################################################################ -# Interactions -# ------------------------------------------------------------------------------ -# Tensor :math:`U` is a series of matrix -# elements -# -# .. math:: -# \begin{equation} -# \langle\psi_{\gamma,\delta}^{\bar{\sigma},\bar{\sigma}^\prime} -# |\hat{H}| -# \psi_{\alpha,\beta}^{\sigma,\sigma^\prime}\rangle -# \end{equation} -# -# the combination of which defines the energetic cost of pairwise -# electron-electron interactions between states :math:`\alpha,\sigma` -# and :math:`\beta,\sigma^\prime`. In EDRIXS we follow the convention of -# summing over all orbital pairs. Some other texts count each pair of -# indices only once. The matrix elements here will consequently -# be half the magnitude of those in other references. -# We can express the interactions in terms of -# the orbitals involved. It is common to distinguish "direct Coulomb" and -# "exchange" interactions. The former come from electrons in the same orbital -# and the later involve swapping orbital labels for electrons. We will use -# :math:`U_0` and :math:`J` as a shorthand for distinguishing these. -# -# Before we describe the different types of interactions, we note that since -# the Coulomb interaction is real, and due to the spin symmmetry properties -# of the process :math:`U` always obeys -# -# .. math:: -# \begin{equation} -# U_{\alpha\sigma,\beta\sigma^\prime,\gamma\sigma^\prime,\delta\sigma} = -# U_{\beta\sigma,\alpha\sigma^\prime,\delta\sigma^\prime,\gamma\sigma} = -# U_{\delta\sigma,\gamma\sigma^\prime,\beta\sigma^\prime,\alpha\sigma} = -# U_{\gamma\sigma,\delta\sigma^\prime,\alpha\sigma^\prime,\beta\sigma}. -# \end{equation} -# -# -# 1. Intra orbital -# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -# The direct Coulomb energy cost to double-occupy an orbital comes from terms -# like :math:`U_{\alpha\sigma,\alpha\bar\sigma,\alpha\bar\sigma,\alpha\sigma}`. -# In this notation, we use :math:`\sigma^\prime` to denote that the matrix -# element is summed over all pairs and :math:`\bar{\sigma}` to denote sums -# over all opposite spin pairs. Due to rotational symmetry, all these -# elements are the same and equal to -# -# .. math:: -# \begin{eqnarray} -# U_0 &=& \frac{A}{2} + 2B + \frac{3C}{2}\\ -# &=& \frac{F_0}{2} + 2F_2 + 18F_4 -# \end{eqnarray} -# -# Let's print these to demonstrate where these live in the array -for i in range(0, 5): - val = umat[i*2, i*2 + 1, i*2 + 1, i*2].real - print(f"{orbitals[i]:<8} \t {val:.3f}") - -################################################################################ -# 2. Inter orbital Coulomb interactions -# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -# Direct Coulomb repulsion between different orbitals depends on terms like -# :math:`U_{\alpha\sigma,\beta\sigma^\prime,\beta\sigma^\prime,\alpha\sigma}`. -# Expresions for these parameters are provided in column :math:`U` in -# :ref:`table_2_orbital`. We can print the values from :code:`umat` -# like this: -for i, j in itertools.combinations(range(5), 2): - val = umat[i*2, j*2 + 1, j*2 + 1, i*2].real - print(f"{orbitals[i]:<8} \t {orbitals[j]:<8} \t {val:.3f}") - -################################################################################ -# 3. Inter-orbital exchange interactions -# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -# Exchange terms exist with the form -# :math:`U_{\alpha\sigma,\beta\sigma^\prime,\alpha\sigma^\prime,\beta\sigma}`. -# Expresions for these parameters are provided in column :math:`J` of -# :ref:`table_2_orbital`. These come from terms like this in the matrix: -for i, j in itertools.combinations(range(5), 2): - val = umat[i*2, j*2 + 1, i*2 + 1, j*2].real - print(f"{orbitals[i]:<8} \t {orbitals[j]:<8} \t {val:.3f}") - -################################################################################ -# 4. Pair hopping term -# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -# Terms that swap pairs of electrons exist as -# :math:`(1-\delta_{\sigma\sigma'})U_{\alpha\sigma,\alpha\bar\sigma,\beta\bar\sigma,\beta\sigma}` -# and depend on exchange interactions column :math:`J` from -# :ref:`table_2_orbital` -# and here in the matrix. -for i, j in itertools.combinations(range(5), 2): - val = umat[i*2, i*2 + 1, j*2 + 1, j*2].real - print(f"{orbitals[i]:<8} \t {orbitals[j]:<8} \t {val:.3f}") - -################################################################################ -# 5. Three orbital -# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -# Another set of terms that one might not immediately anticipate involve three -# orbitals like -# -# .. math:: -# \begin{equation} -# U_{\alpha\sigma, \gamma\sigma', \beta\sigma', \gamma\sigma} \\ -# U_{\alpha\sigma, \gamma\sigma', \gamma\sigma', \beta\sigma} \\ -# (1-\delta_{\sigma\sigma'}) -# U_{\alpha\sigma, \beta\sigma', \gamma\sigma', \gamma\sigma} -# \end{equation} -# -# for :math:`\alpha=3z^2-r^2, \beta=x^2-y^2, \gamma=xz/yz`. -# These are needed to maintain the rotational symmetry of the interations. -# See :ref:`table_3_orbital` for the expressions. We can print some of -# these via: -ijkl = [[0, 1, 3, 1], - [0, 2, 3, 2], - [1, 0, 3, 1], - [1, 1, 3, 0], - [2, 0, 3, 2], - [2, 2, 3, 0]] - -for i, j, k, l in ijkl: - val = umat[i*2, j*2 + 1, k*2 + 1, l*2].real - print(f"{orbitals[i]:<8} \t {orbitals[j]:<8} \t" - f"{orbitals[k]:<8} \t {orbitals[l]:<8} \t {val:.3f}") - -################################################################################ -# 6. Four orbital -# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -# Futher multi-orbital terms include -# :math:`U_{\alpha\sigma,\beta\sigma^\prime,\gamma\sigma^\prime,\delta\sigma}`. -# We can find these here in the matrix: -ijkl = [[0, 1, 2, 4], - [0, 1, 4, 2], - [0, 2, 1, 4], - [0, 2, 4, 1], - [0, 4, 1, 2], - [0, 4, 2, 1], - [3, 1, 4, 2], - [3, 2, 4, 1], - [3, 4, 1, 2], - [3, 4, 2, 1]] - -for i, j, k, l in ijkl: - val = umat[i*2, j*2 + 1, k*2 + 1, l*2].real - print(f"{orbitals[i]:<8} \t {orbitals[j]:<8} \t {orbitals[k]:<8}" - f"\t {orbitals[l]:<8} \t {val:.3f}") - -################################################################################ -# Effects of multi-orbital terms -# ------------------------------------------------------------------------------ -# To test the effects of the multi-orbital terms, let's plot the eigenenergy -# spectra with and without multi-orbital terms switched on for system with and -# without a cubic crystal field. We will use a :math:`d`-shell with two -# electrons. -ten_dqs = [0, 2, 4, 12] - -def diagonalize(ten_dq, umat): - emat = edrixs.cb_op(edrixs.cf_cubic_d(ten_dq), - edrixs.tmat_c2r('d', ispin=True)) - H = (edrixs.build_opers(4, umat, basis) - + edrixs.build_opers(2, emat, basis)) - e, v = scipy.linalg.eigh(H) - return e - e.min() - -basis = edrixs.get_fock_bin_by_N(10, 2) -umat_no_multiorbital = np.copy(umat) -B = F2/49 - 5*F4/441 -for val in [np.sqrt(3)*B/2, np.sqrt(3)*B, 3*B/2]: - umat_no_multiorbital[(np.abs(umat)- val) < 1e-6] = 0 - -fig, axs = plt.subplots(1, len(ten_dqs), figsize=(8, 3)) - -for cind, (ax, ten_dq) in enumerate(zip(axs, ten_dqs)): - ax.hlines(diagonalize(ten_dq, umat), xmin=0, xmax=1, - label='on', color=f'C{cind}') - ax.hlines(diagonalize(ten_dq, umat_no_multiorbital), - xmin=1.5, xmax=2.5, - label='off', - linestyle=':', color=f'C{cind}') - ax.set_title(f"$10D_q={ten_dq}$") - ax.set_ylim([-.5, 20]) - ax.set_xticks([]) - ax.legend() - -fig.suptitle("Eigenvalues with 3&4-orbital effects on/off") -fig.subplots_adjust(wspace=.3) -axs[0].set_ylabel('Eigenvalues (eV)') -fig.subplots_adjust(top=.8) -plt.show() - -################################################################################ -# On the left of the plot Coulomb interactions in spherical symmetry cause -# substantial mxing between :math:`t_{2g}` and :math:`e_{g}` orbitals in the -# eigenstates and 3 & 4 orbital orbital terms are crucial for obtaining the -# the right eigenenergies. As :math:`10D_q` get large, this mixing is switched -# off and the spectra start to become independent of whether the 3 & 4 orbital -# orbital terms are included or not. -# -# -# -# .. _table_2_orbital: -# .. table:: Table of 2 orbital interactions -# -# +-----------------------------+------------------+-----------------------+----------------+--------------------+ -# |Orbitals :math:`\alpha,\beta`|:math:`U_0` Racah | :math:`U_0` Slater |:math:`J` Racah |:math:`J` Slater | -# +=============================+==================+=======================+================+====================+ -# |:math:`3z^2-r^2, xz` |:math:`A/2+B+C/2` |:math:`F_0/2+F_2-12F_4`| :math:`B/2+C/2`|:math:`F_2/2+15F_4` | -# +-----------------------------+------------------+-----------------------+----------------+--------------------+ -# |:math:`3z^2-r^2, yz` |:math:`A/2+B+C/2` |:math:`F_0/2+F_2-12F_4`| :math:`B/2+C/2`|:math:`F_2/2+15F_4` | -# +-----------------------------+------------------+-----------------------+----------------+--------------------+ -# |:math:`3z^2-r^2, x^2-y^2` |:math:`A/2-2B+C/2`|:math:`F_0/2-2F_2+3F_4`|:math:`2B+C/2` |:math:`2F_2+15F_4/2`| -# +-----------------------------+------------------+-----------------------+----------------+--------------------+ -# |:math:`3z^2-r^2, xy` |:math:`A/2-2B+C/2`|:math:`F_0/2-2F_2+3F_4`|:math:`2B+C/2` |:math:`2F_2+15F_4/2`| -# +-----------------------------+------------------+-----------------------+----------------+--------------------+ -# |:math:`xz, yz` |:math:`A/2-B+C/2` |:math:`F_0/2-F_2-12F_4`|:math:`3B/2+C/2`|:math:`3F_2/2+10F_4`| -# +-----------------------------+------------------+-----------------------+----------------+--------------------+ -# |:math:`xz, x^2-y^2` |:math:`A/2-B+C/2` |:math:`F_0/2-F_2-12F_4`|:math:`3B/2+C/2`|:math:`3F_2/2+10F_4`| -# +-----------------------------+------------------+-----------------------+----------------+--------------------+ -# |:math:`xz, xy` |:math:`A/2-B+C/2` |:math:`F_0/2-F_2-12F_4`|:math:`3B/2+C/2`|:math:`3F_2/2+10F_4`| -# +-----------------------------+------------------+-----------------------+----------------+--------------------+ -# |:math:`yz, x^2-y^2` |:math:`A/2-B+C/2` |:math:`F_0/2-F_2-12F_4`|:math:`3B/2+C/2`|:math:`3F_2/2+10F_4`| -# +-----------------------------+------------------+-----------------------+----------------+--------------------+ -# |:math:`yz, xy` |:math:`A/2-B+C/2` |:math:`F_0/2-F_2-12F_4`|:math:`3B/2+C/2`|:math:`3F_2/2+10F_4`| -# +-----------------------------+------------------+-----------------------+----------------+--------------------+ -# |:math:`x^2-y^2, xy` |:math:`A/2+2B+C/2`|:math:`F_0+4F_2-34F_4` | :math:`C/2` |:math:`35F_4/2` | -# +-----------------------------+------------------+-----------------------+----------------+--------------------+ -# -# -# .. _table_3_orbital: -# .. table:: Table of 3 orbital interactions -# -# +-----------------------------+-------------+----------------------------------------------------+-----------------------------------------------------+ -# |Orbitals :math:`\alpha,\beta,\gamma,\delta`|:math:`\langle\alpha\beta|\gamma\delta\rangle` Racah|:math:`\langle\alpha\beta|\gamma\delta\rangle` Slater| -# +=============================+=============+====================================================+=====================================================+ -# |:math:`3z^2-r^2, xz, x^2-y^2, xz` | :math:`\sqrt{3}B/2` | :math:`\sqrt{3}F_2/2-5\sqrt{3}F_4/2` | -# +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ -# |:math:`3z^2-r^2, yz, x^2-y^2, yz` | :math:`-\sqrt{3}B/2` | :math:`-\sqrt{3}F_2/2+5\sqrt{3}F_4/2` | -# +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ -# |:math:`xz, 3z^2-r^2, x^2-y^2, xz` | :math:`-\sqrt{3}B` | :math:`-\sqrt{3}F_2+5\sqrt{3}F_4` | -# +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ -# |:math:`xz, xz, x^2-y^2, 3z^2-r^2` | :math:`\sqrt{3}B/2` | :math:`\sqrt{3}F_2/2-5\sqrt{3}F_4/2` | -# +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ -# |:math:`yz, 3z^2-r^2, x^2-y^2, yz` | :math:`\sqrt{3}B` | :math:`\sqrt{3}F_2-5\sqrt{3}F_4` | -# +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ -# |:math:`yz, yz, x^2-y^2, 3z^2-r^2` | :math:`-\sqrt{3}B/2` | :math:`-\sqrt{3}F_2/2+5\sqrt{3}F_4/2` | -# +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ -# -# -# .. _table_4_orbital: -# .. table:: Table of 4 orbital interactions -# -# +-----------------------------+-------------+----------------------------------------------------+-----------------------------------------------------+ -# |Orbitals :math:`\alpha,\beta,\gamma,\delta`|:math:`\langle\alpha\beta|\gamma\delta\rangle` Racah|:math:`\langle\alpha\beta|\gamma\delta\rangle` Slater| -# +=============================+=============+====================================================+=====================================================+ -# |:math:`3z^2-r^2, xz, yz, xy` | :math:`-\sqrt{3}B` | :math:`-\sqrt{3}F_2+5\sqrt{3}F_4` | -# +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ -# |:math:`3z^2-r^2, xz, xy, yz` | :math:`\sqrt{3}B/2` | :math:`\sqrt{3}F_2/2-5\sqrt{3}F_4/2` | -# +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ -# |:math:`3z^2-r^2, yz, xz, xy` | :math:`-\sqrt{3}B` | :math:`-\sqrt{3}F_2+5\sqrt{3}F_4` | -# +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ -# |:math:`3z^2-r^2, yz, xy, xz` | :math:`\sqrt{3}B/2` | :math:`\sqrt{3}F_2/2-5\sqrt{3}F_4/2` | -# +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ -# |:math:`3z^2-r^2, xy, xz, yz` | :math:`\sqrt{3}B/2` | :math:`\sqrt{3}F_2/2-5\sqrt{3}F_4/2` | -# +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ -# |:math:`3z^2-r^2, xy, yz, xz` | :math:`\sqrt{3}B/2` | :math:`\sqrt{3}F_2/2-5\sqrt{3}F_4/2` | -# +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ -# |:math:`x^2-y^2 , xz, xy, yz` | :math:`-3B/2` | :math:`-3F_2/2+15F_4/2` | -# +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ -# |:math:`x^2-y^2 , yz, xy, xz` | :math:`3B/2` | :math:`3F_2/2-15F_4/2` | -# +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ -# |:math:`x^2-y^2 , xy, xz, yz` | :math:`-3B/2` | :math:`-3F_2/2+15F_4/2` | -# +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ -# |:math:`x^2-y^2 , xy, yz, xz` | :math:`3B/2` | :math:`3F_2/2-15F_4/2` | -# +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ -# -# -# .. rubric:: Footnotes -# -# .. [1] MSugano S, Tanabe Y and Kamimura H. 1970. Multiplets of -# Transition-Metal Ions in Crystals. Academic Press, New York and London. diff --git a/edrixs/_downloads/ea938601a5ea0e41baae4e0ea4cfc9b2/example_5_charge_transfer.ipynb b/edrixs/_downloads/ea938601a5ea0e41baae4e0ea4cfc9b2/example_5_charge_transfer.ipynb deleted file mode 100644 index 8027b44d95..0000000000 --- a/edrixs/_downloads/ea938601a5ea0e41baae4e0ea4cfc9b2/example_5_charge_transfer.ipynb +++ /dev/null @@ -1,154 +0,0 @@ -{ - "cells": [ - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n# Charge-transfer energy for NiO\nThis example follows the `sphx_glr_auto_examples_example_3_AIM_XAS.py`\nexample and considers the same model. This time we outline how to determine\nthe charge transfer energy in the sense defined by Zaanen, Sawatzky, and Allen\n[1]_. That is, a $d^{n_d} \\rightarrow d^{n_d + 1} \\underline{L}$ transition\nin the atomic limit, after considering Coulomb interactions and crystal field. Although\nthis can be determined analytically in some cases, the easiest way is often just to\ncalculate it, as we will do here.\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "import edrixs\nimport numpy as np\nimport matplotlib.pyplot as plt\nimport scipy\nimport example_3_AIM_XAS\nimport importlib\n_ = importlib.reload(example_3_AIM_XAS)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Determine eigenvectors and occupations\nThe first step repeats what was done in\n`sphx_glr_auto_examples_example_4_GS_analysis.py` but it does not apply\nthe hybridization between the impurity and both states.\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "from example_3_AIM_XAS import (F0_dd, F2_dd, F4_dd,\n nd, norb_d, norb_bath, v_noccu,\n imp_mat, bath_level,\n hyb, ext_B, trans_c2n)\nntot = 20\numat_delectrons = edrixs.get_umat_slater('d', F0_dd, F2_dd, F4_dd)\numat = np.zeros((ntot, ntot, ntot, ntot), dtype=complex)\numat[:norb_d, :norb_d, :norb_d, :norb_d] += umat_delectrons\nemat_rhb = np.zeros((ntot, ntot), dtype='complex')\nemat_rhb[0:norb_d, 0:norb_d] += imp_mat\nindx = np.arange(norb_d, norb_d*2)\nemat_rhb[indx, indx] += bath_level[0]\ntmat = np.eye(ntot, dtype=complex)\nfor i in range(2):\n off = i * norb_d\n tmat[off:off+norb_d, off:off+norb_d] = np.conj(np.transpose(trans_c2n))\n\nemat_chb = edrixs.cb_op(emat_rhb, tmat)\nv_orbl = 2\nsx = edrixs.get_sx(v_orbl)\nsy = edrixs.get_sy(v_orbl)\nsz = edrixs.get_sz(v_orbl)\nzeeman = ext_B[0] * (2 * sx) + ext_B[1] * (2 * sy) + ext_B[2] * (2 * sz)\nemat_chb[0:norb_d, 0:norb_d] += zeeman\nbasis = np.array(edrixs.get_fock_bin_by_N(ntot, v_noccu))\nH = (edrixs.build_opers(2, emat_chb, basis)\n + edrixs.build_opers(4, umat, basis))\n\ne, v = scipy.linalg.eigh(H)\ne -= e[0]\n\nnum_d_electrons = basis[:, :norb_d].sum(1)\nalphas = np.sum(np.abs(v[num_d_electrons == 8, :])**2, axis=0)\nbetas = np.sum(np.abs(v[num_d_electrons == 9, :])**2, axis=0)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Energy to lowest energy ligand orbital\nLet's vizualize $\\alpha$ and $\\beta$.\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "fig, ax = plt.subplots()\n\nax.plot(e, alphas, '.-', label=r'$\\alpha$ $d^8L^{10}$')\nax.plot(e, betas, '.-', label=r'$\\beta$ $d^9L^{9}$')\n\nax.set_xlabel('Energy (eV)')\nax.set_ylabel('Population')\nax.set_title('NiO')\nax.legend()\nplt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "One can see that the mixing between impurity and bath states has disappered\nbecause we have turned off the hybridization. The energy required to\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "GS_energy = min(e[np.isclose(alphas, 1)])\nlowest_energy_to_transfer_electron = min(e[np.isclose(betas, 1)])\nE_to_ligand = lowest_energy_to_transfer_electron - GS_energy\nprint(f\"Energy to lowest energy ligand state is {E_to_ligand:.3f} eV\")" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where we have used :code:`np.isclose` to avoid errors from finite numerical\nprecision.\n\n" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Diagonalizing by blocks\nWhen working on a problem with a large basis, one can take advantage of the\nlack of hybridization and separately diagonalize the impurity and bath\nstates\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "energies = []\n\nfor n_ligand_holes in [0, 1]:\n basis_d = edrixs.get_fock_bin_by_N(10, nd + n_ligand_holes)\n Hd = (edrixs.build_opers(2, emat_chb[:10, :10], basis_d)\n + edrixs.build_opers(4, umat[:10, :10, :10, :10], basis_d))\n ed = scipy.linalg.eigh(Hd, eigvals_only=True, subset_by_index=[0, 0])[0]\n\n basis_L = edrixs.get_fock_bin_by_N(10, 10 - n_ligand_holes)\n HL = (edrixs.build_opers(2, emat_chb[10:, 10:], basis_L)\n + edrixs.build_opers(4, umat[10:, 10:, 10:, 10:], basis_L))\n eL = scipy.linalg.eigh(HL, eigvals_only=True, subset_by_index=[0, 0])[0]\n\n energies.append(ed + eL)\n\nprint(f\"Energy to lowest energy ligand state is {energies[1] - energies[0]:.3f} eV\")" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "which yields the same result.\n\n" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Energy splitting in ligand states\nThe last thing to consider is that our definition of the charge transfer\nenergy refers to the atomic limit with all hopping terms switched off, whereas\nthe ligand states in the model are already split by the oxygen-oxygen hopping\nterm $T_{pp}$ as illustrated below. So the final charge transer energy\nneeds to account for this.\n\n .. image:: /_static/energy_level.png\n\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "T_pp = 1\nprint(f\"Charge transfer is {energies[1] - energies[0] + T_pp:.3f} eV\")" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - ".. rubric:: Footnotes\n\n.. [1] J. Zaanen, G. A. Sawatzky, and J. W. Allen,\n [Phys. Rev. Lett. 55, 418 (1985)](https://doi.org/10.1103/PhysRevLett.55.418).\n\n" - ] - } - ], - "metadata": { - "kernelspec": { - "display_name": "Python 3", - "language": "python", - "name": "python3" - }, - "language_info": { - "codemirror_mode": { - "name": "ipython", - "version": 3 - }, - "file_extension": ".py", - "mimetype": "text/x-python", - "name": "python", - "nbconvert_exporter": "python", - "pygments_lexer": "ipython3", - "version": "3.10.16" - } - }, - "nbformat": 4, - "nbformat_minor": 0 -} \ No newline at end of file diff --git a/edrixs/_downloads/ec5be3ffa2b20d2aa5db9e4571690517/example_2_single_atom_RIXS.ipynb b/edrixs/_downloads/ec5be3ffa2b20d2aa5db9e4571690517/example_2_single_atom_RIXS.ipynb deleted file mode 100644 index 9c4374907c..0000000000 --- a/edrixs/_downloads/ec5be3ffa2b20d2aa5db9e4571690517/example_2_single_atom_RIXS.ipynb +++ /dev/null @@ -1,190 +0,0 @@ -{ - "cells": [ - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n# RIXS calculations for an atomic model\nHere we show how to compute RIXS for a single site atomic model with crystal\nfield and electron-electron interactions. We take the case of\nSr\\ :sub:`2`\\ YIrO\\ :sub:`6`\nfrom Ref. [1]_ as the material in question. The aim of this example is to\nillustrate the proceedure and to provide what we hope is useful advice. What is\nwritten is not meant to be a replacement for reading the docstrings of the\nfunctions, which can always be accessed on the\n[edrixs website](https://nsls-ii.github.io/edrixs/reference/index.html) or\nby executing functions with ?? in IPython.\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "import edrixs\nimport numpy as np\nimport matplotlib.pyplot as plt" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Specify active core and valence orbitals\nSr\\ :sub:`2`\\ YIrO\\ :sub:`6`\\ has a $5d^4$ electronic configuration and\nwe want to calculate the $L_3$ edge spectrum i.e. resonating with a\n$2p_{3/2}$ core hole. We will start by including only the\n$t_{2g}$ valance orbitals.\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "shell_name = ('t2g', 'p32')\nv_noccu = 4" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Slater parameters\nHere we want to use Hund's interaction\n$J_H$ and spin orbit coupling $\\lambda$ as adjustable parameters\nto match experiment. We will take\nthe core hole interaction parameter from the Hartree Fock numbers EDRIXS has\nin its database. These need to be converted and arranged into the order\nrequired by EDRIXS.\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "Ud = 2\nJH = 0.25\nlam = 0.42\nF0_d, F2_d, F4_d = edrixs.UdJH_to_F0F2F4(Ud, JH)\ninfo = edrixs.utils.get_atom_data('Ir', '5d', v_noccu, edge='L3')\nG1_dp = info['slater_n'][5][1]\nG3_dp = info['slater_n'][6][1]\nF0_dp = edrixs.get_F0('dp', G1_dp, G3_dp)\nF2_dp = info['slater_n'][4][1]\n\nslater_i = [F0_d, F2_d, F4_d] # Fk for d\nslater_n = [\n F0_d, F2_d, F4_d, # Fk for d\n F0_dp, F2_dp, # Fk for dp\n G1_dp, G3_dp, # Gk for dp\n 0.0, 0.0 # Fk for p\n]\nslater = [slater_i, slater_n]\nv_soc = (lam, lam)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Diagonalization\nWe obtain the ground and intermediate state eigenenergies and the transition\noperators via matrix diagonalization. Note that the calculation does not know\nthe core hole energy, so we need to adjust the energy that the resonance will\nappear at by hand. We know empirically that the resonance is at 11215 eV\nand that putting four electrons into the valance band costs about\n$4 F^0_d\\approx6$ eV. In this case\nwe are assuming a perfectly cubic crystal field, which we have already\nimplemented when we specified the use of the $t_{2g}$ subshell only\nso we do not need to pass an additional :code:`v_cfmat` matrix.\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "off = 11215 - 6\nout = edrixs.ed_1v1c_py(shell_name, shell_level=(0, -off), v_soc=v_soc,\n c_soc=info['c_soc'], v_noccu=v_noccu, slater=slater)\neval_i, eval_n, trans_op = out" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Compute XAS\nTo calculate XAS we need to correctly specify the orientation of the x-rays\nwith respect to the sample. By default, the $x, y, z$ coordinates\nof the sample's crystal field, will be aligned with our lab frame, passing\n:code:`loc_axis` to :code:`ed_1v1c_py` can be used to specify a different\nconvention. The experimental geometry is specified following the angles\nshown in Figure 1 of Y. Wang et al.,\n[Computer Physics Communications 243, 151-165 (2019)](https://doi.org/10.1016/j.cpc.2019.04.018). The default\nsetting has x-rays along $z$ for $\\theta=\\pi/2$ rad\nand the x-ray beam along $-x$ for\n$\\theta=\\phi=0$. Parameter :code:`scatter_axis` can be passed to\n:code:`xas_1v1c_py` to specify a different geometry if desired.\n\nVariable :code:`pol_type` specifies a list of different x-ray\npolarizations to calculate. Here we will use so-called $\\pi$-polarization\nwhere the x-rays are parallel to the plane spanned by the incident\nbeam and the sample $z$-axis.\n\nEDRIXS represents the system's ground state using a set of\nlow energy eigenstates weighted by Boltzmann thermal factors.\nThese eigenstates are specified by :code:`gs_list`,\nwhich is of the form $[0, 1, 2, 3, \\dots]$. In this example, we\ncalculate these states as those that have non-negligible thermal\npopulation. The function :code:`xas_1v1c_py` assumes that the spectral\nbroadening is dominated by the inverse core hole lifetime :code:`gamma_c`,\nwhich is the Lorentzian half width at half maximum.\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "ominc = np.linspace(11200, 11230, 50)\ntemperature = 300 # in K\nprob = edrixs.boltz_dist(eval_i, temperature)\ngs_list = [n for n, prob in enumerate(prob) if prob > 1e-6]\n\nthin = 30*np.pi/180\nphi = 0\npol_type = [('linear', 0)]\n\nxas = edrixs.xas_1v1c_py(\n eval_i, eval_n, trans_op, ominc, gamma_c=info['gamma_c'],\n thin=thin, phi=phi, pol_type=pol_type,\n gs_list=gs_list)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Compute RIXS\nCalculating RIXS is overall similar to XAS, but with a few additional\nconsiderations. The spectral width in the energy loss axis of RIXS it\nnot set by the core hole lifetime, but by either the final state lifetime\nor the experimental resolution and is parameterized by :code:`gamma_f`\n-- the Lorentzian half width at half maximum.\n\nThe angle and polarization of the emitted beam must also be specified, so\nwe pass :code:`pol_type_rixs` to the function, which specifies the\nincludes the incoming and outgoing x-ray states. If, as is common in\nexperiments, the emitted polarization is not resolved\none needs to add both emitted polarization channels, which is what we will\ndo later on in this example.\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "eloss = np.linspace(-.5, 6, 400)\npol_type_rixs = [('linear', 0, 'linear', 0), ('linear', 0, 'linear', np.pi/2)]\n\nthout = 60*np.pi/180\ngamma_f = 0.02\n\nrixs = edrixs.rixs_1v1c_py(\n eval_i, eval_n, trans_op, ominc, eloss,\n gamma_c=info['gamma_c'], gamma_f=gamma_f,\n thin=thin, thout=thout, phi=phi,\n pol_type=pol_type_rixs, gs_list=gs_list,\n temperature=temperature\n)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The array :code:`xas` will have shape\n:code:`(len(ominc_xas), len(pol_type))`\n\n" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Plot XAS and RIXS\nLet's plot everything. We will use a function so we can reuse the code later.\nNote that the rixs array :code:`rixs` has shape\n:code:`(len(ominc_xas), len(ominc_xas), len(pol_type))`. We will use some numpy\ntricks to sum over the two different emitted polarizations.\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "fig, axs = plt.subplots(2, 2, figsize=(10, 10))\n\n\ndef plot_it(axs, ominc, xas, eloss, rixscut, rixsmap=None, label=None):\n axs[0].plot(ominc, xas[:, 0], label=label)\n axs[0].set_xlabel('Energy (eV)')\n axs[0].set_ylabel('Intensity')\n axs[0].set_title('XAS')\n\n axs[1].plot(eloss, rixscut, label=f\"{label}\")\n axs[1].set_xlabel('Energy loss (eV)')\n axs[1].set_ylabel('Intensity')\n axs[1].set_title(f'RIXS at resonance')\n\n if rixsmap is not None:\n art = axs[2].pcolormesh(ominc, eloss, rixsmap.T, shading='auto')\n plt.colorbar(art, ax=axs[2], label='Intensity')\n axs[2].set_xlabel('Incident energy (eV)')\n axs[2].set_ylabel('Energy loss')\n axs[2].set_title('RIXS map')\n\n\nrixs_pol_sum = rixs.sum(-1)\ncut_index = np.argmax(rixs_pol_sum[:, eloss < 2].sum(1))\nrixscut = rixs_pol_sum[cut_index]\n\nplot_it(axs.ravel(), ominc, xas, eloss, rixscut, rixsmap=rixs_pol_sum)\naxs[0, 1].set_xlim(right=3)\naxs[1, 0].set_ylim(top=3)\naxs[1, 1].remove()\n\nplt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Full d shell calculation\nSome researchers have questioned the appropriateness of only including the\n$t_{2g}$ subshell for iridates [2]_. Let's test this. We specify that\nthe full $d$ shell should be used and apply cubic crystal field matrix\n:code:`v_cfmat`. We shift the energy offset by $\\frac{2}{5}10D_q$, which\nis the amount the crystal field moves the $t_{2g}$ subshell.\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "ten_dq = 3.5\nv_cfmat = edrixs.cf_cubic_d(ten_dq)\noff = 11215 - 6 + ten_dq*2/5\nout = edrixs.ed_1v1c_py(('d', 'p32'), shell_level=(0, -off), v_soc=v_soc,\n v_cfmat=v_cfmat,\n c_soc=info['c_soc'], v_noccu=v_noccu, slater=slater)\neval_i, eval_n, trans_op = out\n\nxas_full_d_shell = edrixs.xas_1v1c_py(\n eval_i, eval_n, trans_op, ominc, gamma_c=info['gamma_c'],\n thin=thin, phi=phi, pol_type=pol_type,\n gs_list=gs_list)\n\nrixs_full_d_shell = edrixs.rixs_1v1c_py(\n eval_i, eval_n, trans_op, np.array([11215]), eloss,\n gamma_c=info['gamma_c'], gamma_f=gamma_f,\n thin=thin, thout=thout, phi=phi,\n pol_type=pol_type_rixs, gs_list=gs_list,\n temperature=temperature)\n\nfig, axs = plt.subplots(1, 2, figsize=(10, 4))\nplot_it(axs, ominc, xas, eloss, rixscut, label='$t_{2g}$ subshell')\nrixscut = rixs_full_d_shell.sum((0, -1))\nplot_it(axs, ominc, xas_full_d_shell, eloss, rixscut, label='$d$ shell')\n\naxs[0].legend()\naxs[1].legend()\nplt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "As expected, we see the appearance of excitations on the energy scale of\n$10D_q$ in the XAS and RIXS. The low energy manifold is qualitatively,\nbut not quantiatively similar. This makes it clear that the parameterization\nof Sr\\ :sub:`2`\\ YIrO\\ :sub:`6`\\ is dependent on the model.\n\n" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - ".. rubric:: Footnotes\n\n.. [1] Bo Yuan et al.,\n [Phys. Rev. B 95, 235114 (2017)](https://doi.org/10.1103/PhysRevB.95.235114).\n\n.. [2] Georgios L. Stamokostas and Gregory A. Fiete\n [Phys. Rev. B 97, 085150 (2018)](https://doi.org/10.1103/PhysRevB.97.085150).\n\n" - ] - } - ], - "metadata": { - "kernelspec": { - "display_name": "Python 3", - "language": "python", - "name": "python3" - }, - "language_info": { - "codemirror_mode": { - "name": "ipython", - "version": 3 - }, - "file_extension": ".py", - "mimetype": "text/x-python", - "name": "python", - "nbconvert_exporter": "python", - "pygments_lexer": "ipython3", - "version": "3.10.16" - } - }, - "nbformat": 4, - "nbformat_minor": 0 -} \ No newline at end of file diff --git a/edrixs/_downloads/fbd478f01d8361d8a36f393bd9b6097c/example_7_transitions.zip b/edrixs/_downloads/fbd478f01d8361d8a36f393bd9b6097c/example_7_transitions.zip deleted file mode 100644 index 0c91e884200ed2b324d2923b15b093825aa6ab00..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 19236 zcmds9-EJevbsq0Jx!|=g0_3L5p)GPuQ4+cK>eM-3ojUcAr{8?~ z)1N(~*N0D9|MG92{`bjWfBx(l{rfvS<8x;g1>AmVr?DdvKlMYAbfU%cpC4pNd@%CG z0T-7nTBOrZ)avzm?YsThS+G*=T7So{+S$}km>wB^M}gyugiXUMmWIrA0yhhsl(YHL zNEl7UPcx4vY{b(m&P6SWxa*I7mxb}jPo01{!edF~3znpik#yJ}IZK@j{FrjcPB;t4 z%wcX8#ni{u)OV+B>Le@_yfzI3&d`?X9?zU~dOY;(oB6x${^DKtc3A1eQQd^G$75ei znCHjbCG|2V2=N{=YQEUV7AQ@)6T28D zEWI=7uy3>Mf_*vTB4s>O?l$_(V|v+zT;QzIs0wNw(xd;r!NTn z7Ak{5!hWC!sMzCJotZDl@5m&UZ2K|{F0n$|kxNk4fh<}|2@r)Rx(>MtOJpx3(_zX{ zP+h}-g$FlZ0RmuWSb17rtfYg*o(P@-f5tu!RF@c;G%dIrEvSEup-_hq7Q%cSIAAU{ z^#hOJScyNGIk$b?iRF4>s-6wm2$1Cgoa8)DohR@DWWDNgOOLxyt7gPQq*T(ptu`{67uE1FG6E|?CM63s34;l1s)JxnI1eyFVwMw2 z-LT)jdh`0bw=JnF)0cm(B$HOYOqj%eG^I}5c!B1DwvB`~uT)k}*enAe0qrO7b!bd2 zuQ4V_lIDcbLrTTCjUs*EsoEw|f-zZd?zkyI8GxOhjesPOgd*$!SSM#jUPEa573;Gj zzTbNxqK*?|yKm+;(W<#)yBI0e>(#C~2dJS@lFgdQi;+aqq|HWT8`pl+RM6C!@Dy!* zVb1{9B1PqHtIfV@wQ8Pc2f()=fsmq8a*)YFN;4HmVtE8=h}ar=^Vh8!Tl>Q5V&{^g z(E8kn8ocNZ+RRHA5%23YZq+c6)Q+bkH4a+Crm3H(Qk3ZFxFb8n+;oe!d7R1zUAMXC z?X_Vdm~j6r&UmZEe#MlkYBp+@Fj4gu_}F+R9TZHZ)K;zL@iA;XwM&(oI+C$ziDAG} zq4Mw6u~b0J(<~Mh`RB)N)_qZFh_Mtc2epp^`wDF+kn;frWP|qwO=BMcK!?3SOad?% z$Ki~cz(S8>K0Y2M(R6N4g4lMlG_-Yen3zB8BU^iuEnNdU+z?Go5Gg@NEgS9p%KdWJ zHL;a7gbwzBgMuCCwFtwE1 zYLkL-JzzqIu0q~rIJ`F?BnhWp%`a0>je2h2E*PH5^Fv*||mP4A$4%d8mAjHo2W z#@F-6BArnZM%WW+$1dhcv)iOKpZQ5rZ7WYOWHb~^ODBtAT9+_Ct)$$QG(a)WNgWCs z+8E9S=fEfIdz#t)r?(Wr)0#{m( zk_L?^$bqn25>8i>%$XqmFpb|?o-@I=I;7p3gp^v!_cD+%?I26rLMGjm_LA~phIaKc z2w8tPa~Q<~(Linkvoy0jz>$K#?GtPqM`!@ebC(0plQffjw3ooV&d z;WlS&(dt__<6=Zi#^>x+bJnte02Hh<>!vj1rfBKM6RWI*rD4}z*=HDY5pNKR%OynE z!DECMQLnpK3KCA53h-^etD#EztahNoly*BG1@LCFeXxYfTeR4(8IB)iJ zb5rzweGwI(oTJzG1Jw_v+(a0sf5F(_o1u+6XXshY?A#x;+IPfkBoePBA-diBfs|rq zi};c1>bU3RqwGn=Ce6l~ka?VZw+=@s5>!jMm*yYdo~VNWg;_E~!RX>xx%?I+c%UAdFH)5%)T^ z6B{GcvB61mHEd|h978pfXra{ujn)JbA16xqawQ!Gee66MP$4WVSjGWTjX+5pM)O?Z z*q;+l5@e^?wu18VDh!iItOLRHDzs5Gb)fk#4zZB|Yx9Li$`H{F?TceVV{DO>Lm=o& zOdtv163LKuowul=f@;ckP}+`pAu3|;gxMsN;;4YOG*`iKJcczno+r=q45EBOn)5@JD!i`+{O=hrA>(_3U*LiI|r#}{RUI+R>Kw|TRBe! 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We aim\nto clarify the form of the matrices, how they are parametrized,\nand how the breaking of spherical symmetry can switch on additional elements\nthat one might not anticipate. Our example is based on a $d$ atomic shell.\n" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Create matrix\nThe Coulomb interaction between two particles can be written as\n\n .. math::\n \\begin{equation}\n \\hat{H} = \\frac{1}{2}\n \\int d\\mathbf{r} \\int d\\mathbf{r}^\\prime\n \\Sigma_{\\sigma, \\sigma^\\prime}\n |\\hat{\\psi}^\\sigma(\\mathbf{r})|^2 \\frac{e^2}{R}\n |\\hat{\\psi}^{\\sigma^\\prime}(\\mathbf{r})|^2,\n \\end{equation}\n\nwhere $\\hat{\\psi}^\\sigma(\\mathbf{r})$ is the electron wavefunction, with\nspin $\\sigma$, and $R=|r-r^\\prime|$ is the electron separation.\nSolving our problem in this form is difficult due to the need to symmeterize\nthe wavefunction to follow fermionic statistics.\nUsing second quantization, we can use operators to impose the required\nparticle exchange statistics and write the equation in terms of\na tensor $U$\n\n .. math::\n \\begin{equation}\n \\hat{H} = \\sum_{\\alpha,\\beta,\\gamma,\\delta,\\sigma,\\sigma^\\prime}\n U_{\\alpha\\sigma,\\beta\\sigma^\\prime,\\gamma\\sigma^\\prime,\\delta\\sigma}\n \\hat{f}^{\\dagger}_{\\alpha\\sigma}\n \\hat{f}^{\\dagger}_{\\beta\\sigma^\\prime}\n \\hat{f}_{t\\sigma^\\prime}\\hat{f}_{\\delta\\sigma},\n \\end{equation}\n\nwhere $\\alpha$, $\\beta$, $\\gamma$, $\\delta$ are\norbital indices and $\\hat{f}^{\\dagger}$\n($\\hat{f}$) are the creation (anihilation) operators.\nFor a $d$-electron system, we have $10$ distinct spin-orbitals\n($5$ orbitals each with $2$ spins), which makes matrix the\n$10\\times10\\times10\\times10$ in total size.\nIn EDRIXS the matrix can be created as follows:\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "import edrixs\nimport numpy as np\nimport scipy\nimport matplotlib.pyplot as plt\nimport itertools\n\nF0, F2, F4 = 6.94, 14.7, 4.41\numat_chb = edrixs.get_umat_slater('d', F0, F2, F4)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We stored this under variable :code:`umat_chb` where \"cbh\" stands for\ncomplex harmonic basis, which is the default basis in EDRIXS.\n\n" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Parameterizing interactions\nEDRIXS parameterizes the interactions in $U$ via Slater integral\nparameters $F^{k}$. These relate to integrals of various spherical\nHarmonics as well as Clebsch-Gordon coefficients, Gaunt coefficients,\nand Wigner 3J symbols. Textbooks such as [1]_ can be used for further\nreference. If you are interested in the details of how\nEDRIXS does this (and you probably aren't) function :func:`.umat_slater`,\nconstructs the required matrix via Gaunt coeficents from\n:func:`.get_gaunt`. Two alternative parameterizations are common.\nThe first are the Racah parameters, which are\n\n .. math::\n \\begin{eqnarray}\n A &=& F^0 - \\frac{49}{441} F^4 \\\\\n B &=& \\frac{1}{49}F^2 - \\frac{5}{441}F^4 \\\\\n C &=& \\frac{35}{441}F^4.\n \\end{eqnarray}\n\nor an alternative form for the Slater integrals\n\n .. math::\n \\begin{eqnarray}\n F_0 &=& F^0 \\\\\n F_2 &=& \\frac{1}{49}F^2 \\\\\n F_4 &=& \\frac{1}{441}F^4,\n \\end{eqnarray}\n\nwhich involves different normalization parameters.\n\n" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Basis transform\nIf we want to use the real harmonic basis, we can use a tensor\ntransformation, which imposes the following orbital order\n$3z^2-r^2, xz, yz, x^2-y^2, xy$, each of which involves\n$\\uparrow, \\downarrow$ spin pairs. Let's perform this transformation and\nstore a list of these orbitals.\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "umat = edrixs.transform_utensor(umat_chb, edrixs.tmat_c2r('d', True))\norbitals = ['3z^2-r^2', 'xz', 'yz', 'x^2-y^2', 'xy']" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Interactions\nTensor $U$ is a series of matrix\nelements\n\n .. math::\n \\begin{equation}\n \\langle\\psi_{\\gamma,\\delta}^{\\bar{\\sigma},\\bar{\\sigma}^\\prime}\n |\\hat{H}|\n \\psi_{\\alpha,\\beta}^{\\sigma,\\sigma^\\prime}\\rangle\n \\end{equation}\n\nthe combination of which defines the energetic cost of pairwise\nelectron-electron interactions between states $\\alpha,\\sigma$\nand $\\beta,\\sigma^\\prime$. In EDRIXS we follow the convention of\nsumming over all orbital pairs. Some other texts count each pair of\nindices only once. The matrix elements here will consequently\nbe half the magnitude of those in other references.\nWe can express the interactions in terms of\nthe orbitals involved. It is common to distinguish \"direct Coulomb\" and\n\"exchange\" interactions. The former come from electrons in the same orbital\nand the later involve swapping orbital labels for electrons. We will use\n$U_0$ and $J$ as a shorthand for distinguishing these.\n\nBefore we describe the different types of interactions, we note that since\nthe Coulomb interaction is real, and due to the spin symmmetry properties\nof the process $U$ always obeys\n\n .. math::\n \\begin{equation}\n U_{\\alpha\\sigma,\\beta\\sigma^\\prime,\\gamma\\sigma^\\prime,\\delta\\sigma} =\n U_{\\beta\\sigma,\\alpha\\sigma^\\prime,\\delta\\sigma^\\prime,\\gamma\\sigma} =\n U_{\\delta\\sigma,\\gamma\\sigma^\\prime,\\beta\\sigma^\\prime,\\alpha\\sigma} =\n U_{\\gamma\\sigma,\\delta\\sigma^\\prime,\\alpha\\sigma^\\prime,\\beta\\sigma}.\n \\end{equation}\n\n\n### 1. Intra orbital\nThe direct Coulomb energy cost to double-occupy an orbital comes from terms\nlike $U_{\\alpha\\sigma,\\alpha\\bar\\sigma,\\alpha\\bar\\sigma,\\alpha\\sigma}$.\nIn this notation, we use $\\sigma^\\prime$ to denote that the matrix\nelement is summed over all pairs and $\\bar{\\sigma}$ to denote sums\nover all opposite spin pairs. Due to rotational symmetry, all these\nelements are the same and equal to\n\n .. math::\n \\begin{eqnarray}\n U_0 &=& \\frac{A}{2} + 2B + \\frac{3C}{2}\\\\\n &=& \\frac{F_0}{2} + 2F_2 + 18F_4\n \\end{eqnarray}\n\nLet's print these to demonstrate where these live in the array\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "for i in range(0, 5):\n val = umat[i*2, i*2 + 1, i*2 + 1, i*2].real\n print(f\"{orbitals[i]:<8} \\t {val:.3f}\")" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "### 2. Inter orbital Coulomb interactions\nDirect Coulomb repulsion between different orbitals depends on terms like\n$U_{\\alpha\\sigma,\\beta\\sigma^\\prime,\\beta\\sigma^\\prime,\\alpha\\sigma}$.\nExpresions for these parameters are provided in column $U$ in\n`table_2_orbital`. We can print the values from :code:`umat`\nlike this:\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "for i, j in itertools.combinations(range(5), 2):\n val = umat[i*2, j*2 + 1, j*2 + 1, i*2].real\n print(f\"{orbitals[i]:<8} \\t {orbitals[j]:<8} \\t {val:.3f}\")" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "### 3. Inter-orbital exchange interactions\nExchange terms exist with the form\n$U_{\\alpha\\sigma,\\beta\\sigma^\\prime,\\alpha\\sigma^\\prime,\\beta\\sigma}$.\nExpresions for these parameters are provided in column $J$ of\n`table_2_orbital`. These come from terms like this in the matrix:\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "for i, j in itertools.combinations(range(5), 2):\n val = umat[i*2, j*2 + 1, i*2 + 1, j*2].real\n print(f\"{orbitals[i]:<8} \\t {orbitals[j]:<8} \\t {val:.3f}\")" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "### 4. Pair hopping term\nTerms that swap pairs of electrons exist as\n$(1-\\delta_{\\sigma\\sigma'})U_{\\alpha\\sigma,\\alpha\\bar\\sigma,\\beta\\bar\\sigma,\\beta\\sigma}$\nand depend on exchange interactions column $J$ from\n`table_2_orbital`\nand here in the matrix.\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "for i, j in itertools.combinations(range(5), 2):\n val = umat[i*2, i*2 + 1, j*2 + 1, j*2].real\n print(f\"{orbitals[i]:<8} \\t {orbitals[j]:<8} \\t {val:.3f}\")" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "### 5. Three orbital\nAnother set of terms that one might not immediately anticipate involve three\norbitals like\n\n .. math::\n \\begin{equation}\n U_{\\alpha\\sigma, \\gamma\\sigma', \\beta\\sigma', \\gamma\\sigma} \\\\\n U_{\\alpha\\sigma, \\gamma\\sigma', \\gamma\\sigma', \\beta\\sigma} \\\\\n (1-\\delta_{\\sigma\\sigma'})\n U_{\\alpha\\sigma, \\beta\\sigma', \\gamma\\sigma', \\gamma\\sigma}\n \\end{equation}\n\nfor $\\alpha=3z^2-r^2, \\beta=x^2-y^2, \\gamma=xz/yz$.\nThese are needed to maintain the rotational symmetry of the interations.\nSee `table_3_orbital` for the expressions. We can print some of\nthese via:\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "ijkl = [[0, 1, 3, 1],\n [0, 2, 3, 2],\n [1, 0, 3, 1],\n [1, 1, 3, 0],\n [2, 0, 3, 2],\n [2, 2, 3, 0]]\n\nfor i, j, k, l in ijkl:\n val = umat[i*2, j*2 + 1, k*2 + 1, l*2].real\n print(f\"{orbitals[i]:<8} \\t {orbitals[j]:<8} \\t\"\n f\"{orbitals[k]:<8} \\t {orbitals[l]:<8} \\t {val:.3f}\")" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "### 6. Four orbital\nFuther multi-orbital terms include\n$U_{\\alpha\\sigma,\\beta\\sigma^\\prime,\\gamma\\sigma^\\prime,\\delta\\sigma}$.\nWe can find these here in the matrix:\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "ijkl = [[0, 1, 2, 4],\n [0, 1, 4, 2],\n [0, 2, 1, 4],\n [0, 2, 4, 1],\n [0, 4, 1, 2],\n [0, 4, 2, 1],\n [3, 1, 4, 2],\n [3, 2, 4, 1],\n [3, 4, 1, 2],\n [3, 4, 2, 1]]\n\nfor i, j, k, l in ijkl:\n val = umat[i*2, j*2 + 1, k*2 + 1, l*2].real\n print(f\"{orbitals[i]:<8} \\t {orbitals[j]:<8} \\t {orbitals[k]:<8}\"\n f\"\\t {orbitals[l]:<8} \\t {val:.3f}\")" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Effects of multi-orbital terms\nTo test the effects of the multi-orbital terms, let's plot the eigenenergy\nspectra with and without multi-orbital terms switched on for system with and\nwithout a cubic crystal field. We will use a $d$-shell with two\nelectrons.\n\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "ten_dqs = [0, 2, 4, 12]\n\ndef diagonalize(ten_dq, umat):\n emat = edrixs.cb_op(edrixs.cf_cubic_d(ten_dq),\n edrixs.tmat_c2r('d', ispin=True))\n H = (edrixs.build_opers(4, umat, basis)\n + edrixs.build_opers(2, emat, basis))\n e, v = scipy.linalg.eigh(H)\n return e - e.min()\n\nbasis = edrixs.get_fock_bin_by_N(10, 2)\numat_no_multiorbital = np.copy(umat)\nB = F2/49 - 5*F4/441\nfor val in [np.sqrt(3)*B/2, np.sqrt(3)*B, 3*B/2]:\n umat_no_multiorbital[(np.abs(umat)- val) < 1e-6] = 0\n\nfig, axs = plt.subplots(1, len(ten_dqs), figsize=(8, 3))\n\nfor cind, (ax, ten_dq) in enumerate(zip(axs, ten_dqs)):\n ax.hlines(diagonalize(ten_dq, umat), xmin=0, xmax=1,\n label='on', color=f'C{cind}')\n ax.hlines(diagonalize(ten_dq, umat_no_multiorbital),\n xmin=1.5, xmax=2.5,\n label='off',\n linestyle=':', color=f'C{cind}')\n ax.set_title(f\"$10D_q={ten_dq}$\")\n ax.set_ylim([-.5, 20])\n ax.set_xticks([])\n ax.legend()\n\nfig.suptitle(\"Eigenvalues with 3&4-orbital effects on/off\")\nfig.subplots_adjust(wspace=.3)\naxs[0].set_ylabel('Eigenvalues (eV)')\nfig.subplots_adjust(top=.8)\nplt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "On the left of the plot Coulomb interactions in spherical symmetry cause\nsubstantial mxing between $t_{2g}$ and $e_{g}$ orbitals in the\neigenstates and 3 & 4 orbital orbital terms are crucial for obtaining the\nthe right eigenenergies. As $10D_q$ get large, this mixing is switched\noff and the spectra start to become independent of whether the 3 & 4 orbital\norbital terms are included or not.\n\n\n\n.. table:: Table of 2 orbital interactions\n\n +-----------------------------+------------------+-----------------------+----------------+--------------------+\n |Orbitals $\\alpha,\\beta$|$U_0$ Racah | $U_0$ Slater |$J$ Racah |$J$ Slater |\n +=============================+==================+=======================+================+====================+\n |$3z^2-r^2, xz$ |$A/2+B+C/2$ |$F_0/2+F_2-12F_4$| $B/2+C/2$|$F_2/2+15F_4$ |\n +-----------------------------+------------------+-----------------------+----------------+--------------------+\n |$3z^2-r^2, yz$ |$A/2+B+C/2$ |$F_0/2+F_2-12F_4$| $B/2+C/2$|$F_2/2+15F_4$ |\n +-----------------------------+------------------+-----------------------+----------------+--------------------+\n |$3z^2-r^2, x^2-y^2$ |$A/2-2B+C/2$|$F_0/2-2F_2+3F_4$|$2B+C/2$ |$2F_2+15F_4/2$|\n +-----------------------------+------------------+-----------------------+----------------+--------------------+\n |$3z^2-r^2, xy$ |$A/2-2B+C/2$|$F_0/2-2F_2+3F_4$|$2B+C/2$ |$2F_2+15F_4/2$|\n +-----------------------------+------------------+-----------------------+----------------+--------------------+\n |$xz, yz$ |$A/2-B+C/2$ |$F_0/2-F_2-12F_4$|$3B/2+C/2$|$3F_2/2+10F_4$|\n +-----------------------------+------------------+-----------------------+----------------+--------------------+\n |$xz, x^2-y^2$ |$A/2-B+C/2$ |$F_0/2-F_2-12F_4$|$3B/2+C/2$|$3F_2/2+10F_4$|\n +-----------------------------+------------------+-----------------------+----------------+--------------------+\n |$xz, xy$ |$A/2-B+C/2$ |$F_0/2-F_2-12F_4$|$3B/2+C/2$|$3F_2/2+10F_4$|\n +-----------------------------+------------------+-----------------------+----------------+--------------------+\n |$yz, x^2-y^2$ |$A/2-B+C/2$ |$F_0/2-F_2-12F_4$|$3B/2+C/2$|$3F_2/2+10F_4$|\n +-----------------------------+------------------+-----------------------+----------------+--------------------+\n |$yz, xy$ |$A/2-B+C/2$ |$F_0/2-F_2-12F_4$|$3B/2+C/2$|$3F_2/2+10F_4$|\n +-----------------------------+------------------+-----------------------+----------------+--------------------+\n |$x^2-y^2, xy$ |$A/2+2B+C/2$|$F_0+4F_2-34F_4$ | $C/2$ |$35F_4/2$ |\n +-----------------------------+------------------+-----------------------+----------------+--------------------+\n\n\n.. table:: Table of 3 orbital interactions\n\n +-----------------------------+-------------+----------------------------------------------------+-----------------------------------------------------+\n |Orbitals $\\alpha,\\beta,\\gamma,\\delta$|$\\langle\\alpha\\beta|\\gamma\\delta\\rangle$ Racah|$\\langle\\alpha\\beta|\\gamma\\delta\\rangle$ Slater|\n +=============================+=============+====================================================+=====================================================+\n |$3z^2-r^2, xz, x^2-y^2, xz$ | $\\sqrt{3}B/2$ | $\\sqrt{3}F_2/2-5\\sqrt{3}F_4/2$ |\n +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+\n |$3z^2-r^2, yz, x^2-y^2, yz$ | $-\\sqrt{3}B/2$ | $-\\sqrt{3}F_2/2+5\\sqrt{3}F_4/2$ |\n +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+\n |$xz, 3z^2-r^2, x^2-y^2, xz$ | $-\\sqrt{3}B$ | $-\\sqrt{3}F_2+5\\sqrt{3}F_4$ |\n +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+\n |$xz, xz, x^2-y^2, 3z^2-r^2$ | $\\sqrt{3}B/2$ | $\\sqrt{3}F_2/2-5\\sqrt{3}F_4/2$ |\n +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+\n |$yz, 3z^2-r^2, x^2-y^2, yz$ | $\\sqrt{3}B$ | $\\sqrt{3}F_2-5\\sqrt{3}F_4$ |\n +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+\n |$yz, yz, x^2-y^2, 3z^2-r^2$ | $-\\sqrt{3}B/2$ | $-\\sqrt{3}F_2/2+5\\sqrt{3}F_4/2$ |\n +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+\n\n\n.. table:: Table of 4 orbital interactions\n\n +-----------------------------+-------------+----------------------------------------------------+-----------------------------------------------------+\n |Orbitals $\\alpha,\\beta,\\gamma,\\delta$|$\\langle\\alpha\\beta|\\gamma\\delta\\rangle$ Racah|$\\langle\\alpha\\beta|\\gamma\\delta\\rangle$ Slater|\n +=============================+=============+====================================================+=====================================================+\n |$3z^2-r^2, xz, yz, xy$ | $-\\sqrt{3}B$ | $-\\sqrt{3}F_2+5\\sqrt{3}F_4$ |\n +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+\n |$3z^2-r^2, xz, xy, yz$ | $\\sqrt{3}B/2$ | $\\sqrt{3}F_2/2-5\\sqrt{3}F_4/2$ |\n +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+\n |$3z^2-r^2, yz, xz, xy$ | $-\\sqrt{3}B$ | $-\\sqrt{3}F_2+5\\sqrt{3}F_4$ |\n +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+\n |$3z^2-r^2, yz, xy, xz$ | $\\sqrt{3}B/2$ | $\\sqrt{3}F_2/2-5\\sqrt{3}F_4/2$ |\n +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+\n |$3z^2-r^2, xy, xz, yz$ | $\\sqrt{3}B/2$ | $\\sqrt{3}F_2/2-5\\sqrt{3}F_4/2$ |\n +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+\n |$3z^2-r^2, xy, yz, xz$ | $\\sqrt{3}B/2$ | $\\sqrt{3}F_2/2-5\\sqrt{3}F_4/2$ |\n +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+\n |$x^2-y^2 , xz, xy, yz$ | $-3B/2$ | $-3F_2/2+15F_4/2$ |\n +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+\n |$x^2-y^2 , yz, xy, xz$ | $3B/2$ | $3F_2/2-15F_4/2$ |\n +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+\n |$x^2-y^2 , xy, xz, yz$ | $-3B/2$ | $-3F_2/2+15F_4/2$ |\n +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+\n |$x^2-y^2 , xy, yz, xz$ | $3B/2$ | $3F_2/2-15F_4/2$ |\n +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+\n\n\n.. rubric:: Footnotes\n\n.. [1] MSugano S, Tanabe Y and Kamimura H. 1970. Multiplets of\n Transition-Metal Ions in Crystals. Academic Press, New York and London.\n\n" - ] - } - ], - "metadata": { - "kernelspec": { - "display_name": "Python 3", - "language": "python", - "name": "python3" - }, - "language_info": { - "codemirror_mode": { - "name": "ipython", - "version": 3 - }, - "file_extension": ".py", - "mimetype": "text/x-python", - "name": "python", - "nbconvert_exporter": "python", - "pygments_lexer": "ipython3", - "version": "3.8.16" - } - }, - "nbformat": 4, - "nbformat_minor": 0 -} \ No newline at end of file diff --git a/edrixs/_images/energy_level.png b/edrixs/_images/energy_level.png deleted file mode 100644 index 9878f7d91ddad2c85716f6b62d7f18b90508eb95..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 14301 zcmYLw1yoeu8|@uH8l}Vmq=pa$VF)D!gaK&L%V0biSSV8`kMrU=RM2)koFX=f0<-kH5`h2gu*wU&O`3&D+k#%R$8R zu~Wu|JR1aYLmH^t4+1jRrvn4ne|}Wl+QcTh%RFLiOLw`Fm*TSJ7>3s z$hG>1PnEUy_1@$<yXvpX%z{EiXy^(6%b1*UqhXKzYLjZJXG 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Source code for edrixs.angular_momentum

-__all__ = ['get_ladd', 'get_lminus', 'get_lx', 'get_ly', 'get_lz', 'get_orb_momentum',
-           'get_pauli', 'get_sx', 'get_sy', 'get_sz', 'get_spin_momentum', 'euler_to_rmat',
-           'rmat_to_euler', 'where_is_angle', 'dmat_spinor', 'zx_to_rmat', 'get_wigner_dmat',
-           'cf_cubic_d', 'cf_tetragonal_d', 'cf_square_planar_d', 'cf_trigonal_t2g']
-
-import numpy as np
-from .basis_transform import cb_op, tmat_r2c
-
-
-

-[docs]
-def get_ladd(ll, ispin=False):
-    """
-    Get the matrix form of the raising operator :math:`l^+` in the
-    complex spherical harmonics basis
-
-    .. math::
-
-        l^+|ll,m> = \\sqrt{(ll-m)(ll+m+1)} |ll,m+1>
-
-    Parameters
-    ----------
-    ll: int
-        Orbital angular momentum number.
-    ispin: logical
-        Whether including spin or not (default: False).
-
-    Returns
-    -------
-    ladd: 2d complex array
-        The matrix form of :math:`l^+`.
-
-        If ispin=True, the dimension will be :math:`2(2ll+1) \\times 2(2ll+1)`,
-
-        otherwise, it will be :math:`(2ll+1) \\times (2ll+1)`.
-    """
-
-    norbs = 2 * ll + 1
-    ladd = np.zeros((norbs, norbs), dtype=np.complex128)
-    cone = np.complex128(1.0 + 0.0j)
-    for m in range(-ll, ll):
-        ladd[ll + m + 1, ll + m] = np.sqrt((ll - m) * (ll + m + 1.0) * cone)
-    if ispin:
-        ladd_spin = np.zeros((2 * norbs, 2 * norbs), dtype=np.complex128)
-        ladd_spin[0:2 * norbs:2, 0:2 * norbs:2] = ladd
-        ladd_spin[1:2 * norbs:2, 1:2 * norbs:2] = ladd
-        return ladd_spin
-    else:
-        return ladd

-
-
-
-

-[docs]
-def get_lminus(ll, ispin=False):
-    """
-    Get the matrix form of the lowering operator :math:`l^-` in the
-    complex spherical harmonics basis
-
-    .. math::
-
-        l^-|ll,m> = \\sqrt{(ll+m)(ll-m+1)} |ll,m-1>
-
-    Parameters
-    ----------
-    ll: int
-        Orbital angular momentum number.
-    ispin: logical
-        Whether including spin or not (default: False).
-
-    Returns
-    -------
-    lminus: 2d complex array
-        The matrix form of :math:`l^-`.
-
-        If ispin=True, the dimension will be :math:`2(2ll+1) \\times 2(2ll+1)`,
-
-        otherwise, it will be :math:`(2ll+1) \\times (2ll+1)`.
-    """
-
-    norbs = 2 * ll + 1
-    lminus = np.zeros((norbs, norbs), dtype=np.complex128)
-    cone = np.complex128(1.0 + 0.0j)
-    for m in range(-ll + 1, ll + 1):
-        lminus[ll + m - 1, ll + m] = np.sqrt((ll + m) * (ll - m + 1.0) * cone)
-    if ispin:
-        lminus_spin = np.zeros((2 * norbs, 2 * norbs), dtype=np.complex128)
-        lminus_spin[0:2 * norbs:2, 0:2 * norbs:2] = lminus
-        lminus_spin[1:2 * norbs:2, 1:2 * norbs:2] = lminus
-        return lminus_spin
-    else:
-        return lminus

-
-
-
-

-[docs]
-def get_lx(ll, ispin=False):
-    """
-    Get the matrix form of the orbital angular momentum
-    operator :math:`l_x` in the complex spherical harmonics basis,
-
-    .. math::
-
-        l_x = \\frac{1}{2} (l^+ + l^-)
-
-    Parameters
-    ----------
-    ll: int
-        Orbital angular momentum number.
-    ispin: logical
-        Whether including spin or not (default: False).
-
-    Returns
-    -------
-    lx: 2d complex array
-        The matrix form of :math:`l_x`.
-
-        If ispin=True, the dimension will be :math:`2(2ll+1) \\times 2(2ll+1)`,
-
-        otherwise, it will be :math:`(2ll+1) \\times (2ll+1)`.
-    """
-
-    if ispin:
-        return (get_ladd(ll, True) + get_lminus(ll, True)) / 2.0
-    else:
-        return (get_ladd(ll) + get_lminus(ll)) / 2.0

-
-
-
-

-[docs]
-def get_ly(ll, ispin=False):
-    """
-    Get the matrix form of the orbital angular momentum
-    operator :math:`l_y` in the complex spherical harmonics basis,
-
-    .. math::
-
-        l_y = \\frac{-i}{2} (l^+ - l^-)
-
-    Parameters
-    ----------
-    ll: int
-        Orbital angular momentum number.
-    ispin: logical
-        Whether including spin or not (default: False).
-
-    Returns
-    -------
-    ly: 2d complex array
-        The matrix form of :math:`l_y`.
-
-        If ispin=True, the dimension will be :math:`2(2ll+1) \\times 2(2ll+1)`,
-
-        otherwise, it will be :math:`(2ll+1) \\times (2ll+1)`.
-    """
-
-    if ispin:
-        return (get_ladd(ll, True) - get_lminus(ll, True)) / 2.0 * -1j
-    else:
-        return (get_ladd(ll) - get_lminus(ll)) / 2.0 * -1j

-
-
-
-

-[docs]
-def get_lz(ll, ispin=False):
-    """
-    Get the matrix form of the orbital angular momentum
-    operator :math:`l_z` in the complex spherical harmonics basis.
-
-    Parameters
-    ----------
-    ll: int
-        Orbital angular momentum number.
-    ispin: logical
-        Whether including spin or not (default: False).
-
-    Returns
-    -------
-    lz: 2d complex array
-        The matrix form of :math:`l_z`.
-
-        If ispin=True, the dimension will be :math:`2(2ll+1) \\times 2(2ll+1)`,
-
-        otherwise, it will be :math:`(2ll+1) \\times (2ll+1)`.
-    """
-    norbs = 2 * ll + 1
-    lz = np.zeros((norbs, norbs), dtype=np.complex128)
-    for m in range(-ll, ll + 1):
-        lz[ll + m, ll + m] = m
-    if ispin:
-        lz_spin = np.zeros((2 * norbs, 2 * norbs), dtype=np.complex128)
-        lz_spin[0:2 * norbs:2, 0:2 * norbs:2] = lz
-        lz_spin[1:2 * norbs:2, 1:2 * norbs:2] = lz
-        return lz_spin
-    else:
-        return lz

-
-
-
-

-[docs]
-def get_orb_momentum(ll, ispin=False):
-    """
-    Get the matrix form of the orbital angular momentum
-    operator :math:`l_x, l_y, l_z` in the complex spherical harmonics basis.
-
-    Parameters
-    ----------
-    ll: int
-        Orbital angular momentum number.
-    ispin: logical
-        Whether including spin or not (default: False).
-
-    Returns
-    -------
-    res: 3d complex array
-        The matrix form of
-
-        - res[0]: :math:`l_x`
-
-        - res[1]: :math:`l_y`
-
-        - res[2]: :math:`l_z`
-
-        If ispin=True, the dimension will be :math:`3 \\times 2(2ll+1) \\times 2(2ll+1)`,
-
-        otherwise, it will be :math:`3 \\times (2ll+1) \\times (2ll+1)`.
-    """
-    norbs = 2 * ll + 1
-    if ispin:
-        res = np.zeros((3, 2*norbs, 2*norbs), dtype=np.complex128)
-    else:
-        res = np.zeros((3, norbs, norbs), dtype=np.complex128)
-    res[0] = get_lx(ll, ispin)
-    res[1] = get_ly(ll, ispin)
-    res[2] = get_lz(ll, ispin)
-
-    return res

-
-
-
-

-[docs]
-def get_pauli():
-    """
-    Get the Pauli matrix
-
-    Returns
-    -------
-    sigma: 3d complex array, shape=(3, 2, 2)
-
-        - sigma[0] is :math:`\\sigma_x`,
-
-        - sigma[1] is :math:`\\sigma_y`,
-
-        - sigma[2] is :math:`\\sigma_z`,
-    """
-
-    sigma = np.zeros((3, 2, 2), dtype=np.complex128)
-    sigma[0, 0, 1] = 1.0
-    sigma[0, 1, 0] = 1.0
-    sigma[1, 0, 1] = -1.0j
-    sigma[1, 1, 0] = 1.0j
-    sigma[2, 0, 0] = 1.0
-    sigma[2, 1, 1] = -1.0
-
-    return sigma

-
-
-
-

-[docs]
-def get_sx(ll):
-    """
-    Get the matrix form of spin angular momentum operator :math:`s_x` in the
-    complex spherical harmonics basis.
-
-    Parameters
-    ----------
-    ll: int
-        Quantum number of orbital angular momentum.
-
-    Returns
-    -------
-    sx: 2d complex array.
-        Matrix form of :math:`s_x`, the dimension
-        is :math:`2(2ll+1) \\times 2(2ll+1)`,
-
-        Orbital order is: \\|-ll,up\\>, \\|-ll,down\\>, ...,
-        \\|+ll, up\\>, \\|+ll,down\\>.
-    """
-
-    norbs = 2 * (2 * ll + 1)
-    sx = np.zeros((norbs, norbs), dtype=np.complex128)
-    sigma = get_pauli()
-    for i in range(2 * ll + 1):
-        sx[2 * i:2 * i + 2, 2 * i:2 * i + 2] = sigma[0, :, :] / 2.0
-
-    return sx

-
-
-
-

-[docs]
-def get_sy(ll):
-    """
-    Get the matrix form of spin angular momentum operator :math:`s_y` in the
-    complex spherical harmonics basis.
-
-    Parameters
-    ----------
-    ll: int
-        Quantum number of orbital angular momentum.
-
-    Returns
-    -------
-    sy: 2d complex array.
-        Matrix form of :math:`s_y`, the dimension
-        is :math:`2(2ll+1) \\times 2(2ll+1)`, spin order is:
-
-        Orbital order is: \\|-ll,up\\>, \\|-ll,down\\>, ...,
-        \\|+ll, up\\>, \\|+ll,down\\>
-    """
-
-    norbs = 2 * (2 * ll + 1)
-    sy = np.zeros((norbs, norbs), dtype=np.complex128)
-    sigma = get_pauli()
-    for i in range(2 * ll + 1):
-        sy[2 * i:2 * i + 2, 2 * i:2 * i + 2] = sigma[1, :, :] / 2.0
-
-    return sy

-
-
-
-

-[docs]
-def get_sz(ll):
-    """
-    Get the matrix form of spin angular momentum operator :math:`s_z` in the
-    complex spherical harmonics basis.
-
-    Parameters
-    ----------
-    ll: int
-        Quantum number of orbital angular momentum.
-
-    Returns
-    -------
-    sz: 2d complex array.
-        Matrix form of :math:`s_z`, the dimension
-        is :math:`2(2ll+1) \\times 2(2ll+1)`.
-
-        Orbital order is: \\|-ll,up\\>, \\|-ll,down\\>, ...,
-        \\|+ll, up\\>, \\|+ll,down\\>
-    """
-
-    norbs = 2 * (2 * ll + 1)
-    sz = np.zeros((norbs, norbs), dtype=np.complex128)
-    sigma = get_pauli()
-    for i in range(2 * ll + 1):
-        sz[2 * i:2 * i + 2, 2 * i:2 * i + 2] = sigma[2, :, :] / 2.0
-
-    return sz

-
-
-
-

-[docs]
-def get_spin_momentum(ll):
-    """
-    Get the matrix form of the spin angular momentum
-    operator :math:`s_x, s_y, s_z` in the complex spherical harmonics basis.
-
-    Parameters
-    ----------
-    ll: int
-        Orbital angular momentum number.
-
-    Returns
-    -------
-    res: 3d complex array
-        The matrix form of
-
-        - res[0]: :math:`s_x`
-
-        - res[1]: :math:`s_y`
-
-        - res[2]: :math:`s_z`
-
-        the dimension is :math:`3 \\times 2(2ll+1) \\times 2(2ll+1)`,
-
-        Orbital order is: \\|-ll,up\\>, \\|-ll,down\\>, ...,
-        \\|+ll, up\\>, \\|+ll,down\\>
-    """
-    norbs = 2 * (2 * ll + 1)
-    res = np.zeros((3, norbs, norbs), dtype=np.complex128)
-    res[0] = get_sx(ll)
-    res[1] = get_sy(ll)
-    res[2] = get_sz(ll)
-
-    return res

-
-
-
-

-[docs]
-def euler_to_rmat(alpha, beta, gamma):
-    """
-    Given Euler angle: :math:`\\alpha, \\beta, \\gamma`,
-    generate the :math:`3 \\times 3` rotational matrix :math:`R`.
-
-    Parameters
-    ----------
-    alpha: float
-        Euler angle, in radian, [0, :math:`2\\pi`]
-    beta: float
-        Euler angle, in radian, [0, :math:`\\pi`]
-    gamma: float
-        Euler angle, in radian, [0, :math:`2\\pi`]
-
-    Returns
-    -------
-    rmat: 2d float array
-        The :math:`3 \\times 3` rotational matrix.
-    """
-
-    rmat = np.zeros((3, 3), dtype=np.float64)
-    rmat[0, 0] = np.cos(alpha) * np.cos(beta) * np.cos(gamma) - np.sin(alpha) * np.sin(gamma)
-    rmat[0, 1] = -np.sin(gamma) * np.cos(alpha) * np.cos(beta) - np.sin(alpha) * np.cos(gamma)
-    rmat[0, 2] = np.cos(alpha) * np.sin(beta)
-    rmat[1, 0] = np.sin(alpha) * np.cos(beta) * np.cos(gamma) + np.cos(alpha) * np.sin(gamma)
-    rmat[1, 1] = -np.sin(gamma) * np.sin(alpha) * np.cos(beta) + np.cos(alpha) * np.cos(gamma)
-    rmat[1, 2] = np.sin(alpha) * np.sin(beta)
-    rmat[2, 0] = -np.cos(gamma) * np.sin(beta)
-    rmat[2, 1] = np.sin(beta) * np.sin(gamma)
-    rmat[2, 2] = np.cos(beta)
-    return rmat

-
-
-
-

-[docs]
-def rmat_to_euler(rmat):
-    """
-    Given the :math:`3 \\times 3` rotational matrix :math:`R`, return the Euler
-    angles: :math:`\\alpha, \\beta, \\gamma`.
-
-    Parameters
-    ----------
-    rmat:  2d float array
-        The :math:`3 \\times 3` rotational matrix :math:`R`.
-
-    Returns
-    -------
-    alpha: float
-        Euler angle :math:`\\alpha` in radian, [0, :math:`2\\pi`].
-    beta: float
-        Euler angle :math:`\\beta` in radian, [0, :math:`\\pi`].
-    gamma: float
-        Euler angle :math:`\\gamma` in radian, [0, :math:`2\\pi`]
-
-    """
-
-    if np.abs(rmat[2, 2]) < 1.0:
-        beta = np.arccos(rmat[2, 2])
-
-        cos_gamma = -rmat[2, 0] / np.sin(beta)
-        sin_gamma = rmat[2, 1] / np.sin(beta)
-        gamma = where_is_angle(sin_gamma, cos_gamma)
-
-        cos_alpha = rmat[0, 2] / np.sin(beta)
-        sin_alpha = rmat[1, 2] / np.sin(beta)
-        alpha = where_is_angle(sin_alpha, cos_alpha)
-    else:
-        if rmat[2, 2] > 0:
-            beta = 0.0
-        else:
-            beta = np.pi
-        gamma = 0.0
-        alpha = np.arccos(rmat[1, 1])
-    return alpha, beta, gamma

-
-
-
-

-[docs]
-def where_is_angle(sina, cosa):
-    """
-    Given sine and cosine of an angle :math:`\\alpha`, return the
-    angle :math:`\\alpha` range from [0, :math:`2\\pi`].
-
-    Parameters
-    ----------
-    sina:  float
-        :math:`\\sin(\\alpha)`.
-    cosa:  float
-        :math:`\\cos(\\alpha)`.
-
-    Returns
-    -------
-    alpha: float
-        The angle :math:`\\alpha` in radian [0, :math:`2\\pi`].
-    """
-
-    if cosa > 1.0:
-        cosa = 1.0
-    elif cosa < -1.0:
-        cosa = -1.0
-    alpha = np.arccos(cosa)
-    if sina < 0.0:
-        alpha = 2.0 * np.pi - alpha
-    return alpha

-
-
-
-

-[docs]
-def dmat_spinor(alpha, beta, gamma):
-    """
-    Given three Euler angle: :math:`\\alpha, \\beta, \\gamma`,
-    return the transformation
-    matrix for :math:`\\frac{1}{2}`-spinor.
-
-    Parameters
-    ----------
-    alpha: float
-        Euler angle :math:`\\alpha` in radian [0, :math:`2\\pi`].
-    beta: float
-        Euler angle :math:`\\beta` in radian [0, :math:`\\pi`].
-    gamma: float
-        Euler angle :math:`\\gamma` in radian [0, :math:`2\\pi`].
-
-    Returns
-    -------
-    dmat:  2d complex array
-        The :math:`2 \\times 2` transformation matrix.
-    """
-
-    dmat = np.zeros((2, 2), dtype=np.complex128)
-    dmat[0, 0] = np.exp(-(alpha + gamma) / 2.0 * 1j) * np.cos(beta / 2.0)
-    dmat[0, 1] = -np.exp(-(alpha - gamma) / 2.0 * 1j) * np.sin(beta / 2.0)
-    dmat[1, 0] = np.exp((alpha - gamma) / 2.0 * 1j) * np.sin(beta / 2.0)
-    dmat[1, 1] = np.exp((alpha + gamma) / 2.0 * 1j) * np.cos(beta / 2.0)
-    return dmat

-
-
-
-

-[docs]
-def zx_to_rmat(z, x):
-    """
-    Given :math:`z` vector and :math:`x` vector, calculate :math:`y` vector
-    which satisfies the right-hand Cartesian coordinate and normalize them to
-    unit if needed, and then return the :math:`3 \\times 3`
-    rotational matrix :math:`R`.
-
-    Parameters
-    ----------
-    z: 1d float array
-        The :math:`z` vector.
-    x: 1d float array
-        The :math:`x` vector.
-
-    Returns
-    -------
-    rmat: 2d float array
-        The :math:`3 \\times 3` rotational matrix :math:`R`.
-    """
-
-    z = np.array(z, dtype=np.float64)
-    x = np.array(x, dtype=np.float64)
-    xx = x / np.sqrt(np.dot(x, x))
-    zz = z / np.sqrt(np.dot(z, z))
-    yy = np.cross(zz, xx)
-
-    rmat = np.zeros((3, 3), dtype=np.float64)
-    rmat[:, 0] = xx
-    rmat[:, 1] = yy
-    rmat[:, 2] = zz
-
-    return rmat

-
-
-
-

-[docs]
-def get_wigner_dmat(quant_2j, alpha, beta, gamma):
-    """
-    Given quantum number and Euler angles, return the Wigner-D matrix.
-
-    Parameters
-    ----------
-    quant_2j: int
-        Twice of the quantum number j: 2j, for example, quant_2j=1 means j=1/2,
-        quant_2j=2 means j=1
-    alpha: float number
-        The first Euler angle :math:`\\alpha` in radian [0, :math:`2\\pi`].
-    beta: float number
-        The second Euler angle :math:`\\beta` in radian [0, :math:`\\pi`].
-    gamma: float number
-        The third Euler angle :math:`\\gamma` in radian [0, :math:`2\\pi`].
-
-    Returns
-    -------
-    result: 2d complex array, shape(quant_2j+1, quant_2j+1)
-        The Wigner D-matrix.
-        For :math:`j=1/2`, the orbital order is: +1/2 (spin up), -1/2 (spin down).
-        For :math:`j>1/2`, the orbital order is: :math:`-j, -j+1, ..., +j`
-
-    Examples
-    --------
-    >>> import edrixs
-    spin-1/2 D-matrix
-    >>> edrixs.get_wigner_dmat(1, 1, 2, 3)
-    array([[-0.224845-0.491295j, -0.454649-0.708073j],
-           [ 0.454649-0.708073j, -0.224845+0.491295j]])
-    j=1 D-matrix
-    >>> edrixs.get_wigner_dmat(2, 1, 2, 3)
-    array([[-0.190816-0.220931j,  0.347398+0.541041j, -0.294663-0.643849j],
-           [ 0.636536-0.090736j, -0.416147+0.j      , -0.636536-0.090736j],
-           [-0.294663+0.643849j, -0.347398+0.541041j, -0.190816+0.220931j]])
-    """
-
-    from sympy.physics.quantum.spin import Rotation
-    from sympy import N, S
-    ndim = quant_2j + 1
-    result = np.zeros((ndim, ndim), dtype=complex)
-    # For j=1/2, we use different orbital order: first +1/2, then -1/2
-    if quant_2j == 1:
-        for i, mi in enumerate(range(quant_2j, -quant_2j-1, -2)):
-            for j, mj in enumerate(range(quant_2j, -quant_2j-1, -2)):
-                rot = Rotation.D(S(quant_2j)/2, S(mi)/2, S(mj)/2, alpha, beta, gamma)
-                result[i, j] = N(rot.doit())
-    # For j > 1/2, the order is -j, -j+1, ..., +j
-    else:
-        for i, mi in enumerate(range(-quant_2j, quant_2j+1, 2)):
-            for j, mj in enumerate(range(-quant_2j, quant_2j+1, 2)):
-                rot = Rotation.D(S(quant_2j)/2, S(mi)/2, S(mj)/2, alpha, beta, gamma)
-                result[i, j] = N(rot.doit())
-
-    return result

-
-
-
-

-[docs]
-def cf_cubic_d(ten_dq):
-    """
-    Given 10Dq, return cubic crystal field matrix for d orbitals
-    in the complex harmonics basis.
-
-    Parameters
-    ----------
-    ten_dq: float scalar
-        The splitting between :math:`eg` and :math:`t2g` orbitals.
-
-    Returns
-    -------
-    cf: 2d complex array, shape=(10, 10)
-        The matrix form of crystal field Hamiltonian in complex harmonics basis.
-    """
-
-    tmp = np.zeros((5, 5), dtype=complex)
-    tmp[0, 0] = 0.6 * ten_dq  # dz2
-    tmp[1, 1] = -0.4 * ten_dq  # dzx
-    tmp[2, 2] = -0.4 * ten_dq  # dzy
-    tmp[3, 3] = 0.6 * ten_dq  # dx2-y2
-    tmp[4, 4] = -0.4 * ten_dq  # dxy
-    cf = np.zeros((10, 10), dtype=complex)
-    cf[0:10:2, 0:10:2] = tmp
-    cf[1:10:2, 1:10:2] = tmp
-
-    cf[:, :] = cb_op(cf, tmat_r2c('d', True))
-
-    return cf

-
-
-
-

-[docs]
-def cf_tetragonal_d(ten_dq, d1, d3):
-    """
-    Given 10Dq, d1, d3, return tetragonal crystal field matrix for d orbitals
-    in the complex harmonics basis.
-
-    Parameters
-    ----------
-    ten_dq: float scalar
-        Parameter used to label cubic crystal splitting.
-    d1: float scalar
-        Paramter used to label tetragonal splitting.
-    d3: float scalar
-        Paramter used to label tetragonal splitting.
-
-    Returns
-    -------
-    cf: 2d complex array, shape=(10, 10)
-        The matrix form of crystal field Hamiltonian in complex harmonics basis.
-    """
-
-    dt = (3.0 * d3 - 4.0 * d1) / 35
-    ds = (d3 + d1) / 7.0
-    dq = ten_dq / 10.0
-
-    tmp = np.zeros((5, 5), dtype=complex)
-    tmp[0, 0] = 6 * dq - 2 * ds - 6 * dt  # d3z2-r2
-    tmp[1, 1] = -4 * dq - 1 * ds + 4 * dt  # dzx
-    tmp[2, 2] = -4 * dq - 1 * ds + 4 * dt  # dzy
-    tmp[3, 3] = 6 * dq + 2 * ds - 1 * dt  # dx2-y2
-    tmp[4, 4] = -4 * dq + 2 * ds - 1 * dt  # dxy
-
-    cf = np.zeros((10, 10), dtype=complex)
-    cf[0:10:2, 0:10:2] = tmp
-    cf[1:10:2, 1:10:2] = tmp
-
-    cf[:, :] = cb_op(cf, tmat_r2c('d', True))
-
-    return cf

-
-
-
-

-[docs]
-def cf_trigonal_t2g(delta):
-    """
-    Given delta, return trigonal crystal field matrix for t2g orbitals
-    in the complex harmonics basis.
-
-    Parameters
-    ----------
-    delta: float scalar
-        Parameter used to label trigonal crystal splitting.
-
-    Returns
-    -------
-    cf: 2d complex array, shape=(6, 6)
-        The matrix form of crystal field Hamiltonian in complex harmonics basis.
-    """
-
-    tmp = np.array([[0, delta, delta],
-                    [delta, 0, delta],
-                    [delta, delta, 0]])
-    cf = np.zeros((6, 6), dtype=complex)
-    cf[0:6:2, 0:6:2] += tmp
-    cf[1:6:2, 1:6:2] += tmp
-    cf[:, :] = cb_op(cf, tmat_r2c('t2g', True))
-
-    return cf

-
-
-
-

-[docs]
-def cf_square_planar_d(ten_dq, ds):
-    """
-    Given 10Dq, ds, return square planar crystal field matrix for d orbitals
-    in the complex harmonics basis. This is the limit of strong tetragonal
-    distortion with two axial ligands at infinity. Note that in this case the
-    three parameters, ten_dq, ds, and dt, are no longer independent:
-    dt = 2/35*ten_dq and the levels depend on only two parameters
-    ten_dq and ds.
-
-    Parameters
-    ----------
-    ten_dq: float scalar
-        Parameter associated with eg-t2g splitting.
-    ds: float scalar
-        Paramter associated with splitting orbitals with
-        z-components.
-
-    Returns
-    -------
-    cf: 2d complex array, shape=(10, 10)
-        The matrix form of crystal field Hamiltonian in complex harmonics basis.
-    """
-    tmp = np.zeros((5, 5), dtype=complex)
-    tmp[0, 0] = 9/35*ten_dq - 2*ds  # d3z2-r2
-    tmp[1, 1] = -6/35*ten_dq - ds  # dzx
-    tmp[2, 2] = -6/35*ten_dq - ds  # dzy
-    tmp[3, 3] = 19/35*ten_dq + 2*ds  # dx2-y2
-    tmp[4, 4] = -16/35*ten_dq + 2*ds  # dxy
-
-    cf = np.zeros((10, 10), dtype=complex)
-    cf[0:10:2, 0:10:2] = tmp
-    cf[1:10:2, 1:10:2] = tmp
-
-    cf[:, :] = cb_op(cf, tmat_r2c('d', True))
-
-    return cf

-
-
-
-def cf_tetragonal_t2g(ten_dq, d1, d3):
-    """
-    Given 10Dq, d1, d3, return tetragonal crystal field matrix for t2g orbitals
-    in the complex harmonics basis.
-
-    Parameters
-    ----------
-    ten_dq: float scalar
-        Parameter used to label cubic crystal splitting.
-    d1: float scalar
-        Paramter used to label tetragonal splitting.
-    d3: float scalar
-        Paramter used to label tetragonal splitting.
-
-    Returns
-    -------
-    cf: 2d complex array, shape=(6, 6)
-        The matrix form of crystal field Hamiltonian in complex harmonics basis.
-    """
-    dt = (3.0 * d3 - 4.0 * d1) / 35
-    ds = (d3 + d1) / 7.0
-    dq = ten_dq / 10.0
-
-    tmp = np.zeros((3, 3), dtype=complex)
-    tmp[0, 0] = -4 * dq - 1 * ds + 4 * dt  # dxz
-    tmp[1, 1] = -4 * dq - 1 * ds + 4 * dt  # dyz
-    tmp[2, 2] = -4 * dq + 2 * ds - 1 * dt  # dxy
-
-    cf = np.zeros((6, 6), dtype=complex)
-    cf[0:6:2, 0:6:2] += tmp
-    cf[1:6:2, 1:6:2] += tmp
-    cf[:, :] = cb_op(cf, tmat_r2c('t2g', True))
-    return cf
-
- -
-
-
- -
- -
-

© Copyright 2019, Brookhaven National Lab.

-
- - Built with Sphinx using a - theme - provided by Read the Docs. - - -
-
-
-
-
- - - - \ No newline at end of file diff --git a/edrixs/_modules/edrixs/basis_transform.html b/edrixs/_modules/edrixs/basis_transform.html deleted file mode 100644 index aeb00c772c..0000000000 --- a/edrixs/_modules/edrixs/basis_transform.html +++ /dev/null @@ -1,646 +0,0 @@ - - - - - - - - edrixs.basis_transform — edrixs documentation - - - - - - - - - - - - - - - - - - - - - -
- - -
- -
-
-
- -
-
-
-
- -

Source code for edrixs.basis_transform

-__all__ = ['cb_op', 'cb_op2', 'tmat_c2r', 'tmat_r2c', 'tmat_r2cub_f',
-           'tmat_cub2r_f', 'tmat_c2j', 'transform_utensor', 'fourier_hr2hk']
-
-import numpy as np
-
-
-

-[docs]
-def cb_op(oper_O, TL, TR=None):
-    """
-    Change the basis of an operator :math:`\\hat{O}`.
-
-    .. math::
-
-        O^{\\prime} = (T_L)^{\\dagger} O (T_R),
-
-    Parameters
-    ----------
-    oper_O: array-like
-        At least 2-dimension, the last 2-dimension is the dimension of the
-        matrix form of operator :math:`\\hat{O}` in basis :math:`A`.
-        For example:
-
-        - oper_O.shape = (3, 10, 10), means 3 operators with dimension :math:`10 \\times 10`
-
-        - oper_O.shape = (2, 3, 10, 10), means :math:`2 \\times 3=6` operators with
-          dimension :math:`10 \\times 10`
-    TL: 2d array
-        The unitary transformation matrix from basis :math:`A` to
-        basis :math:`B`,  namely,
-
-        :math:`TL_{ij} = <\\psi^{A}_{i}|\\phi^{B}_{j}>`.
-
-    TR: 2d array
-        The unitary transformation matrix from basis :math:`A` to
-        basis :math:`B`,  namely,
-
-        :math:`TR_{ij} = <\\psi^{A}_{i}|\\phi^{B}_{j}>`.
-
-        if TR = None, TR = TL
-
-    Returns
-    -------
-    res: same shape as oper_O
-        The matrices form of operators :math:`\\hat{O}` in new basis.
-    """
-    oper_O = np.array(oper_O, order='C')
-    dim = oper_O.shape
-    if TR is None:
-        TR = TL
-    if len(dim) < 2:
-        raise Exception("Dimension of oper_O should be at least 2")
-    elif len(dim) == 2:
-        res = np.dot(np.dot(np.conj(np.transpose(TL)), oper_O), TR)
-    else:
-        tot = np.prod(dim[0:-2])
-        tmp_oper = oper_O.reshape((tot, dim[-2], dim[-1]))
-        for i in range(tot):
-            tmp_oper[i] = np.dot(np.dot(np.conj(np.transpose(TL)), tmp_oper[i]), TR)
-        res = tmp_oper.reshape(dim)
-
-    return res

-
-
-
-

-[docs]
-def cb_op2(oper_O, TL, TR):
-    """
-    Change the basis of an operator :math:`\\hat{O}`.
-
-    .. math::
-
-        O^{\\prime} = (T_L)^{\\dagger} O (T_R),
-
-
-    Parameters
-    ----------
-    oper_O: array-like
-        At least 2-dimension, the last 2-dimension is the dimension of the
-        matrix form of operator :math:`\\hat{O}` in basis :math:`A`.
-        For example:
-
-        - oper_O.shape = (3, 10, 10), means 3 operatos with dimension :math:`10 \\times 10`
-
-        - oper_O.shape = (2, 3, 10, 10), means :math:`2 \\times 3=6` operators with
-          dimension :math:`10 \\times 10`
-    TL: 2d array
-        The unitary transformation matrix applied on the left.
-    TR: 2d array
-        The unitary transformation matrix applied on the right.
-
-    Returns
-    -------
-    res: same shape as oper_O
-        The matrices form of operators :math:`\\hat{O}` in new basis.
-    """
-    oper_O = np.array(oper_O, order='C')
-    dim = oper_O.shape
-    if len(dim) < 2:
-        raise Exception("Dimension of oper_O should be at least 2")
-    elif len(dim) == 2:
-        res = np.dot(np.dot(np.conj(np.transpose(TL)), oper_O), TR)
-    else:
-        tot = np.prod(dim[0:-2])
-        tmp_oper = oper_O.reshape((tot, dim[-2], dim[-1]))
-        for i in range(tot):
-            tmp_oper[i] = np.dot(np.dot(np.conj(np.transpose(TL)), tmp_oper[i]), TR)
-        res = tmp_oper.reshape(dim)
-
-    return res

-
-
-
-

-[docs]
-def tmat_c2r(case, ispin=False):
-    """
-    Get the unitary transformation matrix from the basis of complex
-    spherical harmonics to real spherical harmonics.
-
-    Parameters
-    ----------
-    case: string
-        Label for different systems.
-
-        - 'p': for :math:`p`-shell
-        - 't2g': for :math:`t_{2g}`-shell
-        - 'd': for :math:`d`-shell
-        - 'f': for :math:`f`-shell
-    ispin: logical
-        Whether including spin degree of freedom or not (default: False).
-
-    Returns
-    -------
-    t_c2r: 2d complex array
-        The transformation matrix.
-    """
-
-    sqrt2 = np.sqrt(2.0)
-    ci = np.complex128(0.0 + 1.0j)
-    cone = np.complex128(1.0 + 0.0j)
-
-    # p orbitals: px, py, pz
-    if case.strip() == 'p':
-        norbs = 3
-        t_c2r = np.zeros((norbs, norbs), dtype=np.complex128)
-        # px=1/sqrt(2)( |1,-1> - |1,1> )
-        t_c2r[0, 0] = cone / sqrt2
-        t_c2r[2, 0] = -cone / sqrt2
-        # py=i/sqrt(2)( |1,-1> + |1,1> )
-        t_c2r[0, 1] = ci / sqrt2
-        t_c2r[2, 1] = ci / sqrt2
-        # pz=|1,0>
-        t_c2r[1, 2] = cone
-
-    # t2g orbitals in the t2g subspace, here, we use the so-called
-    # T-P equivalence, t2g orbitals behave like the effective orbital
-    # angular momentum leff=1
-    # dzx ~ py,  dzy ~ px, dxy ~ pz
-    elif case.strip() == 't2g':
-        norbs = 3
-        t_c2r = np.zeros((norbs, norbs), dtype=np.complex128)
-        # dzx --> py=i/sqrt(2)( |1,-1> + |1,1> )
-        t_c2r[0, 0] = ci / sqrt2
-        t_c2r[2, 0] = ci / sqrt2
-        # dzy --> px=1/sqrt(2)( |1,-1> - |1,1> )
-        t_c2r[0, 1] = cone / sqrt2
-        t_c2r[2, 1] = -cone / sqrt2
-        # dxy --> pz=|1,0>
-        t_c2r[1, 2] = cone
-
-    # d orbitals: dz2, dzx, dzy, dx2-y2, dxy
-    elif case.strip() == 'd':
-        norbs = 5
-        t_c2r = np.zeros((norbs, norbs), dtype=np.complex128)
-        # dz2=|2,0>
-        t_c2r[2, 0] = cone
-        # dzx=1/sqrt(2)( |2,-1> - |2,1> )
-        t_c2r[1, 1] = cone / sqrt2
-        t_c2r[3, 1] = -cone / sqrt2
-        # dzy=i/sqrt(2)( |2,-1> + |2,1> )
-        t_c2r[1, 2] = ci / sqrt2
-        t_c2r[3, 2] = ci / sqrt2
-        # dx2-y2=1/sqrt(2)( |2,-2> + |2,2> )
-        t_c2r[0, 3] = cone / sqrt2
-        t_c2r[4, 3] = cone / sqrt2
-        # dxy=i/sqrt(2)( |2,-2> - |2,2> )
-        t_c2r[0, 4] = ci / sqrt2
-        t_c2r[4, 4] = -ci / sqrt2
-
-    # f orbitals, please NOTE that this real form of the f orbitals is not the
-    # basis of the representation of the cubic point group, please call the
-    # function ``tmat_r2cub" to get the transformation matrix from this basis
-    # to the cubic basis that is the representation of the cubic point group.
-    elif case.strip() == 'f':
-        norbs = 7
-        t_c2r = np.zeros((norbs, norbs), dtype=np.complex128)
-        # fz3 = |3,0>
-        t_c2r[3, 0] = cone
-        # fxz2 = 1/sqrt(2)( |3,-1> - |3,1> )
-        t_c2r[2, 1] = cone / sqrt2
-        t_c2r[4, 1] = -cone / sqrt2
-        # fyz2 = i/sqrt(2)( |3,-1> + |3,1> )
-        t_c2r[2, 2] = ci / sqrt2
-        t_c2r[4, 2] = ci / sqrt2
-        # fz(x2-y2) = 1/sqrt(2)( |3,-2> + |3,2> )
-        t_c2r[1, 3] = cone / sqrt2
-        t_c2r[5, 3] = cone / sqrt2
-        # fxyz = i/sqrt(2)( |3,-2> - |3,2> )
-        t_c2r[1, 4] = ci / sqrt2
-        t_c2r[5, 4] = -ci / sqrt2
-        # fx(x2-3y2) = 1/sqrt(2) ( |3,-3> - |3,3> )
-        t_c2r[0, 5] = cone / sqrt2
-        t_c2r[6, 5] = -cone / sqrt2
-        # fy(3x2-y2) = i/sqrt(2) ( |3,-3> + |3,3> )
-        t_c2r[0, 6] = ci / sqrt2
-        t_c2r[6, 6] = ci / sqrt2
-    else:
-        raise Exception("error in tmat_c2r: Do NOT support tmat_c2r for this case: ", case)
-
-    # the spin order is: up dn up dn ... up dn
-    if ispin:
-        ntot_orbs = 2 * norbs
-        t_c2r_spin = np.zeros((ntot_orbs, ntot_orbs), dtype=np.complex128)
-        # spin up
-        t_c2r_spin[0:ntot_orbs:2, 0:ntot_orbs:2] = t_c2r
-        # spin dn
-        t_c2r_spin[1:ntot_orbs:2, 1:ntot_orbs:2] = t_c2r
-        return t_c2r_spin
-    else:
-        return t_c2r

-
-
-
-

-[docs]
-def tmat_r2c(case, ispin=False):
-    """
-    Get the unitary transformation matrix from the basis of real
-    spherical harmonics to complex spherical harmonics.
-
-    Parameters
-    ----------
-    case: string
-        Label for different systems.
-
-        - 'p': for :math:`p`-shell
-        - 't2g': for :math:`t_{2g}`-shell
-        - 'd': for :math:`d`-shell
-        - 'f': for :math:`f`-shell
-    ispin: logical
-        Whether including spin degree of freedom or not (default: False).
-
-    Returns
-    -------
-    t_r2c: 2d complex array
-        The transformation matrix.
-    """
-
-    t_r2c = np.conj(np.transpose(tmat_c2r(case, ispin)))
-    return t_r2c

-
-
-
-

-[docs]
-def tmat_r2cub_f(ispin=False):
-    """
-    Get the transformation matrix from real spherical harmonics to the
-    cubic spherical harmonics that is the representation of the cubic
-    point group, only for :math:`f` system.
-
-    Parameters
-    ----------
-    ispin: logical
-        Whether including spin degree of freedom or not (default: False).
-
-    Returns
-    -------
-    t_r2cub: 2d complex array
-        The transformation matrix.
-    """
-
-    a = np.sqrt(10.0) / 4.0 + 0.0j
-    b = np.sqrt(6.0) / 4.0 + 0.0j
-    c = 1.0 + 0.0j
-
-    norbs = 7
-    t_r2cub = np.zeros((norbs, norbs), dtype=np.complex128)
-
-    # T1u
-    # fx3 = -sqrt(6)/4 fxz2 + sqrt(10)/4 fx(x2-3y2)
-    t_r2cub[1, 0] = -b
-    t_r2cub[5, 0] = a
-    # fy3 = -sqrt(6)/4 fyz2 - sqrt(10)/4 fy(3x2-y2)
-    t_r2cub[2, 1] = -b
-    t_r2cub[6, 1] = -a
-    # fz3 = fz3
-    t_r2cub[0, 2] = c
-
-    # T2u
-    # fx(y2-z2) = -sqrt(10)/4 fxz2 - sqrt(6)/4 fx(x2-3y2)
-    t_r2cub[1, 3] = -a
-    t_r2cub[5, 3] = -b
-    # fy(z2-x2) = sqrt(10)/4 fyz2 - sqrt(6)/4 fy(3x2-y2)
-    t_r2cub[2, 4] = a
-    t_r2cub[6, 4] = -b
-    # fz(x2-y2) = fz(x2-y2)
-    t_r2cub[3, 5] = c
-
-    # A2u
-    # fxyz = fxyz
-    t_r2cub[4, 6] = c
-
-    if ispin:
-        ntot_orbs = 2 * norbs
-        t_r2cub_spin = np.zeros((ntot_orbs, ntot_orbs), dtype=np.complex128)
-        # spin up
-        t_r2cub_spin[0:ntot_orbs:2, 0:ntot_orbs:2] = t_r2cub
-        # spin dn
-        t_r2cub_spin[1:ntot_orbs:2, 1:ntot_orbs:2] = t_r2cub
-        return t_r2cub_spin
-    else:
-        return t_r2cub

-
-
-
-

-[docs]
-def tmat_cub2r_f(ispin=False):
-    """
-    Get the transformation matrix from the cubic spherical harmonics to
-    real spherical harmonics, only for :math:`f` system.
-
-    Parameters
-    ----------
-    ispin: logical
-        Whether including spin degree of freedom or not (default: False).
-
-    Returns
-    -------
-    t_cub2r: 2d complex array
-        The transformation matrix.
-    """
-
-    t_cub2r = np.conj(np.transpose(tmat_r2cub_f(ispin)))
-    return t_cub2r

-
-
-
-

-[docs]
-def tmat_c2j(orb_l):
-    """
-    Get the transformation matrix from the complex spherical harmonics to
-    the :math:`|j^2,j_z>` basis in which the spin-oribt coupling Hamiltonian
-    is diagonal. The orbital order is:
-
-    :math:`|j=l-1/2, -j>, |j=l-1/2, -j+1>, ... |j=l-1/2, +j>,`
-
-    :math:`|j=l+1/2, -j>, |j=l+1/2, -j+1>, ..., |j=l+1/2, +j>`.
-
-    Parameters
-    ----------
-    orb_l: int
-        Quantum number of orbital angular momentum.
-
-    Returns
-    -------
-    t_c2j: 2d complex array
-        The transformation matrix.
-    """
-
-    if orb_l == 1:
-        t_c2j = np.zeros((6, 6), dtype=np.complex128)
-        t_c2j[0, 0] = -np.sqrt(2.0 / 3.0)
-        t_c2j[3, 0] = np.sqrt(1.0 / 3.0)
-        t_c2j[2, 1] = -np.sqrt(1.0 / 3.0)
-        t_c2j[5, 1] = np.sqrt(2.0 / 3.0)
-        t_c2j[1, 2] = 1.0
-        t_c2j[0, 3] = np.sqrt(1.0 / 3.0)
-        t_c2j[3, 3] = np.sqrt(2.0 / 3.0)
-        t_c2j[2, 4] = np.sqrt(2.0 / 3.0)
-        t_c2j[5, 4] = np.sqrt(1.0 / 3.0)
-        t_c2j[4, 5] = 1.0
-
-        return t_c2j
-
-    elif orb_l == 2:
-        t_c2j = np.zeros((10, 10), dtype=np.complex128)
-        t_c2j[0, 0] = -np.sqrt(4.0 / 5.0)
-        t_c2j[3, 0] = np.sqrt(1.0 / 5.0)
-        t_c2j[2, 1] = -np.sqrt(3.0 / 5.0)
-        t_c2j[5, 1] = np.sqrt(2.0 / 5.0)
-        t_c2j[4, 2] = -np.sqrt(2.0 / 5.0)
-        t_c2j[7, 2] = np.sqrt(3.0 / 5.0)
-        t_c2j[6, 3] = -np.sqrt(1.0 / 5.0)
-        t_c2j[9, 3] = np.sqrt(4.0 / 5.0)
-        t_c2j[1, 4] = 1.0
-        t_c2j[0, 5] = np.sqrt(1.0 / 5.0)
-        t_c2j[3, 5] = np.sqrt(4.0 / 5.0)
-        t_c2j[2, 6] = np.sqrt(2.0 / 5.0)
-        t_c2j[5, 6] = np.sqrt(3.0 / 5.0)
-        t_c2j[4, 7] = np.sqrt(3.0 / 5.0)
-        t_c2j[7, 7] = np.sqrt(2.0 / 5.0)
-        t_c2j[6, 8] = np.sqrt(4.0 / 5.0)
-        t_c2j[9, 8] = np.sqrt(1.0 / 5.0)
-        t_c2j[8, 9] = 1.0
-
-        return t_c2j
-
-    elif orb_l == 3:
-        t_c2j = np.zeros((14, 14), dtype=np.complex128)
-        t_c2j[0, 0] = -np.sqrt(6.0 / 7.0)
-        t_c2j[3, 0] = np.sqrt(1.0 / 7.0)
-        t_c2j[2, 1] = -np.sqrt(5.0 / 7.0)
-        t_c2j[5, 1] = np.sqrt(2.0 / 7.0)
-        t_c2j[4, 2] = -np.sqrt(4.0 / 7.0)
-        t_c2j[7, 2] = np.sqrt(3.0 / 7.0)
-        t_c2j[6, 3] = -np.sqrt(3.0 / 7.0)
-        t_c2j[9, 3] = np.sqrt(4.0 / 7.0)
-        t_c2j[8, 4] = -np.sqrt(2.0 / 7.0)
-        t_c2j[11, 4] = np.sqrt(5.0 / 7.0)
-        t_c2j[10, 5] = -np.sqrt(1.0 / 7.0)
-        t_c2j[13, 5] = np.sqrt(6.0 / 7.0)
-        t_c2j[1, 6] = 1.0
-        t_c2j[0, 7] = np.sqrt(1.0 / 7.0)
-        t_c2j[3, 7] = np.sqrt(6.0 / 7.0)
-        t_c2j[2, 8] = np.sqrt(2.0 / 7.0)
-        t_c2j[5, 8] = np.sqrt(5.0 / 7.0)
-        t_c2j[4, 9] = np.sqrt(3.0 / 7.0)
-        t_c2j[7, 9] = np.sqrt(4.0 / 7.0)
-        t_c2j[6, 10] = np.sqrt(4.0 / 7.0)
-        t_c2j[9, 10] = np.sqrt(3.0 / 7.0)
-        t_c2j[8, 11] = np.sqrt(5.0 / 7.0)
-        t_c2j[11, 11] = np.sqrt(2.0 / 7.0)
-        t_c2j[10, 12] = np.sqrt(6.0 / 7.0)
-        t_c2j[13, 12] = np.sqrt(1.0 / 7.0)
-        t_c2j[12, 13] = 1.0
-
-        return t_c2j
-
-    else:
-        raise Exception("error in tmat_c2j: Have NOT implemented for this case: ", orb_l)

-
-
-
-

-[docs]
-def transform_utensor(umat, tmat):
-    """
-    Transform the rank-4 Coulomb interaction tensor from one basis to
-    another basis.
-
-    Parameters
-    ----------
-    umat: 4d array
-        Coulomb interaction tensor (rank-4).
-    tmat: 2d array
-        The transformation matrix.
-
-    Returns
-    -------
-    umat_new: 4d complex array
-        The Coulomb interaction tensor in the new basis.
-    """
-
-    n = umat.shape[0]
-    umat_new = np.zeros((n, n, n, n), dtype=np.complex128)
-
-    a1, a2, a3, a4 = np.nonzero(abs(umat) > 1E-16)
-    nonzero = np.stack((a1, a2, a3, a4), axis=-1)
-
-    for ii, jj, kk, mm in nonzero:
-        for i in range(n):
-            if abs(tmat[ii, i]) < 1E-16:
-                continue
-            else:
-                for j in range(n):
-                    if abs(tmat[jj, j]) < 1E-16:
-                        continue
-                    else:
-                        for k in range(n):
-                            if abs(tmat[kk, k]) < 1E-16:
-                                continue
-                            else:
-                                for m in range(n):
-                                    umat_new[i, j, k, m] += (np.conj(tmat[ii, i]) *
-                                                             np.conj(tmat[jj, j]) *
-                                                             umat[ii, jj, kk, mm] *
-                                                             tmat[kk, k] *
-                                                             tmat[mm, m])
-
-    return umat_new

-
-
-
-

-[docs]
-def fourier_hr2hk(norbs, nkpt, kvec, nrpt, rvec, deg_rpt, hr):
-    """
-    Fourier transform a tight-binding Hamiltonian :math:`H(r)` from
-    real space to :math:`k` space :math:`H(k)`,
-    for Wannier90.
-
-    Parameters
-    ----------
-    norbs: int
-        Number of orbitals.
-    nkpt: int
-        Number of :math:`k`-points.
-    kvec: 2d float array
-        Fractional coordinate for k-points.
-    nrpt: int
-        Number of r-points.
-    rvec: 2d float array
-        Fractional coordinate for r-points.
-    deg_rpt: int
-        The degenerancy for each r-point.
-    hr: 3d complex array
-        Hamiltonian in r-space.
-
-    Returns
-    -------
-    hk: 3d complex array
-        Hamiltonian in k-space.
-    """
-
-    hk = np.zeros((nkpt, norbs, norbs), dtype=np.complex128)
-    for i in range(nkpt):
-        for j in range(nrpt):
-            coef = -2 * np.pi * np.dot(kvec[i, :], rvec[j, :])
-            ratio = (np.cos(coef) + np.sin(coef) * 1j) / float(deg_rpt[j])
-            hk[i, :, :] = hk[i, :, :] + ratio * hr[j, :, :]
-    return hk

-
-
- -
-
-
- -
- -
-

© Copyright 2019, Brookhaven National Lab.

-
- - Built with Sphinx using a - theme - provided by Read the Docs. - - -
-
-
-
-
- - - - \ No newline at end of file diff --git a/edrixs/_modules/edrixs/coulomb_utensor.html b/edrixs/_modules/edrixs/coulomb_utensor.html deleted file mode 100644 index 54d346d5f7..0000000000 --- a/edrixs/_modules/edrixs/coulomb_utensor.html +++ /dev/null @@ -1,871 +0,0 @@ - - - - - - - - edrixs.coulomb_utensor — edrixs documentation - - - - - - - - - - - - - - - - - - - - - -
- - -
- -
-
-
- -
-
-
-
- -

Source code for edrixs.coulomb_utensor

-__all__ = ['get_gaunt', 'umat_slater', 'get_umat_kanamori_ge', 'get_F0',
-           'get_umat_slater', 'get_umat_kanamori', 'get_umat_slater_3shells']
-
-import numpy as np
-from sympy.physics.wigner import gaunt
-from .basis_transform import tmat_c2r, tmat_r2c, tmat_c2j, transform_utensor
-from .utils import info_atomic_shell, case_to_shell_name, slater_integrals_name
-
-
-

-[docs]
-def get_gaunt(l1, l2):
-    """
-    Calculate the Gaunt coefficents :math:`C_{l_1,l_2}(k,m_1,m_2)`
-
-    .. math::
-
-        C_{l_1,l_2}(k,m_1,m_2)=\\sqrt{\\frac{4\\pi}{2k+1}} \\int
-        \\mathop{d\\phi} \\mathop{d\\theta} sin(\\theta)
-        Y_{l_1}^{m_1\\star}(\\theta,\\phi) Y_{k}^{m_1-m_2}(\\theta,\\phi)
-        Y_{l_2}^{m_2}(\\theta,\\phi)
-
-    Parameters
-    ----------
-    l1: int
-        The first quantum number of angular momentum.
-    l2: int
-        The second quantum number of angular momentum.
-
-    Returns
-    -------
-    res: 3d float array
-        The calculated Gaunt coefficents.
-
-        The 1st index (:math:`= 0, 1, ..., l_1+l_2+1`) is the order :math:`k`.
-
-        The 2nd index (:math:`= 0, 1, ... ,2l_1`) is the magnetic quantum
-        number :math:`m_1` plus :math:`l_1`
-
-        The 3nd index (:math:`= 0, 1, ... ,2l_2`) is the magnetic quantum
-        number :math:`m_2` plus :math:`l_2`
-
-    Notes
-    -----
-    It should be noted that :math:`C_{l_1,l_2}(k,m_1,m_2)` is
-    nonvanishing only when
-
-    :math:`k + l_1 + l_2 = \\text{even}`,
-
-    and
-
-    :math:`|l_1 -  l_2| \\leq k \\leq l_1 + l_2`.
-
-    Please see Ref. [1]_ p. 10 for more details.
-
-    References
-    ----------
-    .. [1] Sugano S, Tanabe Y and Kamimura H. 1970. Multiplets of
-       Transition-Metal Ions in Crystals. Academic Press, New York and London.
-
-    Examples
-    --------
-    >>> import edrixs
-
-    Get gaunt coefficients between :math:`p`-shell and :math:`d`-shell
-
-    >>> g = edrixs.get_gaunt(1, 2)
-
-    """
-
-    from sympy import N
-    res = np.zeros((l1 + l2 + 1, 2 * l1 + 1, 2 * l2 + 1), dtype=np.float64)
-    for k in range(l1 + l2 + 1):
-        if not (np.mod(l1 + l2 + k, 2) == 0 and np.abs(l1 - l2) <= k <= l1 + l2):
-            continue
-        for i1, m1 in enumerate(range(-l1, l1 + 1)):
-            for i2, m2 in enumerate(range(-l2, l2 + 1)):
-                res[k, i1, i2] = (N(gaunt(l1, k, l2, -m1, m1 - m2, m2)) *
-                                  (-1.0)**m1 * np.sqrt(4 * np.pi / (2 * k + 1)))
-    return res

-
-
-
-

-[docs]
-def umat_slater(l_list, fk):
-    """
-    Calculate the Coulomb interaction tensor which is parameterized by
-    Slater integrals :math:`F^{k}`:
-
-    .. math::
-
-        U_{m_{l_i}m_{s_i}, m_{l_j}m_{s_j}, m_{l_t}m_{s_t},
-        m_{l_u}m_{s_u}}^{i,j,t,u}
-        =\\frac{1}{2} \\delta_{m_{s_i},m_{s_t}}\\delta_{m_{s_j},m_{s_u}}
-        \\delta_{m_{l_i}+m_{l_j}, m_{l_t}+m_{l_u}}
-        \\sum_{k}C_{l_i,l_t}(k,m_{l_i},m_{l_t})C_{l_u,l_j}
-        (k,m_{l_u},m_{l_j})F^{k}_{i,j,t,u}
-
-    where :math:`m_s` is the magnetic quantum number for spin
-    and :math:`m_l` is the magnetic quantum number for orbital.
-    :math:`F^{k}_{i,j,t,u}` are Slater integrals.
-    :math:`C_{l_i,l_j}(k,m_{l_i},m_{l_j})` are Gaunt coefficients.
-
-    Parameters
-    ----------
-    l_list: list of int
-        contains the quantum number of orbital angular momentum
-        :math:`l` for each shell.
-    fk: dict of float
-        contains all the possible Slater integrals between the shells
-        in l_list, the key is a tuple of 5 ints (:math:`k,i,j,t,u`),
-        where :math:`k` is the order,
-        :math:`i,j,t,u` are the shell indices begin with 1.
-
-    Returns
-    -------
-    umat: 4d array of complex
-        contains the Coulomb interaction tensor.
-
-    Examples
-    --------
-    >>> import edrixs
-
-    For only one :math:`d`-shell
-
-    >>> l_list = [2]
-    >>> fk={}
-    >>> F0, F2, F4 = 5.0, 4.0 2.0
-    >>> fk[(0,1,1,1,1)] = F0
-    >>> fk[(2,1,1,1,1)] = F2
-    >>> fk[(4,1,1,1,1)] = F4
-    >>> umat_d = edrixs.umat_slater(l_list, fk)
-
-    For one :math:`d`-shell and one :math:`p`-shell
-
-    >>> l_list = [2,1]
-    >>> fk={}
-    >>> F0_dd, F2_dd, F4_dd = 5.0, 4.0, 2.0
-    >>> F0_dp, F2_dp = 4.0, 2.0
-    >>> G1_dp, G3_dp = 2.0, 1.0
-    >>> F0_pp, F2_pp = 2.0, 1.0
-
-    >>> fk[(0,1,1,1,1)] = F0_dd
-    >>> fk[(2,1,1,1,1)] = F2_dd
-    >>> fk[(4,1,1,1,1)] = F4_dd
-
-    >>> fk[(0,1,2,1,2)] = F0_dp
-    >>> fk[(0,2,1,2,1)] = F0_dp
-    >>> fk[(2,1,2,1,2)] = F2_dp
-    >>> fk[(2,2,1,2,1)] = F2_dp
-
-    >>> fk[(1,1,2,2,1)] = G1_dp
-    >>> fk[(1,2,1,1,2)] = G1_dp
-    >>> fk[(3,1,2,2,1)] = G3_dp
-    >>> fk[(3,2,1,1,2)] = G3_dp
-
-    >>> fk[(0,2,2,2,2)] = F0_pp
-    >>> fk[(2,2,2,2,2)] = F2_pp
-
-    >>> umat_dp = edrixs.umat_slater(l_list, fk)
-
-    See also
-    --------
-    coulomb_utensor.get_umat_slater
-    coulomb_utensor.get_umat_kanamori
-    coulomb_utensor.get_umat_kanamori_ge
-    """
-    k_list = list(range(0, 2 * max(l_list) + 1))
-    ck = {}
-    orb_label = []
-    # for each shell
-    for i, l in enumerate(l_list):
-        # magnetic quantum number
-        for m in range(-l, l + 1):
-            # spin part, up,dn
-            for spin in range(2):
-                orb_label.append((i + 1, l, m, spin))
-
-    # all the possible gaunt coefficient
-    for l1 in l_list:
-        for l2 in l_list:
-            ck_tmp = get_gaunt(l1, l2)
-            for m1 in range(-l1, l1 + 1):
-                for m2 in range(-l2, l2 + 1):
-                    for k in k_list:
-                        if (np.mod(l1 + l2 + k, 2) == 0 and np.abs(l1 - l2) <= k <= l1 + l2):
-                            ck[(k, l1, m1, l2, m2)] = ck_tmp[k, m1 + l1, m2 + l2]
-                        else:
-                            ck[(k, l1, m1, l2, m2)] = 0
-
-    # build the coulomb interaction tensor
-    norbs = len(orb_label)
-    umat = np.zeros((norbs, norbs, norbs, norbs), dtype=np.complex128)
-    for orb1 in range(norbs):
-        for orb2 in range(norbs):
-            for orb3 in range(norbs):
-                for orb4 in range(norbs):
-                    i1, l1, m1, sigma1 = orb_label[orb1]
-                    i2, l2, m2, sigma2 = orb_label[orb2]
-                    i3, l3, m3, sigma3 = orb_label[orb3]
-                    i4, l4, m4, sigma4 = orb_label[orb4]
-                    if (m1 + m2) != (m3 + m4):
-                        continue
-                    if (sigma1 != sigma3) or (sigma2 != sigma4):
-                        continue
-                    res = 0.0
-                    for k in k_list:
-                        tmp_key = (k, i1, i2, i3, i4)
-                        if tmp_key in list(fk.keys()):
-                            res += ck[(k, l1, m1, l3, m3)] * ck[(k, l4, m4, l2, m2)] * fk[tmp_key]
-                    umat[orb1, orb2, orb4, orb3] = res
-
-    umat = umat / 2.0
-    return umat

-
-
-
-

-[docs]
-def get_umat_kanamori_ge(norbs, U1, U2, J, Jx, Jp):
-    """
-    Calculate the Coulomb interaction tensor for a Kanamori-type interaction.
-    For the general case, it is parameterized by :math:`U_1, U_2, J, J_x, J_p`.
-
-    Parameters
-    ----------
-    norbs: int
-        number of orbitals (including spin).
-    U1: float
-        Hubbard :math:`U` for electrons residing on the same orbital with opposite spin.
-    U2: float
-        Hubbard :math:`U` for electrons residing on different orbitals.
-    J: float
-        Hund's coupling for density-density interaction.
-    Jx: float
-        Hund's coupling for spin flip.
-    Jp: float
-        Hund's coupling for pair-hopping.
-
-    Returns
-    -------
-    umat: 4d complex array
-        The calculated Coulomb interaction tensor
-
-    Notes
-    -----
-    The order of spin index is: up, down, up, down, ..., up, down.
-
-    See Also
-    --------
-    coulomb_utensor.get_umat_kanamori
-    coulomb_utensor.get_umat_slater
-    coulomb_utensor.umat_slater
-
-    """
-
-    umat = np.zeros((norbs, norbs, norbs, norbs), dtype=np.complex128)
-    for alpha in range(0, norbs - 1):
-        for beta in range(alpha + 1, norbs):
-            for gamma in range(0, norbs - 1):
-                for delta in range(gamma + 1, norbs):
-                    aband = alpha // 2
-                    aspin = alpha % 2
-                    bband = beta // 2
-                    bspin = beta % 2
-                    gband = gamma // 2
-                    gspin = gamma % 2
-                    dband = delta // 2
-                    dspin = delta % 2
-
-                    dtmp = 0.0
-                    if ((alpha == gamma) and (beta == delta)):
-                        if ((aband == bband) and (aspin != bspin)):
-                            dtmp = dtmp + U1
-
-                    if ((alpha == gamma) and (beta == delta)):
-                        if (aband != bband):
-                            dtmp = dtmp + U2
-
-                    if ((alpha == gamma) and (beta == delta)):
-                        if ((aband != bband) and (aspin == bspin)):
-                            dtmp = dtmp - J
-
-                    if ((aband == gband) and (bband == dband)):
-                        if ((aspin != gspin) and (bspin != dspin)
-                                and (aspin != bspin)):
-                            dtmp = dtmp - Jx
-
-                    if ((aband == bband) and (dband == gband)
-                            and (aband != dband)):
-                        if ((aspin != bspin) and (dspin != gspin)
-                                and (aspin == gspin)):
-                            dtmp = dtmp + Jp
-
-                    umat[alpha, beta, delta, gamma] = dtmp
-
-    return umat

-
-
-
-def get_F0(case, *args):
-    case = case.strip()
-    if case == 's':
-        return 0.0
-
-    elif case == 'p':
-        F2 = args[0]
-        return 2.0 / 25.0 * F2
-
-    elif case == 'd':
-        F2, F4 = args[0:2]
-        return 2.0 / 63.0 * F2 + 2.0 / 63.0 * F4
-
-    elif case == 'f':
-        F2, F4, F6 = args[0:3]
-        return 4.0 / 195.0 * F2 + 2.0 / 143.0 * F4 + 100.0 / 5577.0 * F6
-
-    elif case == 'ss':
-        G0 = args[0]
-        return 0.5 * G0
-
-    elif case == 'sp' or case == 'ps':
-        G1 = args[0]
-        return 1.0 / 6.0 * G1
-
-    elif case == 'sd' or case == 'ds':
-        G2 = args[0]
-        return 1.0 / 10.0 * G2
-
-    elif case == 'sf' or case == 'fs':
-        G3 = args[0]
-        return 1.0 / 14.0 * G3
-
-    elif case == 'pp':
-        G0, G2 = args[0:2]
-        return 1.0 / 6.0 * G0 + 1.0 / 15.0 * G2
-
-    elif case == 'pd' or case == 'dp':
-        G1, G3 = args[0:2]
-        return 1.0 / 15.0 * G1 + 3.0 / 70.0 * G3
-
-    elif case == 'pf' or case == 'fp':
-        G2, G4 = args[0:2]
-        return 3.0 / 70.0 * G2 + 2.0 / 63.0 * G4
-
-    elif case == 'dd':
-        G0, G2, G4 = args[0:3]
-        return 1.0 / 10.0 * G0 + 1.0 / 35.0 * G2 + 1.0 / 35.0 * G4
-
-    elif case == 'df' or case == 'fd':
-        G1, G3, G5 = args[0:3]
-        return 3.0 / 70.0 * G1 + 2.0 / 105.0 * G3 + 5.0 / 231.0 * G5
-
-    elif case == 'ff':
-        G0, G2, G4, G6 = args[0:4]
-        return 1.0 / 14.0 * G0 + 2.0 / 105.0 * G2 + 1.0 / 77.0 * G4 + 50.0 / 3003.0 * G6
-
-    else:
-        raise Exception("error in get_F0():  Unknown case name:", case)
-
-
-

-[docs]
-def get_umat_slater(case, *args):
-    """
-    Convenient adapter function to return the Coulomb interaction tensor for common case.
-
-    Parameters
-    ----------
-    case: string
-        Indicates atomic shells, should be one of
-
-        For single shell:
-
-        - 's':    single :math:`s`-shell (:math:`l=0`)
-        - 'p':    single :math:`p`-shell (:math:`l=1`)
-        - 'p12':  single :math:`p_{1/2}`-shell (:math:`l=1`)
-        - 'p32':  single :math:`p_{3/2}`-shell (:math:`l=1`)
-        - 't2g':  single :math:`t_{2g}`-shell (:math:`l_{\\text{eff}}=1`)
-        - 'd':    single :math:`d`-shell (:math:`l=2`)
-        - 'd32':  single :math:`d_{3/2}`-shell (:math:`l=2`)
-        - 'd52':  single :math:`d_{5/2}`-shell (:math:`l=2`)
-        - 'f':    single :math:`f`-shell (:math:`l=3`)
-        - 'f52':  single :math:`f_{5/2}`-shell (:math:`l=3`)
-        - 'f72':  single :math:`f_{7/2}`-shell (:math:`l=3`)
-
-        For two shells:
-
-        case = str1 + str2
-
-        where, str1 and str2 are strings and they can be any of
-
-        ['s', 'p', 'p12', 'p32', 't2g', 'd', 'd32', 'd52', 'f', 'f52', 'f72']
-
-        For examples,
-
-        - 'dp':     1st :math:`d`-shell and 2nd :math:`p`-shell
-        - 'dp12':   1st :math:`d`-shell and 2nd :math:`p_{1/2}`-shell
-        - 'f52p32': 1st :math:`f_{5/2}`-shell and 2nd :math:`p_{3/2}`-shell
-        - 't2gp':   1st :math:`t_{2g}`-shell and 2nd :math:`p`-shell
-
-    *args: floats
-        Variable length argument list. Slater integrals.
-        The order of these integrals shoule be
-
-        For only one shell case,
-
-        args = [F0, F2, F4, F6, ....]
-
-        For two shells case,
-
-        args = [FX_11, FX_12, GX_12, FX_22]
-
-        where, 1 (2) means 1st (2nd)-shell, and X=0, 2, 4, ... or X=1, 3, 5 ...,
-        and X should be in ascending order. The following are possible cases:
-
-        - 's':
-
-        args = [F0]
-
-        - 'p', 'p12', 'p32':
-
-        args = [F0, F2]
-
-        - 'd', 'd32', 'd52', 't2g':
-
-        args = [F0, F2, F4]
-
-        - 'f', 'f52', 'f72':
-
-        args = [F0, F2, F4, F6]
-
-        - 'ss':
-
-        args = [F0_11, F0_12, G0_12, F0_22]
-
-        - 'ps', 'p12s', 'p32s':
-
-        args = [F0_11, F2_11, F0_12, G1_12, F0_22]
-
-        - 'ds', 'd32s', 'd52s', 't2gs':
-
-        args = [F0_11, F2_11, F4_11, F0_12, G2_12, F0_22]
-
-        - 'fs', 'f52s', 'f72s':
-
-        args = [F0_11, F2_11, F4_11, F6_11, F0_12, G3_12, F0_22]
-
-        - 'sp', 'sp12', 'sp32':
-
-        args = [F0_11, F0_12, G1_12, F0_22, F2_22]
-
-        - 'pp', 'pp12', 'pp32', 'p12p', 'p12p12', 'p12p32', 'p32p', 'p32p12', 'p32p32':
-
-        args = [F0_11, F2_11, F0_12, F2_12, G0_12, G2_12, F0_22, F2_22]
-
-        - 'dp', 'dp12', 'dp32', 'd32p', 'd32p12', 'd32p32', 'd52p', 'd52p12', 'd52p32',
-          't2gp', 't2gp12', t2gp32':
-
-        args = [F0_11, F2_11, F4_11, F0_12, F2_12, G1_12, G3_12, F0_22, F2_22]
-
-        - 'fp', 'fp12', 'fp32', 'f52p', 'f52p12', 'f52p32', 'f72p', 'f72p12', 'f72p32':
-
-        args = [F0_11, F2_11, F4_11, F6_11, F0_12, F2_12, G2_12, G4_12, F0_22, F2_22]
-
-        - 'sd', 'sd32', 'sd52':
-
-        args = [F0_11, F0_12, G2_12, F0_22, F2_22, F4_22]
-
-        - 'pd', 'pd32', 'pd52', 'p12d', 'p12d32', 'p12d52', 'p32d', 'p32d32', 'p32d52':
-
-        args = [F0_11, F2_11, F0_12, F2_12, G1_12, G3_12, F0_22, F2_22, F4_22]
-
-        - 'dd', 'dd32', 'dd52', 'd32d', 'd32d32', 'd32d52', 'd52d', 'd52d32', 'd52d52',
-          't2gd', 't2gd32', 't2gd52':
-
-        args = [F0_11, F2_11, F4_11, F0_12, F2_12, F4_12, G0_12, G2_12, G4_12, F0_22, F2_22, F4_22]
-
-        - 'fd', 'fd32', 'fd52', 'f52d', 'f52d32', 'f52d52', 'f72d', 'f72d32', 'f72d52':
-
-        args = [F0_11, F2_11, F4_11, F6_11, F0_12, F2_12, F4_12, G1_12, G3_12, G5_12,
-        F0_22, F2_22, F4_22]
-
-        - 'sf', 'sf52', 'sf72':
-
-        args = [F0_11, F0_12, G3_12, F0_22, F2_22, F4_22, F6_22]
-
-        - 'pf', 'pf52', 'pf72', 'p12f', 'p12f52', 'p12f72', 'p32f', 'p32f52', 'p32f72':
-
-        args = [F0_11, F2_11, F0_12, F2_12, G2_12, G4_12, F0_22, F2_22, F4_22, F6_22]
-
-        - 'df', 'df52', 'df72', 'd32f', 'd32f52', 'd32f72', 'd52f', 'd52f52', 'd52f72',
-          't2gf', 't2gf52', 't2gf72':
-
-        args = [F0_11, F2_11, F4_11, F0_12, F2_12, F4_12, G1_12, G3_12, G5_12,
-        F0_22, F2_22, F4_22, F6_22]
-
-        - 'ff', 'ff52', 'ff72', 'f52f', 'f52f52', 'f52f72', 'f72f', 'f72f52', 'f72f72':
-
-        args = [F0_11, F2_11, F4_11, F6_11, F0_12, F2_12, F4_12, F6_12,
-        G0_12, G2_12, G4_12, G6_12, F0_22, F2_22, F4_22, F6_22]
-
-    Returns
-    -------
-    umat: 4d array of complex
-        the Coulomb interaction tensor
-
-    Examples
-    --------
-    >>> import edrixs
-    >>> F0_dd, F2_dd, F4_dd = 3.0, 1.0, 0.5
-    >>> F0_dp, F2_dp, G1_dp, G3_dp = 2.0, 1.0, 0.2, 0.1
-    >>> F0_pp, F2_pp = 0.0, 0.0
-    >>> slater = [F0_dd, F2_dd, F4_dd, F0_dp, F2_dp, G1_dp, G3_dp, F0_pp, F2_pp]
-    >>> umat_d = edrixs.get_umat_slater('d', F0_dd, F2_dd, F4_dd)
-    >>> umat_dp = edrixs.get_umat_slater('dp', *slater)
-    >>> umat_t2gp = edrixs.get_umat_slater('t2gp', *slater)
-    >>> umat_dp32 = edrixs.get_umat_slater('dp32', *slater)
-
-    See Also
-    --------
-    coulomb_utensor.umat_slater
-    coulomb_utensor.get_umat_kanamori
-    coulomb_utensor.get_umat_kanamori_ge
-
-    """
-    info = info_atomic_shell()
-    shells = case_to_shell_name(case)
-    nslat = len(slater_integrals_name(shells))
-    if nslat != len(args):
-        raise Exception("Number of Slater integrals", len(args), " is not equal to ", nslat)
-
-    special_shell = ['t2g', 'p12', 'p32', 'd32', 'd52', 'f52', 'f72']
-    orb_indx = {
-        special_shell[0]: [2, 3, 4, 5, 8, 9],
-        special_shell[1]: [0, 1],
-        special_shell[2]: [2, 3, 4, 5],
-        special_shell[3]: [0, 1, 2, 3],
-        special_shell[4]: [4, 5, 6, 7, 8, 9],
-        special_shell[5]: [0, 1, 2, 3, 4, 5],
-        special_shell[6]: [6, 7, 8, 9, 10, 11, 12, 13]
-    }
-    # only one shell
-    if len(shells) == 1:
-        orbl = info[shells[0]][0]
-        l_list = [orbl]
-        fk = {}
-        it = 0
-        for rank in range(0, 2*orbl+1, 2):
-            fk[(rank, 1, 1, 1, 1)] = args[it]
-            it += 1
-        umat = umat_slater(l_list, fk)
-        # truncate to a sub-shell if necessary
-        if shells[0] in special_shell:
-            if shells[0] == 't2g':
-                tmat = tmat_c2r('d', True)
-            else:
-                tmat = tmat_c2j(orbl)
-            umat = transform_utensor(umat, tmat)
-            indx = orb_indx[shells[0]]
-            umat_tmp = np.zeros((len(indx), len(indx), len(indx), len(indx)), dtype=complex)
-            umat_tmp[:, :, :, :] = umat[indx][:, indx][:, :, indx][:, :, :, indx]
-            if shells[0] == 't2g':
-                umat_tmp[:, :, :, :] = transform_utensor(umat_tmp, tmat_r2c('t2g', True))
-            umat = umat_tmp
-    # two shells
-    elif len(shells) == 2:
-        name1, name2 = shells
-        l1, l2 = info[name1][0], info[name2][0]
-        n1, n2 = 2*(2*l1+1), 2*(2*l2+1)
-        ntot = n1 + n2
-        l_list = [l1, l2]
-        fk = {}
-        it = 0
-        for rank in range(0, 2 * l1 + 1, 2):
-            fk[(rank, 1, 1, 1, 1)] = args[it]
-            it += 1
-        for rank in range(0, min(2 * l1, 2 * l2) + 1, 2):
-            fk[(rank, 1, 2, 1, 2)] = args[it]
-            fk[(rank, 2, 1, 2, 1)] = args[it]
-            it += 1
-        for rank in range(abs(l1 - l2), l1 + l2 + 1, 2):
-            fk[(rank, 1, 2, 2, 1)] = args[it]
-            fk[(rank, 2, 1, 1, 2)] = args[it]
-            it += 1
-        for rank in range(0, 2 * l2 + 1, 2):
-            fk[(rank, 2, 2, 2, 2)] = args[it]
-            it += 1
-        umat = umat_slater(l_list, fk)
-        # truncate to a sub-shell if necessary
-        if (name1 in special_shell) or (name2 in special_shell):
-            tmat = np.eye(ntot, dtype=complex)
-            indx1 = list(range(0, n1))
-            if name1 in special_shell:
-                if name1 == 't2g':
-                    tmat[0:n1, 0:n1] = tmat_c2r('d', True)
-                else:
-                    tmat[0:n1, 0:n1] = tmat_c2j(l1)
-                indx1 = orb_indx[name1]
-
-            indx2 = [n1 + i for i in range(0, n2)]
-            if name2 in special_shell:
-                if name2 == 't2g':
-                    tmat[n1:ntot, n1:ntot] = tmat_c2r('d', True)
-                else:
-                    tmat[n1:ntot, n1:ntot] = tmat_c2j(l2)
-                indx2 = [n1 + i for i in orb_indx[name2]]
-
-            indx = indx1 + indx2
-            umat = transform_utensor(umat, tmat)
-            umat_tmp = np.zeros((len(indx), len(indx), len(indx), len(indx)), dtype=complex)
-            umat_tmp[:, :, :, :] = umat[indx][:, indx][:, :, indx][:, :, :, indx]
-            if name1 == 't2g' or name2 == 't2g':
-                tmat = np.eye(len(indx), dtype=np.complex128)
-                if name1 == 't2g':
-                    tmat[0:6, 0:6] = tmat_r2c('t2g', True)
-                if name2 == 't2g':
-                    tmat[-6:, -6:] = tmat_r2c('t2g', True)
-                umat_tmp[:, :, :, :] = transform_utensor(umat_tmp, tmat)
-            umat = umat_tmp
-    else:
-        raise Exception("Not implemented for this case: ", shells)
-
-    return umat

-
-
-
-

-[docs]
-def get_umat_slater_3shells(shell_name, *args):
-    """
-    Given three shells, build the slater type of Coulomb tensors among
-    the three shells.
-
-    Parameters
-    ----------
-    shell_name: tuple of three strings
-        Shells names.
-    *args: floats
-        Slater integrals. The order should be
-
-        FX_11, FX_12, GX_12, FX_22, FX_13, GX_13, FX_23, GX_23, FX_33
-
-        where, 1, 2, 3 means 1st, 2nd, 3rd shell, and X=0, 2, 4, ... or X=1, 3, 5 ...,
-        and X should be in ascending order.
-
-    Returns
-    -------
-    umat: 4d complex array
-        Rank-4 Coulomb tensors.
-    """
-
-    v1_name = shell_name[0].strip()
-    v2_name = shell_name[1].strip()
-    v3_name = shell_name[2].strip()
-    info_shell = info_atomic_shell()
-
-    v1_orbl = info_shell[v1_name][0]
-    v2_orbl = info_shell[v2_name][0]
-    v3_orbl = info_shell[v3_name][0]
-
-    v1_norb = info_shell[v1_name][1]
-    v2_norb = info_shell[v2_name][1]
-    v3_norb = info_shell[v3_name][1]
-
-    # total number of orbitals
-    ntot = v1_norb + v2_norb + v3_norb
-    v1v2_norb = v1_norb + v2_norb
-
-    indx = []
-    it = 0
-    it += len(range(0, 2*v1_orbl+1, 2))
-    indx.append(it)
-    it += len(range(0, min(2*v1_orbl, 2*v2_orbl)+1, 2))
-    it += len(range(abs(v1_orbl-v2_orbl), v1_orbl+v2_orbl+1, 2))
-    indx.append(it)
-    it += len(range(0, 2*v2_orbl+1, 2))
-    indx.append(it)
-    it += len(range(0, min(2*v1_orbl, 2*v3_orbl)+1, 2))
-    it += len(range(abs(v1_orbl-v3_orbl), v1_orbl+v3_orbl+1, 2))
-    indx.append(it)
-    it += len(range(0, min(2*v2_orbl, 2*v3_orbl)+1, 2))
-    it += len(range(abs(v2_orbl-v3_orbl), v2_orbl+v3_orbl+1, 2))
-    indx.append(it)
-    it += len(range(0, 2*v3_orbl+1, 2))
-    indx.append(it)
-
-    if it != len(args):
-        raise Exception("Number of Slater integrals", len(args), " is not equal to ", it)
-
-    umat = np.zeros((ntot, ntot, ntot, ntot), dtype=complex)
-
-    # v1-v2
-    case = v1_name + v2_name
-    arg_list = list(args[0:indx[0]]) + list(args[indx[0]:indx[1]]) + list(args[indx[1]:indx[2]])
-    umat_tmp = get_umat_slater(case, *arg_list)
-    umat[0:v1v2_norb, 0:v1v2_norb, 0:v1v2_norb, 0:v1v2_norb] = umat_tmp
-
-    # v1-v3
-    case = v1_name + v3_name
-    arg_list = [0.0] * indx[0] + list(args[indx[2]:indx[3]]) + list(args[indx[4]:indx[5]])
-    umat_tmp = get_umat_slater(case, *arg_list)
-    aa = list(range(0, v1_norb)) + list(range(v1v2_norb, ntot))
-    for i in range(v1_norb + v3_norb):
-        for j in range(v1_norb + v3_norb):
-            for k in range(v1_norb + v3_norb):
-                for m in range(v1_norb + v3_norb):
-                    umat[aa[i], aa[j], aa[k], aa[m]] += umat_tmp[i, j, k, m]
-
-    # v2-v3
-    case = v2_name + v3_name
-    arg_list = ([0.0] * (indx[2] - indx[1]) +
-                list(args[indx[3]:indx[4]]) +
-                [0.0] * (indx[5] - indx[4]))
-    umat_tmp = get_umat_slater(case, *arg_list)
-    aa = list(range(v1_norb, ntot))
-    for i in range(v2_norb + v3_norb):
-        for j in range(v2_norb + v3_norb):
-            for k in range(v2_norb + v3_norb):
-                for m in range(v2_norb + v3_norb):
-                    umat[aa[i], aa[j], aa[k], aa[m]] += umat_tmp[i, j, k, m]
-
-    return umat

-
-
-
-

-[docs]
-def get_umat_kanamori(norbs, U, J):
-    """
-    Calculate the Coulomb interaction tensor for a Kanamori-type interaction.
-    For the :math:`t2g`-shell case, it is parameterized by :math:`U, J`.
-
-    Parameters
-    ----------
-    norbs: int
-        number of orbitals (including spin).
-    U: float
-        Hubbard :math:`U` for electrons residing on the same orbital with opposite spin.
-    J: float
-        Hund's coupling.
-
-    Returns
-    -------
-    umat: 4d complex array
-        The calculated Coulomb interaction tensor
-
-    Notes
-    -----
-    The order of spin index is: up, down, up, down, ..., up, down.
-
-    See Also
-    --------
-    coulomb_utensor.get_umat_kanamori_ge
-    coulomb_utensor.get_umat_slater
-    coulomb_utensor.umat_slater
-
-    """
-
-    return get_umat_kanamori_ge(norbs, U, U - 2 * J, J, J, J)

-
-
- -
-
-
- -
- -
-

© Copyright 2019, Brookhaven National Lab.

-
- - Built with Sphinx using a - theme - provided by Read the Docs. - - -
-
-
-
-
- - - - \ No newline at end of file diff --git a/edrixs/_modules/edrixs/fit_hyb.html b/edrixs/_modules/edrixs/fit_hyb.html deleted file mode 100644 index 2f7b4aca6c..0000000000 --- a/edrixs/_modules/edrixs/fit_hyb.html +++ /dev/null @@ -1,237 +0,0 @@ - - - - - - - - edrixs.fit_hyb — edrixs documentation - - - - - - - - - - - - - - - - - - - - - -
- - -
- -
-
-
- -
-
-
-
- -

Source code for edrixs.fit_hyb

-__all__ = ['fit_func', 'fit_hyb', 'get_hyb']
-
-import numpy as np
-from scipy.optimize import curve_fit
-
-
-

-[docs]
-def fit_func(x, *args):
-    """
-    Given frequency :math:`\\omega`, bath energy level :math:`\\epsilon_{l}` and
-    the hybridization strength :math:`V_{l}`,
-    return the hybridization function,
-
-    .. math::
-
-        \\Delta(\\omega)=\\sum_{l=1}^{N}\\frac{|V_{l}|^2}{\\omega-\\epsilon_{l}}.
-
-    Parameters
-    ----------
-    x: 1d float array
-        Frequency :math:`\\omega`, the first half is the real part and
-        the second half is the imaginary part.
-    args: 1d float array
-        The first half is the bath energy level :math:`\\epsilon_{l}` and the
-        second half if the hybridization strength :math:`V_{l}`.
-
-    Returns
-    -------
-    y: 1d float array
-        The calculated hybridization function :math:`\\Delta(\\omega)`, the
-        first half is the real part and the second half is the imaginary part.
-    """
-
-    m = len(x) // 2
-    n = len(args) // 2
-    y = np.zeros(len(x), dtype=np.float64)
-
-    tmp_x = np.zeros(m, dtype=np.complex128)
-    tmp_y = np.zeros(m, dtype=np.complex128)
-    tmp_x[:] = x[0:m] + 1j * x[m:2 * m]
-
-    for i in range(n):
-        tmp_y[:] += args[n + i]**2 / (tmp_x[:] - args[i])
-
-    y[0:m] = tmp_y.real
-    y[m:2 * m] = tmp_y.imag
-
-    return y

-
-
-
-

-[docs]
-def fit_hyb(x, y, N, p0):
-    """
-    Given the hybridization function :math:`\\Delta(\\omega)`,
-    call function curve_fit in scipy to
-    fit bath energy levels :math:`\\epsilon_{l}` and
-    hybridization strength :math:`V_{l}`.
-
-    .. math::
-
-        \\Delta(\\omega)=\\sum_{l=1}^{N}\\frac{|V_{l}|^2}{\\omega-\\epsilon_{l}}.
-
-
-    Parameters
-    ----------
-    x: 1d complex array
-       Frequency :math:`\\omega`.
-    y: 1d complex array
-       Hybridization function :math:`\\Delta(\\omega)`.
-    N: int
-        Number of bath sites
-    p0: N-length 1d float array
-        Initial guess, the first half is :math:`\\epsilon_{l}` and
-        the second half is :math:`V_{l}`.
-
-    Returns
-    -------
-    e: N-length 1d float array
-        The fitted bath energy levels :math:`\\epsilon_{l}`.
-    v: N-length 1d float array
-        The fitted hybridization strength :math:`V_{l}`.
-    """
-
-    m = len(x)
-    xdata = np.zeros(2 * m, dtype=np.float64)
-    ydata = np.zeros(2 * m, dtype=np.float64)
-    xdata[0:m], xdata[m:2 * m] = x.real, x.imag
-    ydata[0:m], ydata[m:2 * m] = y.real, y.imag
-    popt, pcov = curve_fit(fit_func, xdata, ydata, p0)
-    e, v = popt[0:N], popt[N:2 * N]
-    return e, v

-
-
-
-

-[docs]
-def get_hyb(x, e, v):
-    """
-    Given the fitted :math:`\\epsilon_{l}` and :math:`V_{l}`, calcualte the
-    hybridization function :math:`\\Delta(\\omega)`,
-
-    .. math::
-
-        \\Delta(\\omega)=\\sum_{l=1}^{N}\\frac{|V_{l}|^2}{\\omega-\\epsilon_{l}}.
-
-    Parameters
-    ----------
-    x: 1d complex array
-       Frequency :math:`\\omega`.
-    e:  N-length 1d float array
-       The fitted bath energy levels.
-    v:  N-length 1d float array
-        The fitted hybridization strength.
-
-    Returns
-    -------
-    y: 1d complex array
-        The calculated hybridization function :math:`\\Delta(\\omega)`.
-    """
-
-    y = np.zeros(len(x), dtype=np.complex128)
-    for i in range(len(e)):
-        y[:] += v[i]**2 / (x[:] - e[i])
-    return y

-
-
- -
-
-
- -
- -
-

© Copyright 2019, Brookhaven National Lab.

-
- - Built with Sphinx using a - theme - provided by Read the Docs. - - -
-
-
-
-
- - - - \ No newline at end of file diff --git a/edrixs/_modules/edrixs/fock_basis.html b/edrixs/_modules/edrixs/fock_basis.html deleted file mode 100644 index de7e235ae4..0000000000 --- a/edrixs/_modules/edrixs/fock_basis.html +++ /dev/null @@ -1,638 +0,0 @@ - - - - - - - - edrixs.fock_basis — edrixs documentation - - - - - - - - - - - - - - - - - - - - - -
- - -
- -
-
-
- -
-
-
-
- -

Source code for edrixs.fock_basis

-#!/usr/bin/env python
-
-__all__ = ['combination', 'fock_bin', 'get_fock_bin_by_N', 'get_fock_half_N',
-           'get_fock_full_N', 'get_fock_basis_by_NLz', 'get_fock_basis_by_NSz',
-           'get_fock_basis_by_NJz', 'get_fock_basis_by_N_abelian',
-           'get_fock_basis_by_N_LzSz', 'write_fock_dec_by_N']
-
-import numpy as np
-import itertools
-
-
-

-[docs]
-def combination(n, m):
-    """
-    Calculate the combination :math:`C_{n}^{m}`,
-
-    .. math::
-
-        C_{n}^{m} = \\frac{n!}{m!(n-m)!}.
-
-    Parameters
-    ----------
-    n: int
-       Number n.
-    m: int
-        Number m.
-
-    Returns
-    -------
-    res: int
-        The calculated result.
-
-    Examples
-    --------
-    >>> import edrixs
-    >>> edrixs.combination(6, 2)
-    15
-
-    """
-
-    if m > n or n < 0 or m < 0:
-        print("wrong number in combination")
-        return
-    if m == 0 or n == m:
-        return 1
-
-    largest = max(m, n - m)
-    smallest = min(m, n - m)
-    numer = 1.0
-    for i in range(largest + 1, n + 1):
-        numer *= i
-
-    denom = 1.0
-    for i in range(1, smallest + 1):
-        denom *= i
-
-    res = int(numer / denom)
-    return res

-
-
-
-

-[docs]
-def fock_bin(n, k):
-    """
-    Return all the possible :math:`n`-length binary
-    where :math:`k` of :math:`n` digitals are set to 1.
-
-    Parameters
-    ----------
-    n: int
-        Binary length :math:`n`.
-    k: int
-        How many digitals are set to be 1.
-
-    Returns
-    -------
-    res: list of int-lists
-        A list of list containing the binary digitals.
-
-    Examples
-    --------
-    >>> import edrixs
-    >>> edrixs.fock_bin(4, 2)
-    [[1, 1, 0, 0],
-     [1, 0, 1, 0],
-     [1, 0, 0, 1],
-     [0, 1, 1, 0],
-     [0, 1, 0, 1],
-     [0, 0, 1, 1]]
-
-    """
-
-    if n == 0:
-        return [[0]]
-
-    res = []
-    for bits in itertools.combinations(list(range(n)), k):
-        s = [0] * n
-        for bit in bits:
-            s[bit] = 1
-        res.append(s)
-    return res

-
-
-
-

-[docs]
-def get_fock_bin_by_N(*args):
-    """
-    Get binary form to represent a Fock state.
-
-    Parameters
-    ----------
-    args: ints
-        args[0]: number of orbitals for 1st-shell,
-
-        args[1]: number of occupancy for 1st-shell,
-
-        args[2]: number of orbitals for 2nd-shell,
-
-        args[3]: number of occupancy for 2nd-shell,
-
-        ...
-
-        args[ :math:`2N-2`]: number of orbitals for :math:`N` th-shell,
-
-        args[ :math:`2N-1`]: number of occupancy for :math:`N` th-shell.
-
-    Returns
-    -------
-    result: list of int list
-        The binary form of Fock states.
-
-    Examples
-    --------
-    >>> import edrixs
-    >>> edrixs.get_fock_bin_by_N(4, 2)
-    [[1, 1, 0, 0],
-     [1, 0, 1, 0],
-     [1, 0, 0, 1],
-     [0, 1, 1, 0],
-     [0, 1, 0, 1],
-     [0, 0, 1, 1]]
-
-    >>> edrixs.get_fock_bin_by_N(4, 2, 2, 1)
-    [[1, 1, 0, 0, 1, 0],
-     [1, 0, 1, 0, 1, 0],
-     [1, 0, 0, 1, 1, 0],
-     [0, 1, 1, 0, 1, 0],
-     [0, 1, 0, 1, 1, 0],
-     [0, 0, 1, 1, 1, 0],
-     [1, 1, 0, 0, 0, 1],
-     [1, 0, 1, 0, 0, 1],
-     [1, 0, 0, 1, 0, 1],
-     [0, 1, 1, 0, 0, 1],
-     [0, 1, 0, 1, 0, 1],
-     [0, 0, 1, 1, 0, 1]]
-
-    """
-
-    n = len(args)
-
-    if n % 2 != 0:
-        print("Error: number of arguments is not even")
-        return
-
-    if n == 2:
-        return fock_bin(args[0], args[1])
-    else:
-        result = []
-        res1 = fock_bin(args[0], args[1])
-        res2 = get_fock_bin_by_N(*args[2:])
-        for ifock in res2:
-            for jfock in res1:
-                result.append(jfock + ifock)
-        return result

-
-
-
-def get_fock_half_N(N):
-    res = [[] for i in range(N + 1)]
-    for i in range(2**N):
-        occu = bin(i).count('1')
-        res[occu].append(i)
-    return res
-
-
-

-[docs]
-def get_fock_full_N(norb, N):
-    """
-    Get the decimal digitals to represent Fock states.
-
-    Parameters
-    ----------
-    norb: int
-        Number of orbitals.
-    N: int
-        Number of occupancy.
-
-    Returns
-    -------
-    res: list of int
-        The decimal digitals to represent Fock states.
-
-    Examples
-    --------
-    >>> import edrixs
-    >>> edrixs.fock_bin(4,2)
-    [[1, 1, 0, 0],
-     [1, 0, 1, 0],
-     [0, 1, 1, 0],
-     [1, 0, 0, 1],
-     [0, 1, 0, 1],
-     [0, 0, 1, 1]]
-
-    >>> import edrixs
-    >>> edrixs.get_fock_full_N(4,2)
-    [3, 5, 6, 9, 10, 12]
-
-    """
-
-    res = []
-    half_N = get_fock_half_N(norb // 2)
-    for m in range(norb // 2 + 1):
-        n = N - m
-        if n >= 0 and n <= norb // 2:
-            res.extend([i * 2**(norb // 2) + j for i in half_N[m] for j in half_N[n]])
-    return res

-
-
-
-

-[docs]
-def get_fock_basis_by_NLz(norb, N, lz_list):
-    """
-    Get decimal digitals to represent Fock states, use good quantum number:
-
-    - orbital angular momentum :math:`L_{z}`
-
-    Parameters
-    ----------
-    norb: int
-        Number of orbitals.
-    N: int
-        Number of total occupancy.
-    lz_list: list of int
-        Quantum number :math:`l_{z}` for each orbital.
-
-    Returns
-    -------
-    res: dict
-        A dictionary containing the decimal digitals, the key is good
-        quantum numbers :math:`L_{z}`, the value is a list of int.
-
-    Examples
-    --------
-    >>> import edrixs
-    >>> edrixs.get_fock_basis_by_NLz(6, 2, [-1, -1, 0, 0, 1, 1])
-    {
-     -2: [3],
-     -1: [5, 6, 9, 10],
-      0: [12, 17, 18, 33, 34],
-      1: [20, 36, 24, 40],
-      2: [48]
-    }
-    """
-
-    res = get_fock_basis_by_N_abelian(norb, N, lz_list)
-    return res

-
-
-
-

-[docs]
-def get_fock_basis_by_NSz(norb, N, sz_list):
-    """
-    Get decimal digitals to represent Fock states, use good quantum number:
-
-    - spin angular momentum :math:`S_{z}`
-
-    Parameters
-    ----------
-    norb: int
-        Number of orbitals.
-    N: int
-        Number of total occupancy.
-    sz_list: list of int
-        Quantum number :math:`s_{z}` for each orbital.
-
-    Returns
-    -------
-    res: dict
-        A dictionary containing the decimal digitals, the key is good quantum
-        numbers :math:`S_{z}`, the value is a list of int.
-
-    Examples
-    --------
-    >>> import edrixs
-    >>> edrixs.get_fock_basis_by_NSz(6, 2, [1, -1, 1, -1, 1, -1])
-    {
-     -2: [10, 34, 40],
-     -1: [],
-      0: [3, 6, 9, 12, 18, 33, 36, 24, 48],
-      1: [],
-      2: [5, 17, 20]
-    }
-    """
-
-    res = get_fock_basis_by_N_abelian(norb, N, sz_list)
-    return res

-
-
-
-

-[docs]
-def get_fock_basis_by_NJz(norb, N, jz_list):
-    """
-    Get decimal digitals to represent Fock states, use good quantum number:
-
-    - total angular momentum :math:`J_{z}`
-
-    Parameters
-    ----------
-    norb: int
-        Number of orbitals.
-    N: int
-        Number of total occupancy.
-    jz_list: list of int
-        Quantum number :math:`j_{z}` for each orbital.
-
-    Returns
-    -------
-    res: dict
-        A dictionary containing the decimal digitals, the key is good quantum
-        numbers :math:`j_{z}`, the value is a list of int.
-
-    Examples
-    --------
-    >>> import edrixs
-    >>> edrixs.get_fock_basis_by_NJz(6, 2, [-1, 1, -3, -1, 1, 3])
-    {
-     -6: [],
-     -5: [],
-     -4: [5, 12],
-     -3: [],
-     -2: [6, 9, 20],
-     -1: [],
-      0: [3, 10, 17, 36, 24],
-      1: [],
-      2: [18, 33, 40],
-      3: [],
-      4: [34, 48],
-      5: [],
-      6: []
-    }
-    """
-
-    res = get_fock_basis_by_N_abelian(norb, N, jz_list)
-    return res

-
-
-
-

-[docs]
-def get_fock_basis_by_N_abelian(norb, N, a_list):
-    """
-    Get decimal digitals to represent Fock states, use some Abelian good quantum number.
-
-    Parameters
-    ----------
-    norb: int
-        Number of orbitals.
-    N: int
-        Number of total occupancy.
-    a_list: list of int
-        Quantum number of the Abelian symmetry for each orbital.
-
-    Returns
-    -------
-    basis: dict
-        A dictionary containing the decimal digitals, the key is good quantum numbers,
-        the value is a list of int.
-    """
-
-    result = get_fock_full_N(norb, N)
-    min_a, max_a = min(a_list) * N, max(a_list) * N
-    basis = {}
-    for i in range(min_a, max_a + 1):
-        basis[i] = []
-    for n in result:
-        a = sum([a_list[i] for i in range(0, n.bit_length()) if (n >> i & 1)])
-        basis[a].append(n)
-    return basis

-
-
-
-

-[docs]
-def get_fock_basis_by_N_LzSz(norb, N, lz_list, sz_list):
-    """
-    Get decimal digitals to represent Fock states, use good quantum number:
-
-    - orbital angular momentum :math:`L_{z}`
-    - spin angular momentum :math:`S_{z}`
-
-    Parameters
-    ----------
-    norb: int
-        Number of orbitals.
-    N: int
-        Number of total occupancy.
-    lz_list: list of int
-        Quantum number :math:`l_{z}` for each orbital.
-    sz_list: list of int
-        Quantum number :math:`s_{z}` for each orbital.
-
-    Returns
-    -------
-    basis: dict
-        A dictionary containing the decimal digitals, the key is a tuple containing good quantum
-        numbers ( :math:`l_{z}`, :math:`s_{z}`), the value is a list of int.
-
-    Examples
-    --------
-    >>> import edrixs
-    >>> edrixs.get_fock_basis_by_N_LzSz(6, 2, [-1, -1, 0, 0, 1, 1], [1, -1, 1, -1, 1, -1])
-    {
-     (-2, -2): [],
-     (-2, -1): [],
-      (-2, 0): [3],
-      (-2, 1): [],
-      (-2, 2): [],
-     (-1, -2): [10],
-     (-1, -1): [],
-      (-1, 0): [6, 9],
-      (-1, 1): [],
-      (-1, 2): [5],
-      (0, -2): [34],
-      (0, -1): [],
-       (0, 0): [12, 18, 33],
-       (0, 1): [],
-       (0, 2): [17],
-      (1, -2): [40],
-      (1, -1): [],
-       (1, 0): [36, 24],
-       (1, 1): [],
-       (1, 2): [20],
-      (2, -2): [],
-      (2, -1): [],
-       (2, 0): [48],
-       (2, 1): [],
-       (2, 2): []
-    }
-    """
-    result = get_fock_full_N(norb, N)
-    min_Lz, max_Lz = min(lz_list) * N, max(lz_list) * N
-    min_Sz, max_Sz = min(sz_list) * N, max(sz_list) * N
-    basis = {}
-    for i in range(min_Lz, max_Lz + 1):
-        for j in range(min_Sz, max_Sz + 1):
-            basis[(i, j)] = []
-    for n in result:
-        Lz, Sz = np.sum([[lz_list[i], sz_list[i]] for i in range(0, n.bit_length())
-                         if (n >> i & 1)], axis=0)
-        basis[(Lz, Sz)].append(n)
-    return basis

-
-
-
-

-[docs]
-def write_fock_dec_by_N(N, r, fname='fock_i.in'):
-    """
-    Get decimal digitals to represent Fock states, sort them by
-    ascending order and then write them to file.
-
-    Parameters
-    ----------
-    N: int
-       Number of orbitals.
-    r: int
-        Number of occuancy.
-    fname: string
-        File name.
-
-    Returns
-    -------
-    ndim: int
-        The dimension of the Hilbert space
-
-    Examples
-    --------
-    >>> import edrixs
-    >>> edrixs.write_fock_dec_by_N(4, 2, 'fock_i.in')
-    file fock_i.in looks like
-    15
-    3
-    5
-    6
-    9
-    10
-    12
-    17
-    18
-    20
-    24
-    33
-    34
-    36
-    40
-    48
-
-    where, the first line is the total numer of Fock states,
-    and the following lines are the Fock states in decimal form.
-    """
-
-    res = get_fock_full_N(N, r)
-    res.sort()
-    ndim = len(res)
-    f = open(fname, 'w')
-    print(ndim, file=f)
-    for item in res:
-        print(item, file=f)
-    f.close()
-    return ndim

-
-
- -
-
-
- -
- -
-

© Copyright 2019, Brookhaven National Lab.

-
- - Built with Sphinx using a - theme - provided by Read the Docs. - - -
-
-
-
-
- - - - \ No newline at end of file diff --git a/edrixs/_modules/edrixs/iostream.html b/edrixs/_modules/edrixs/iostream.html deleted file mode 100644 index f78e7e15b5..0000000000 --- a/edrixs/_modules/edrixs/iostream.html +++ /dev/null @@ -1,487 +0,0 @@ - - - - - - - - edrixs.iostream — edrixs documentation - - - - - - - - - - - - - - - - - - - - - -
- - -
- -
-
-
- -
-
-
-
- -

Source code for edrixs.iostream

-__all__ = ['write_tensor', 'write_emat', 'write_umat', 'write_config',
-           'read_poles_from_file', 'dump_poles', 'load_poles']
-
-import numpy as np
-import json
-
-
-def write_tensor_1(tensor, fname, only_nonzeros=False, tol=1E-10, fmt_int='{:10d}',
-                   fmt_float='{:.15f}'):
-    (n1, ) = tensor.shape
-    is_cmplx = False
-    if tensor.dtype in (complex, np.complex128):
-        is_cmplx = True
-    space = "    "
-    f = open(fname, 'w')
-    for i in range(n1):
-        if only_nonzeros and abs(tensor[i]) < tol:
-            continue
-        if is_cmplx:
-            fmt_string = fmt_int + space + (fmt_float + space) * 2 + '\n'
-            f.write(fmt_string.format(i + 1, tensor[i].real, tensor[i].imag))
-        else:
-            fmt_string = fmt_int + space + fmt_float + space + '\n'
-            f.write(fmt_string.format(i + 1, tensor[i]))
-    f.close()
-
-
-def write_tensor_2(tensor, fname, only_nonzeros=False, tol=1E-10, fmt_int='{:10d}',
-                   fmt_float='{:.15f}'):
-    (n1, n2) = tensor.shape
-    is_cmplx = False
-    if tensor.dtype in (complex, np.complex128):
-        is_cmplx = True
-    space = "    "
-    f = open(fname, 'w')
-    for i in range(n1):
-        for j in range(n2):
-            if only_nonzeros and abs(tensor[i, j]) < tol:
-                continue
-            if is_cmplx:
-                fmt_string = (fmt_int + space) * 2 + (fmt_float + space) * 2 + '\n'
-                f.write(fmt_string.format(i + 1, j + 1, tensor[i, j].real, tensor[i, j].imag))
-            else:
-                fmt_string = (fmt_int + space) * 2 + fmt_float + space + '\n'
-                f.write(fmt_string.format(i + 1, j + 1, tensor[i, j]))
-    f.close()
-
-
-def write_tensor_3(tensor, fname, only_nonzeros=False, tol=1E-10, fmt_int='{:10d}',
-                   fmt_float='{:.15f}'):
-    (n1, n2, n3) = tensor.shape
-    is_cmplx = False
-    if tensor.dtype in (complex, np.complex128):
-        is_cmplx = True
-    space = "    "
-    f = open(fname, 'w')
-    for i in range(n1):
-        for j in range(n2):
-            for k in range(n3):
-                if only_nonzeros and abs(tensor[i, j, k]) < tol:
-                    continue
-                if is_cmplx:
-                    fmt_string = (fmt_int + space) * 3 + (fmt_float + space) * 2 + '\n'
-                    f.write(fmt_string.format(i + 1, j + 1, k + 1, tensor[i, j, k].real,
-                                              tensor[i, j, k].imag))
-                else:
-                    fmt_string = (fmt_int + space) * 3 + fmt_float + space + '\n'
-                    f.write(fmt_string.format(i + 1, j + 1, k + 1, tensor[i, j, k]))
-    f.close()
-
-
-def write_tensor_4(tensor, fname, only_nonzeros=False, tol=1E-10, fmt_int='{:10d}',
-                   fmt_float='{:.15f}'):
-    (n1, n2, n3, n4) = tensor.shape
-    is_cmplx = False
-    if tensor.dtype in (complex, np.complex128):
-        is_cmplx = True
-    space = "    "
-    f = open(fname, 'w')
-    for i in range(n1):
-        for j in range(n2):
-            for k in range(n3):
-                for m in range(n4):
-                    if only_nonzeros and abs(tensor[i, j, k, m]) < tol:
-                        continue
-                    if is_cmplx:
-                        fmt_string = (fmt_int + space) * 4 + (fmt_float + space) * 2 + '\n'
-                        f.write(fmt_string.format(i + 1, j + 1, k + 1, m + 1,
-                                tensor[i, j, k, m].real, tensor[i, j, k, m].imag))
-                    else:
-                        fmt_string = (fmt_int + space) * 4 + fmt_float + space + '\n'
-                        f.write(fmt_string.format(i + 1, j + 1, k + 1, m + 1, tensor[i, j, k, m]))
-    f.close()
-
-
-def write_tensor_5(tensor, fname, only_nonzeros=False, tol=1E-10, fmt_int='{:10d}',
-                   fmt_float='{:.15f}'):
-    (n1, n2, n3, n4, n5) = tensor.shape
-    is_cmplx = False
-    if tensor.dtype in (complex, np.complex128):
-        is_cmplx = True
-    space = "    "
-    f = open(fname, 'w')
-    for i in range(n1):
-        for j in range(n2):
-            for k in range(n3):
-                for r in range(n4):
-                    for m in range(n5):
-                        if only_nonzeros and abs(tensor[i, j, k, r, m]) < tol:
-                            continue
-                        if is_cmplx:
-                            fmt_string = (fmt_int + space) * 5 + (fmt_float + space) * 2 + '\n'
-                            f.write(fmt_string.format(i + 1, j + 1, k + 1, r + 1, m + 1,
-                                    tensor[i, j, k, r, m].real, tensor[i, j, k, r, m].imag))
-                        else:
-                            fmt_string = (fmt_int + space) * 5 + fmt_float + space + '\n'
-                            f.write(fmt_string.format(i + 1, j + 1, k + 1, r + 1, m + 1,
-                                    tensor[i, j, k, r, m]))
-    f.close()
-
-
-

-[docs]
-def write_tensor(tensor, fname, only_nonzeros=False, tol=1E-10, fmt_int='{:10d}',
-                 fmt_float='{:.15f}'):
-    """
-    Write :math:`n` -dimension numpy array to file, currently, :math:`n` can be 1, 2, 3, 4, 5.
-
-    Parameters
-    ----------
-    tensor: :math:`n` d float or complex array
-        The array needs to be written.
-    fname: str
-        File name.
-    only_nonzeros: logical (default: False)
-        Only write nonzero elements.
-    tol: float (default: 1E-10)
-        Only write the elements when their absolute value are larger than tol
-        and only_nonzeros=True.
-    fmt_int: str (default: '{:10d}')
-        The format for printing integer numbers.
-    fmt_float: str (default: '{:.15f}')
-        The format for printing float numbers.
-
-    """
-
-    ndim = tensor.ndim
-    if ndim == 1:
-        write_tensor_1(tensor, fname, only_nonzeros=only_nonzeros, tol=tol,
-                       fmt_int=fmt_int, fmt_float=fmt_float)
-    elif ndim == 2:
-        write_tensor_2(tensor, fname, only_nonzeros=only_nonzeros, tol=tol,
-                       fmt_int=fmt_int, fmt_float=fmt_float)
-    elif ndim == 3:
-        write_tensor_3(tensor, fname, only_nonzeros=only_nonzeros, tol=tol,
-                       fmt_int=fmt_int, fmt_float=fmt_float)
-    elif ndim == 4:
-        write_tensor_4(tensor, fname, only_nonzeros=only_nonzeros, tol=tol,
-                       fmt_int=fmt_int, fmt_float=fmt_float)
-    elif ndim == 5:
-        write_tensor_5(tensor, fname, only_nonzeros=only_nonzeros, tol=tol,
-                       fmt_int=fmt_int, fmt_float=fmt_float)
-    else:
-        raise Exception("error in write_tensor: ndim >5, not implemented !")

-
-
-
-

-[docs]
-def write_emat(emat, fname, tol=1E-12, fmt_int='{:10d}', fmt_float='{:.15f}'):
-    """
-    Write the nonzeros of the rank-2 hopping matrices to file.
-    The first line is the number of nonzeros, and the following lines are the nonzero elements.
-    This file will be read by ed.x, xas.x or rixs.x.
-
-    Parameters
-    ----------
-    emat: 2d complex array
-        The array to be written.
-    fname: str
-        File name.
-    tol: float
-        Precision.
-    fmt_int: str (default: '{:10d}')
-        Format for printing integer numbers.
-    fmt_float: str (default: '{:.15f}')
-        Format for printing float numbers.
-    """
-
-    a1, a2 = np.nonzero(abs(emat) > tol)
-    nonzero = np.stack((a1, a2), axis=-1)
-
-    space = "    "
-    fmt_string = (fmt_int + space) * 2 + (fmt_float + space) * 2 + '\n'
-    f = open(fname, 'w')
-    if len(nonzero) == 0:
-        f.write("{:10d}\n".format(1))
-        f.write(fmt_string.format(1, 1, 0.0, 0.0))
-    else:
-        f.write("{:20d}\n".format(len(nonzero)))
-        for i, j in nonzero:
-            f.write(fmt_string.format(i + 1, j + 1, emat[i, j].real, emat[i, j].imag))
-    f.close()

-
-
-
-

-[docs]
-def write_umat(umat, fname, tol=1E-12, fmt_int='{:10d}', fmt_float='{:.15f}'):
-    """
-    Write the nonzeros of the rank-4 Coulomb U tensor to file.
-    The first line is the number of nonzeros, and the following lines are the nonzero elements.
-    This file will be read by ed.x, xas.x or rixs.x.
-
-    Parameters
-    ----------
-    umat: 4d complex array
-        The array to be written.
-    fname: str
-        File name.
-    tol: float (default: 1E-12)
-        Precision.
-    fmt_int: str (default: '{:10d}')
-        Format for printing integer numbers.
-    fmt_float: str (default: '{:.15f}')
-        Format for printing float numbers.
-    """
-
-    a1, a2, a3, a4 = np.nonzero(abs(umat) > tol)
-    nonzero = np.stack((a1, a2, a3, a4), axis=-1)
-
-    space = "    "
-    fmt_string = (fmt_int + space) * 4 + (fmt_float + space) * 2 + '\n'
-    f = open(fname, 'w')
-    if len(nonzero) == 0:
-        f.write("{:10d}\n".format(1))
-        f.write(fmt_string.format(1, 1, 1, 1, 0.0, 0.0))
-    else:
-        f.write("{:20d}\n".format(len(nonzero)))
-        for i, j, k, l in nonzero:
-            f.write(fmt_string.format(i + 1, j + 1, k + 1, l + 1,
-                                      umat[i, j, k, l].real, umat[i, j, k, l].imag))
-    f.close()

-
-
-
-

-[docs]
-def write_config(
-        directory='.', ed_solver=1, num_val_orbs=2, num_core_orbs=2,
-        neval=1, nvector=1, ncv=1, idump=True, num_gs=1, maxiter=500,
-        linsys_max=1000, min_ndim=1000, nkryl=500, eigval_tol=1e-8,
-        linsys_tol=1e-10, omega_in=0.0, gamma_in=0.1
-        ):
-    """
-    Write control parameters in config.in file for ed_fsolver.
-    """
-    if idump:
-        dump_vector = '.true.'
-    else:
-        dump_vector = '.false.'
-
-    config = [
-        "&control",
-        "ed_solver=" + str(ed_solver),
-        "num_val_orbs=" + str(num_val_orbs),
-        "num_core_orbs=" + str(num_core_orbs),
-        "neval=" + str(neval),
-        "nvector=" + str(nvector),
-        "ncv=" + str(ncv),
-        "idump=" + str(dump_vector),
-        "num_gs=" + str(num_gs),
-        "maxiter=" + str(maxiter),
-        "linsys_max=" + str(linsys_max),
-        "min_ndim=" + str(min_ndim),
-        "nkryl=" + str(nkryl),
-        "eigval_tol=" + str(eigval_tol),
-        "linsys_tol=" + str(linsys_tol),
-        "omega_in=" + str(omega_in),
-        "gamma_in=" + str(gamma_in),
-        "&end"
-    ]
-
-    f = open(directory + '/config.in', 'w')
-    for item in config:
-        f.write(item + "\n")
-    f.close()

-
-
-
-

-[docs]
-def read_poles_from_file(file_list):
-    """
-    Read informations in files xas_poles.n or rixs_poles.n to a dict.
-
-    Parameters
-    ----------
-    file_list: list of strings
-        Names of pole files.
-
-    pole_dict: dict
-        A dict containing information of poles.
-    """
-    pole_dict = {
-        'npoles': [],
-        'eigval': [],
-        'norm': [],
-        'alpha': [],
-        'beta': []
-    }
-    for fname in file_list:
-        f = open(fname, 'r')
-        line = f.readline()
-        neff = int(line.strip().split()[1])
-        pole_dict['npoles'].append(neff)
-
-        line = f.readline()
-        eigval = float(line.strip().split()[1])
-        pole_dict['eigval'].append(eigval)
-
-        line = f.readline()
-        norm = float(line.strip().split()[1])
-        pole_dict['norm'].append(norm)
-
-        alpha = []
-        beta = []
-        for i in range(neff):
-            line = f.readline()
-            line = line.strip().split()
-            alpha.append(float(line[1]))
-            beta.append(float(line[2]))
-        pole_dict['alpha'].append(alpha)
-        pole_dict['beta'].append(beta)
-        f.close()
-
-    return pole_dict

-
-
-
-

-[docs]
-def dump_poles(obj, file_name="poles"):
-    """
-    Dump the objects of poles returned from XAS or RIXS calculations to file for later plotting.
-
-    Parameters
-    ----------
-    obj: Python object
-        Object of poles, a dict or a list of dicts.
-    file_name: string
-        File name.
-    """
-    with open(file_name+'.json', 'w') as f:
-        json.dump(obj, f, indent=2)

-
-
-
-

-[docs]
-def load_poles(file_name='poles'):
-    """
-    Load the objects of poles from file.
-
-    Parameters
-    ----------
-    file_name: string
-
-    Returns
-    -------
-    obj: Python objects
-        Poles object.
-    """
-    with open(file_name+'.json', 'r') as f:
-        obj = json.load(f)
-
-    return obj

-
-
- -
-
-
- -
- -
-

© Copyright 2019, Brookhaven National Lab.

-
- - Built with Sphinx using a - theme - provided by Read the Docs. - - -
-
-
-
-
- - - - \ No newline at end of file diff --git a/edrixs/_modules/edrixs/manybody_operator.html b/edrixs/_modules/edrixs/manybody_operator.html deleted file mode 100644 index 465dcc10a3..0000000000 --- a/edrixs/_modules/edrixs/manybody_operator.html +++ /dev/null @@ -1,447 +0,0 @@ - - - - - - - - edrixs.manybody_operator — edrixs documentation - - - - - - - - - - - - - - - - - - - - - -
- - -
- -
-
-
- -
-
-
-
- -

Source code for edrixs.manybody_operator

-__all__ = ['one_fermion_annihilation', 'two_fermion', 'four_fermion',
-           'build_opers', 'density_matrix']
-
-import numpy as np
-from collections import defaultdict
-
-
-

-[docs]
-def one_fermion_annihilation(iorb, lb, rb):
-    """
-    Build matrix form of a fermionic annihilation operator in the given Fock basis.
-
-    .. math::
-
-        <F_{l}|\\hat{f}_{i}|F_{r}>
-
-    Parameters
-    ----------
-    iorb: int
-        Which orbital.
-    lb: list or array
-        Left fock basis :math:`<F_{l}|`.
-    rb: list of array
-        Right fock basis :math:`|F_{r}>`.
-
-    Returns
-    -------
-    hmat: 2d complex array
-        The matrix form of :math:`\\hat{f}_{i}`.
-    """
-
-    lb, rb = np.array(lb), np.array(rb)
-    nr, nl, norbs = len(rb), len(lb), len(rb[0])
-    indx = defaultdict(lambda: -1)
-    for i, j in enumerate(lb):
-        indx[tuple(j)] = i
-
-    hmat = np.zeros((nl, nr), dtype=np.complex128)
-    tmp_basis = np.zeros(norbs)
-    for icfg in range(nr):
-        tmp_basis[:] = rb[icfg]
-        if tmp_basis[iorb] == 0:
-            continue
-        else:
-            sign = (-1)**np.count_nonzero(tmp_basis[0:iorb])
-            tmp_basis[iorb] = 0
-        jcfg = indx[tuple(tmp_basis)]
-        if jcfg != -1:
-            hmat[jcfg, icfg] += sign
-    return hmat

-
-
-
-

-[docs]
-def two_fermion(emat, lb, rb=None, tol=1E-10):
-    """
-    Build matrix form of a two-fermionic operator in the given Fock basis,
-
-    .. math::
-
-        <F_{l}|\\sum_{ij}E_{ij}\\hat{f}_{i}^{\\dagger}\\hat{f}_{j}|F_{r}>
-
-    Parameters
-    ----------
-    emat: 2d complex array
-        The impurity matrix.
-    lb: list of array
-        Left fock basis :math:`<F_{l}|`.
-    rb: list of array
-        Right fock basis :math:`|F_{r}>`.
-        rb = lb if rb is None
-    tol: float (default: 1E-10)
-        Only consider the elements of emat that are larger than tol.
-
-    Returns
-    -------
-    hmat: 2d complex array
-        The matrix form of the two-fermionic operator.
-    """
-    if rb is None:
-        rb = lb
-    lb, rb = np.array(lb), np.array(rb)
-    nr, nl, norbs = len(rb), len(lb), len(rb[0])
-    indx = defaultdict(lambda: -1)
-    for i, j in enumerate(lb):
-        indx[tuple(j)] = i
-
-    a1, a2 = np.nonzero(abs(emat) > tol)
-    nonzero = np.stack((a1, a2), axis=-1)
-
-    hmat = np.zeros((nl, nr), dtype=np.complex128)
-    tmp_basis = np.zeros(norbs)
-    for iorb, jorb in nonzero:
-        for icfg in range(nr):
-            tmp_basis[:] = rb[icfg]
-            if tmp_basis[jorb] == 0:
-                continue
-            else:
-                s1 = (-1)**np.count_nonzero(tmp_basis[0:jorb])
-                tmp_basis[jorb] = 0
-            if tmp_basis[iorb] == 1:
-                continue
-            else:
-                s2 = (-1)**np.count_nonzero(tmp_basis[0:iorb])
-                tmp_basis[iorb] = 1
-            jcfg = indx[tuple(tmp_basis)]
-            if jcfg != -1:
-                hmat[jcfg, icfg] += emat[iorb, jorb] * s1 * s2
-    return hmat

-
-
-
-

-[docs]
-def four_fermion(umat, lb, rb=None, tol=1E-10):
-    """
-    Build matrix form of a four-fermionic operator in the given Fock basis,
-
-    .. math::
-
-        <F_l|\\sum_{ij}U_{ijkl}\\hat{f}_{i}^{\\dagger}\\hat{f}_{j}^{\\dagger}
-        \\hat{f}_{k}\\hat{f}_{l}|F_r>
-
-    Parameters
-    ----------
-    umat: 4d complex array
-        The 4 index Coulomb interaction tensor.
-    lb: list of array
-        Left fock basis :math:`<F_{l}|`.
-    rb: list of array
-        Right fock basis :math:`|F_{r}>`.
-        rb = lb if rb is None
-    tol: float (default: 1E-10)
-        Only consider the elements of umat that are larger than tol.
-
-    Returns
-    -------
-    hmat: 2d complex array
-        The matrix form of the four-fermionic operator.
-    """
-    if rb is None:
-        rb = lb
-    lb, rb = np.array(lb), np.array(rb)
-    nr, nl, norbs = len(rb), len(lb), len(rb[0])
-    indx = defaultdict(lambda: -1)
-    for i, j in enumerate(lb):
-        indx[tuple(j)] = i
-
-    a1, a2, a3, a4 = np.nonzero(abs(umat) > tol)
-    nonzero = np.stack((a1, a2, a3, a4), axis=-1)
-
-    hmat = np.zeros((nl, nr), dtype=np.complex128)
-    tmp_basis = np.zeros(norbs)
-    for lorb, korb, jorb, iorb in nonzero:
-        if iorb == jorb or korb == lorb:
-            continue
-        for icfg in range(nr):
-            tmp_basis[:] = rb[icfg]
-            if tmp_basis[iorb] == 0:
-                continue
-            else:
-                s1 = (-1)**np.count_nonzero(tmp_basis[0:iorb])
-                tmp_basis[iorb] = 0
-            if tmp_basis[jorb] == 0:
-                continue
-            else:
-                s2 = (-1)**np.count_nonzero(tmp_basis[0:jorb])
-                tmp_basis[jorb] = 0
-            if tmp_basis[korb] == 1:
-                continue
-            else:
-                s3 = (-1)**np.count_nonzero(tmp_basis[0:korb])
-                tmp_basis[korb] = 1
-            if tmp_basis[lorb] == 1:
-                continue
-            else:
-                s4 = (-1)**np.count_nonzero(tmp_basis[0:lorb])
-                tmp_basis[lorb] = 1
-            jcfg = indx[tuple(tmp_basis)]
-            if jcfg != -1:
-                hmat[jcfg, icfg] += umat[lorb, korb, jorb, iorb] * s1 * s2 * s3 * s4
-    return hmat

-
-
-
-

-[docs]
-def build_opers(nfermion, coeff, lb, rb=None, tol=1E-10):
-    """
-    Build matrix form of many-body operators in the given Fock basis,
-
-    .. math::
-
-        <F_{l}|\\sum_{ij}E_{ij}\\hat{f}_{i}^{\\dagger}\\hat{f}_{j}|F_{r}>
-
-    or
-
-    .. math::
-
-        <F_l|\\sum_{ij}U_{ijkl}\\hat{f}_{i}^{\\dagger}\\hat{f}_{j}^{\\dagger}
-        \\hat{f}_{k}\\hat{f}_{l}|F_r>
-
-    Parameters
-    ----------
-    nfermion: int
-        Number of fermion operators. Options can only be 2 or 4 now.
-    coeff: array-like
-        The coefficients.
-
-        - if nfermion=2, coeff should be at least 2-dimension, and the last 2-dimension is
-          the matrix of coefficients of an operator. For examples,
-
-          - coeff.shape = (3, 10, 10), means 3 operators with :math:`10 \\times 10`
-            coefficients matrix.
-
-          - coeff.shape = (2, 3, 10, 10), means :math:`2 \\times 3=6` operators with
-            :math:`10 \\times 10` coefficients matrix.
-
-        - if nfermion=4, coeff should be at least 4-dimension, and the last 4-dimension is
-          the rank-4 tensor of coefficients of an operator. For examples,
-
-          - coeff.shape = (3, 10, 10, 10, 10), means 3 operators with
-            :math:`10 \\times 10 \\times 10 \\times 10` coefficients tensor.
-
-          - coeff.shape = (2, 3, 10, 10, 10, 10), means :math:`2 \\times 3=6` operators with
-            :math:`10 \\times 10 \\times 10 \\times 10` coefficients tensor.
-    lb: list of array
-        Left fock basis :math:`<F_{l}|`.
-    rb: list of array
-        Right fock basis :math:`|F_{r}>`.
-        rb = lb if rb is None
-    tol: float (default: 1E-10)
-        Only consider the elements of emat that are larger than tol.
-
-    Returns
-    -------
-    hmat: array-like
-        At least 2-dimension and the last 2-dimension is matrix form of operators,
-        For examples,
-
-        - if nfermion=2, coeff.shape=(2, 3, 10, 10), hmat.shape=(2, 3, len(lb), len(rb))
-
-        - if nfermion=4, coeff.shape=(2, 3, 10, 10, 10, 10), hmat.shape=(2, 3, len(lb), len(rb))
-    """
-    if nfermion not in [2, 4]:
-        raise Exception("nfermion is not 2 or 4")
-    nl = len(lb)
-    if rb is None:
-        nr = nl
-    else:
-        nr = len(rb)
-    coeff = np.array(coeff, order='C')
-    if nfermion == 2:
-        dim = coeff.shape
-        if len(dim) < 2:
-            raise Exception("Dimension of coeff should be at least 2 when nfermion=2")
-        elif len(dim) == 2:
-            hmat = two_fermion(coeff, lb, rb, tol)
-        else:
-            tot = np.prod(dim[0:-2])
-            hmat_tmp = np.zeros((tot, nl, nr), dtype=complex)
-            coeff_tmp = coeff.reshape((tot, dim[-2], dim[-1]))
-            for i in range(tot):
-                hmat_tmp[i] = two_fermion(coeff_tmp[i], lb, rb, tol)
-            hmat = hmat_tmp.reshape(dim[0:-2] + (nl, nr))
-    if nfermion == 4:
-        dim = coeff.shape
-        if len(dim) < 4:
-            raise Exception("Dimension of coeff should be at least 4 when nfermion=4")
-        elif len(dim) == 4:
-            hmat = four_fermion(coeff, lb, rb, tol)
-        else:
-            tot = np.prod(dim[0:-4])
-            hmat_tmp = np.zeros((tot, nl, nr), dtype=complex)
-            coeff_tmp = coeff.reshape((tot, dim[-4], dim[-3], dim[-2], dim[-1]))
-            for i in range(tot):
-                hmat_tmp[i] = four_fermion(coeff_tmp[i], lb, rb, tol)
-            hmat = hmat_tmp.reshape(dim[0:-4] + (nl, nr))
-
-    return hmat

-
-
-
-

-[docs]
-def density_matrix(iorb, jorb, lb, rb):
-    """
-    Calculate the matrix form of density operators :math:`\\hat{f}_{i}^{\\dagger}\\hat{f}_{j}`
-    in the given Fock basis,
-
-    .. math::
-
-        <F_{l}|\\hat{f}_{i}^{\\dagger}\\hat{f}_{j}|F_{r}>
-
-    Parameters
-    ----------
-    iorb: int
-        Orbital index.
-    jorb: int
-        Orbital index.
-    lb: list or array
-        Left fock basis :math:`<F_{l}|`.
-    rb: list or array
-        Right fock basis :math:`|F_{r}>`.
-
-    Returns
-    -------
-    hmat: 2d complex array
-        The calculated matrix form of the density operator.
-    """
-
-    lb, rb = np.array(lb), np.array(rb)
-    nr, nl, norbs = len(rb), len(lb), len(rb[0])
-    indx = defaultdict(lambda: -1)
-    for i, j in enumerate(lb):
-        indx[tuple(j)] = i
-
-    hmat = np.zeros((nl, nr), dtype=np.complex128)
-    tmp_basis = np.zeros(norbs)
-    for icfg in range(nr):
-        tmp_basis[:] = rb[icfg]
-        if tmp_basis[jorb] == 0:
-            continue
-        else:
-            s1 = (-1)**np.count_nonzero(tmp_basis[0:jorb])
-            tmp_basis[jorb] = 0
-        if tmp_basis[iorb] == 1:
-            continue
-        else:
-            s2 = (-1)**np.count_nonzero(tmp_basis[0:iorb])
-            tmp_basis[iorb] = 1
-        jcfg = indx[tuple(tmp_basis)]
-        if jcfg != -1:
-            hmat[jcfg, icfg] += s1 * s2
-    return hmat

-
-
- -
-
-
- -
- -
-

© Copyright 2019, Brookhaven National Lab.

-
- - Built with Sphinx using a - theme - provided by Read the Docs. - - -
-
-
-
-
- - - - \ No newline at end of file diff --git a/edrixs/_modules/edrixs/photon_transition.html b/edrixs/_modules/edrixs/photon_transition.html deleted file mode 100644 index 695fb301ca..0000000000 --- a/edrixs/_modules/edrixs/photon_transition.html +++ /dev/null @@ -1,689 +0,0 @@ - - - - - - - - edrixs.photon_transition — edrixs documentation - - - - - - - - - - - - - - - - - - - - - -
- - -
- -
-
-
- -
-
-
-
- -

Source code for edrixs.photon_transition

-__all__ = ['dipole_trans_oper', 'quadrupole_trans_oper', 'get_trans_oper',
-           'unit_wavevector', 'wavevector_with_length', 'get_wavevector_rixs',
-           'linear_polvec', 'dipole_polvec_rixs', 'dipole_polvec_xas',
-           'quadrupole_polvec']
-
-import numpy as np
-from sympy.physics.wigner import clebsch_gordan
-from .basis_transform import tmat_c2r, tmat_r2c, tmat_c2j, cb_op2
-from .utils import case_to_shell_name, info_atomic_shell
-
-
-def dipole_trans_oper(l1, l2):
-    from sympy import N
-
-    n1, n2 = 2 * l1 + 1, 2 * l2 + 1
-    op = np.zeros((3, n1, n2), dtype=np.complex128)
-    for i1, m1 in enumerate(range(-l1, l1 + 1)):
-        for i2, m2 in enumerate(range(-l2, l2 + 1)):
-            tmp1 = clebsch_gordan(l2, 1, l1, m2, -1, m1)
-            tmp2 = clebsch_gordan(l2, 1, l1, m2, 1, m1)
-            tmp3 = clebsch_gordan(l2, 1, l1, m2, 0, m1)
-            tmp1, tmp2, tmp3 = N(tmp1), N(tmp2), N(tmp3)
-            op[0, i1, i2] = (tmp1 - tmp2) * np.sqrt(2.0) / 2.0
-            op[1, i1, i2] = (tmp1 + tmp2) * 1j * np.sqrt(2.0) / 2.0
-            op[2, i1, i2] = tmp3
-    op_spin = np.zeros((3, 2 * n1, 2 * n2), dtype=np.complex128)
-    for i in range(3):
-        op_spin[i, 0:2 * n1:2, 0:2 * n2:2] = op[i]
-        op_spin[i, 1:2 * n1:2, 1:2 * n2:2] = op[i]
-
-    return op_spin
-
-
-def quadrupole_trans_oper(l1, l2):
-    from sympy import N
-    n1, n2 = 2 * l1 + 1, 2 * l2 + 1
-    op = np.zeros((5, n1, n2), dtype=np.complex128)
-    for i1, m1 in enumerate(range(-l1, l1 + 1)):
-        for i2, m2 in enumerate(range(-l2, l2 + 1)):
-            t1 = clebsch_gordan(l2, 2, l1, m2, -2, m1)
-            t2 = clebsch_gordan(l2, 2, l1, m2, 2, m1)
-            t3 = clebsch_gordan(l2, 2, l1, m2, 0, m1)
-            t4 = clebsch_gordan(l2, 2, l1, m2, -1, m1)
-            t5 = clebsch_gordan(l2, 2, l1, m2, 1, m1)
-            t1, t2, t3, t4, t5 = N(t1), N(t2), N(t3), N(t4), N(t5)
-
-            op[0, i1, i2] = t3
-            op[1, i1, i2] = (t4 - t5) / np.sqrt(2.0)
-            op[2, i1, i2] = (t4 + t5) * 1j / np.sqrt(2.0)
-            op[3, i1, i2] = (t1 + t2) / np.sqrt(2.0)
-            op[4, i1, i2] = (t1 - t2) * 1j / np.sqrt(2.0)
-
-    op_spin = np.zeros((5, 2 * n1, 2 * n2), dtype=np.complex128)
-    for i in range(5):
-        op_spin[i, 0:2 * n1:2, 0:2 * n2:2] = op[i]
-        op_spin[i, 1:2 * n1:2, 1:2 * n2:2] = op[i]
-
-    return op_spin
-
-
-

-[docs]
-def get_trans_oper(case):
-    """
-    Get the matrix of transition operators between two atomic shell in the complex
-    spherical harmonics basis.
-
-    Parameters
-    ----------
-    case: string
-        A string indicating the two atomic shells, possible transitions are:
-
-        -   'ss':     :math:`s \\rightarrow s`
-        -   'ps':     :math:`s \\rightarrow p`
-        -   't2gs':   :math:`s \\rightarrow t2g`
-        -   'ds':     :math:`s \\rightarrow d`
-        -   'fs':     :math:`s \\rightarrow f`
-        -   'sp':     :math:`p \\rightarrow s`
-        -   'sp12':   :math:`p_{1/2} \\rightarrow s`
-        -   'sp32':   :math:`p_{3/2} \\rightarrow s`
-        -   'pp':     :math:`p \\rightarrow p`
-        -   'pp12':   :math:`p_{1/2} \\rightarrow p`
-        -   'pp32':   :math:`p_{3/2} \\rightarrow p`
-        -   't2gp':   :math:`p \\rightarrow t_{2g}`
-        -   't2gp12': :math:`p_{1/2} \\rightarrow t_{2g}`
-        -   't2gp32': :math:`p_{3/2} \\rightarrow t_{2g}`
-        -   'dp':     :math:`p \\rightarrow d`
-        -   'dp12':   :math:`p_{1/2} \\rightarrow d`
-        -   'dp32':   :math:`p_{3/2} \\rightarrow d`
-        -   'fp':     :math:`p \\rightarrow f`
-        -   'fp12':   :math:`p_{1/2} \\rightarrow f`
-        -   'fp32':   :math:`p_{3/2} \\rightarrow f`
-        -   'sd':     :math:`d \\rightarrow s`
-        -   'sd32':   :math:`d_{3/2} \\rightarrow s`
-        -   'sd52':   :math:`d_{5/2} \\rightarrow s`
-        -   'pd':     :math:`d \\rightarrow p`
-        -   'pd32':   :math:`d_{3/2} \\rightarrow p`
-        -   'pd52':   :math:`d_{5/2} \\rightarrow p`
-        -   't2gd':   :math:`d \\rightarrow t_{2g}`
-        -   't2gd32': :math:`d_{3/2} \\rightarrow t_{2g}`
-        -   't2gd52': :math:`d_{5/2} \\rightarrow t_{2g}`
-        -   'dd':     :math:`d \\rightarrow d`
-        -   'dd32':   :math:`d_{3/2} \\rightarrow d`
-        -   'dd52':   :math:`d_{5/2} \\rightarrow d`
-        -   'fd':     :math:`d \\rightarrow f`
-        -   'fd32':   :math:`d_{3/2} \\rightarrow f`
-        -   'fd52':   :math:`d_{5/2} \\rightarrow f`
-        -   'sf':     :math:`f \\rightarrow s`
-        -   'sf52':   :math:`f_{5/2} \\rightarrow s`
-        -   'sf72':   :math:`f_{7/2} \\rightarrow s`
-        -   'pf':     :math:`f \\rightarrow p`
-        -   'pf52':   :math:`f_{5/2} \\rightarrow p`
-        -   'pf72':   :math:`f_{7/2} \\rightarrow p`
-        -   't2gf':   :math:`f \\rightarrow t_{2g}`
-        -   't2gf52': :math:`f_{5/2} \\rightarrow t_{2g}`
-        -   't2gf72': :math:`f_{7/2} \\rightarrow t_{2g}`
-        -   'df':     :math:`f \\rightarrow d`
-        -   'df52':   :math:`f_{5/2} \\rightarrow d`
-        -   'df72':   :math:`f_{7/2} \\rightarrow d`
-        -   'ff':     :math:`f \\rightarrow f`
-        -   'ff52':   :math:`f_{5/2} \\rightarrow f`
-        -   'ff72':   :math:`f_{7/2} \\rightarrow f`
-
-    Returns
-    -------
-    res: 2d complex array
-        The calculated transition matrix.
-
-    Examples
-    --------
-    >>> import edrixs
-    p to d transition
-    >>> trans_dp = get_trans_oper('dp')
-    p to t2g transition
-    >>> trans_t2gp = get_trans_oper('t2gp')
-    p_{3/2} to d transition
-    >>> trans_dp32 = get_trans_oper('dp32')
-    """
-    info = info_atomic_shell()
-    v_name, c_name = case_to_shell_name(case.strip())
-    v_orbl, c_orbl = info[v_name][0], info[c_name][0]
-    v_norb, c_norb = 2 * (2 * v_orbl + 1), 2 * (2 * c_orbl + 1)
-    if (v_orbl + c_orbl) % 2 == 0:
-        op = quadrupole_trans_oper(v_orbl, c_orbl)
-    else:
-        op = dipole_trans_oper(v_orbl, c_orbl)
-
-    # truncate to a sub-shell if necessary
-    special_shell = ['t2g', 'p12', 'p32', 'd32', 'd52', 'f52', 'f72']
-    orb_indx = {
-        special_shell[0]: [2, 3, 4, 5, 8, 9],
-        special_shell[1]: [0, 1],
-        special_shell[2]: [2, 3, 4, 5],
-        special_shell[3]: [0, 1, 2, 3],
-        special_shell[4]: [4, 5, 6, 7, 8, 9],
-        special_shell[5]: [0, 1, 2, 3, 4, 5],
-        special_shell[6]: [6, 7, 8, 9, 10, 11, 12, 13]
-    }
-    left_tmat = np.eye(v_norb, dtype=complex)
-    right_tmat = np.eye(c_norb, dtype=complex)
-    indx1 = list(range(0, v_norb))
-    indx2 = list(range(0, c_norb))
-    if v_name in special_shell:
-        if v_name == 't2g':
-            left_tmat[0:v_norb, 0:v_norb] = tmat_c2r('d', True)
-        else:
-            left_tmat[0:v_norb, 0:v_norb] = tmat_c2j(v_orbl)
-        indx1 = orb_indx[v_name]
-    if c_name in special_shell[1:]:
-        right_tmat[0:c_norb, 0:c_norb] = tmat_c2j(c_orbl)
-        indx2 = orb_indx[c_name]
-
-    if (v_orbl + c_orbl) % 2 == 0:
-        npol = 5
-    else:
-        npol = 3
-
-    op_tmp = np.zeros((npol, len(indx1), len(indx2)), dtype=complex)
-    for i in range(npol):
-        op[i] = cb_op2(op[i], left_tmat, right_tmat)
-        op_tmp[i] = op[i, indx1][:, indx2]
-        if v_name == 't2g':
-            op_tmp[i] = np.dot(np.conj(np.transpose(tmat_r2c('t2g', True))), op_tmp[i])
-    res = op_tmp
-    return res

-
-
-
-

-[docs]
-def unit_wavevector(theta, phi, local_axis=None, direction='in'):
-    """
-    Given incident or scattered angle, and azimuthal angle, return
-    unit wavevector with respect to global :math:`xyz`-axis.
-
-    Parameters
-    ----------
-    theta: float number
-        Incident or scattered angle (in radian), with respect to local_aixs.
-    phi: float number
-        Azimuthal angle (in radian), with respect to the :math:`x` of local_axis.
-    local_axis: 3*3 float array
-        The local axis defining the scattering geometry.
-
-        - :math:`x`-axis: local_axis[:,0]
-
-        - :math:`y`-axis: local_axis[:,1]
-
-        - :math:`z`-axis: local_axis[:,2]
-
-        It will be an identity matrix if not provided.
-    direction: string
-        The direction of photon wave, options can be
-
-        - 'in': incident photon
-
-        - 'out': scattered photon
-
-    Returns
-    -------
-    unit_k: list of 3 float numbers
-        The unit wavevector.
-    """
-    if local_axis is None:
-        local_axis = np.eye(3)
-    else:
-        local_axis = np.array(local_axis)
-
-    if direction.strip() == 'in':
-        unit_k = np.array([-np.cos(theta) * np.cos(phi),
-                           -np.cos(theta) * np.sin(phi),
-                           -np.sin(theta)])
-        unit_k = np.dot(local_axis, unit_k)
-    elif direction.strip() == 'out':
-        unit_k = np.array([-np.cos(theta) * np.cos(phi),
-                           -np.cos(theta) * np.sin(phi),
-                           np.sin(theta)])
-        unit_k = np.dot(local_axis, unit_k)
-    else:
-        raise Exception("Unknown direction in unit_wavevector: ", direction)
-
-    return unit_k

-
-
-
-

-[docs]
-def wavevector_with_length(theta, phi, energy, local_axis=None, direction='in'):
-    """
-    Given incident or scattered angle, azimuthal angle, energy of photon, return
-    wavevector with respect to global :math:`xyz`-axis.
-
-    Parameters
-    ----------
-    theta: float number
-        Incident or scattered angle (in radian), with respect to local_aixs.
-    phi: float number
-        Azimuthal angle (in radian), with respect to the :math:`x` of local_axis.
-    energy: float number
-        Energy of photon (in eV).
-    local_axis: 3*3 float array
-        The local axis defining the scattering geometry.
-
-        - :math:`x`-axis: local_axis[:,0]
-
-        - :math:`y`-axis: local_axis[:,1]
-
-        - :math:`z`-axis: local_axis[:,2]
-
-        It will be an identity matrix if not provided.
-    direction: string
-        The direction of photon wave, options can be
-
-        - 'in': incident photon
-
-        - 'out': scattered photon
-
-    Returns
-    -------
-    k_with_length: list of 3 float numbers
-        The wavevector with length.
-    """
-
-    hbarc = 1.973270533 * 1000  # eV*A
-    k_len = energy / hbarc
-    if local_axis is None:
-        local_axis = np.eye(3)
-    else:
-        local_axis = np.array(local_axis)
-
-    k_with_length = k_len * unit_wavevector(theta, phi, local_axis, direction)
-
-    return k_with_length

-
-
-
-

-[docs]
-def get_wavevector_rixs(thin, thout, phi, ein, eout, local_axis=None):
-    """
-    Return the wave vector of incident and scattered photons, for RIXS calculation.
-
-    Parameters
-    ----------
-    thin: float
-        The incident angle in radian.
-    thout: float
-        The scattered angle in radian.
-    phi: float
-        The azimuthal angle in radian.
-    ein: float
-        Energy of the incident photon (eV).
-    eout: float
-        Energy of the scattered photon (eV).
-    local_axis: :math:`3 \\times 3` float array
-        The local :math:`z` -axis, the angle thin and thout are defined with respect to this axis.
-
-        - :math:`x`-axis: local_axis[:,0]
-
-        - :math:`y`-axis: local_axis[:,1]
-
-        - :math:`z`-axis: local_axis[:,2]
-
-        It will be an identity matrix if not provided.
-
-    Returns
-    -------
-    k_in_global: 3-length float array
-        The wave vector of the incident photon, with respect to the global :math:`xyz` -axis.
-    k_out_global: 3-length float array
-        The wave vector of the scattered photon, with respect to the global :math:`xyz` -axis.
-    """
-    if local_axis is None:
-        local_axis = np.eye(3)
-    else:
-        local_axis = np.array(local_axis)
-
-    k_in_global = wavevector_with_length(thin, phi, ein, local_axis, direction='in')
-    k_out_global = wavevector_with_length(thout, phi, eout, local_axis, direction='out')
-
-    return k_in_global, k_out_global

-
-
-
-

-[docs]
-def linear_polvec(theta, phi, alpha, local_axis=None, direction='in'):
-    """
-    Return linear polarization vector.
-
-    Parameters
-    ----------
-    theta: float number
-        Incident or scattered angle (in radian) with respect to local_axis.
-    phi: float number
-        Azimuthal angle (in radian) with respect to the :math:`x` of local_axis.
-    alpha: float number
-        The angle (in radian) between the polarization vector and the scattering plane.
-
-    local_axis: 3*3 float array
-        The local axis defining the scattering geometry.
-
-        - :math:`x`-axis: local_axis[:,0]
-
-        - :math:`y`-axis: local_axis[:,1]
-
-        - :math:`z`-axis: local_axis[:,2]
-
-        It will be an identity matrix if not provided.
-    direction: string
-        The direction of photon wave, options can be
-
-        - 'in': incident photon
-
-        - 'out': scattered photon
-
-    Returns
-    -------
-    polvec: list of 3 float number
-        The polarization vector.
-    """
-    if local_axis is None:
-        local_axis = np.eye(3)
-    else:
-        local_axis = np.array(local_axis)
-
-    if direction.strip() == 'in':
-        polvec = (
-            np.array([-np.cos(phi) * np.cos(np.pi / 2.0 - theta),
-                      -np.sin(phi) * np.cos(np.pi / 2.0 - theta),
-                      +np.sin(np.pi / 2.0 - theta)]) * np.cos(alpha) +
-            np.array([-np.sin(phi), np.cos(phi), 0]) * np.sin(alpha)
-        )
-        polvec = np.dot(local_axis, polvec)
-    elif direction.strip() == 'out':
-        polvec = (
-            np.array([+np.cos(phi) * np.cos(np.pi / 2.0 - theta),
-                      +np.sin(phi) * np.cos(np.pi / 2.0 - theta),
-                      +np.sin(np.pi / 2.0 - theta)]) * np.cos(alpha) +
-            np.array([-np.sin(phi), np.cos(phi), 0]) * np.sin(alpha)
-        )
-        polvec = np.dot(local_axis, polvec)
-    else:
-        raise Exception("Unknown direction in linear_polvec: ", direction)
-
-    return polvec

-
-
-
-

-[docs]
-def dipole_polvec_rixs(thin, thout, phi=0, alpha=0, beta=0, local_axis=None, pol_type=None):
-    """
-    Return polarization vector of incident and scattered photons, for RIXS calculation.
-
-    Parameters
-    ----------
-    thin: float
-        The incident angle  (radian).
-    thout: float
-        The scattered angle (radian).
-    phi: float
-        The azimuthal angle (radian).
-    alpha: float
-        The angle between the polarization vector of the incident photon and
-        the scattering plane (radian)
-    beta: float
-        The angle between the polarization vector of the scattered photon and
-        the scattering plane (radian)
-    local_axis: 3*3 float array
-        The local axis defining the scattering geometry.
-
-        - :math:`x`-axis: local_axis[:,0]
-
-        - :math:`y`-axis: local_axis[:,1]
-
-        - :math:`z`-axis: local_axis[:,2]
-
-        It will be an identity matrix if not provided.
-    pol_type: tuple of two strings
-        Specify types of polarization for incident and scattered photons.
-        case[0] for incident photon, case[1] for scattered photon. Options can be
-
-        - 'linear': Linear polarization
-        - 'left'  : Left-circular polarization.
-        - 'right' : Right-circular polarization.
-
-        It will set pol_type=('linear', 'linear') if not provided.
-
-    Returns
-    -------
-    ei_in_global: 3-length complex array
-        The linear polarization vector of the incident photon,
-        with respect to the global :math:`xyz` -axis.
-    ef_out_global: 3-length complex array
-        The linear polarization vector of the scattered photon
-        with respect to the global :math:`xyz` -axis.
-    """
-    if local_axis is None:
-        local_axis = np.eye(3)
-    else:
-        local_axis = np.array(local_axis)
-
-    if pol_type is None:
-        pol_type = ('linear', 'linear')
-
-    ex = linear_polvec(thin, phi, 0, local_axis, direction='in')
-    ey = linear_polvec(thin, phi, np.pi/2.0, local_axis, direction='in')
-    if pol_type[0].strip() == 'linear':
-        ei_global = linear_polvec(thin, phi, alpha, local_axis, direction='in')
-    elif pol_type[0].strip() == 'left':
-        ei_global = (ex + 1j * ey) / np.sqrt(2.0)
-    elif pol_type[0].strip() == 'right':
-        ei_global = (ex - 1j * ey) / np.sqrt(2.0)
-    else:
-        raise Exception("Unknown polarization type for incident photon: ", pol_type[0])
-
-    ex = linear_polvec(thout, phi, 0, local_axis, direction='out')
-    ey = linear_polvec(thout, phi, np.pi/2.0, local_axis, direction='out')
-    if pol_type[1].strip() == 'linear':
-        ef_global = linear_polvec(thout, phi, beta, local_axis, direction='out')
-    elif pol_type[1].strip() == 'left':
-        ef_global = (ex + 1j * ey) / np.sqrt(2.0)
-    elif pol_type[1].strip() == 'right':
-        ef_global = (ex - 1j * ey) / np.sqrt(2.0)
-    else:
-        raise Exception("Unknown polarization type for scattered photon: ", pol_type[1])
-
-    return ei_global, ef_global

-
-
-
-

-[docs]
-def dipole_polvec_xas(thin, phi=0, alpha=0, local_axis=None, pol_type='linear'):
-    """
-    Return the linear polarization vector of incident photons, for XAS calculation.
-
-    Parameters
-    ----------
-    thin: float
-        The incident angle  (radian).
-    phi: float
-        The azimuthal angle (radian).
-    alpha: float
-        The angle between the polarization vector of the incident photon and
-        the scattering plane (radian)
-    local_axis: 3*3 float array
-        The local axis defining the scattering geometry.
-
-         - :math:`x`-axis: local_axis[:,0]
-
-        - :math:`y`-axis: local_axis[:,1]
-
-        - :math:`z`-axis: local_axis[:,2]
-
-        It will be an identity matrix if not provided.
-    pol_type: string
-
-        - 'linear': Linear polarization.
-        - 'left'  : Left-circular polarization.
-        - 'right' : Right-circular polarization.
-
-    Returns
-    -------
-    ei_global: 3-length float array
-        The linear polarization vector of the incident photon, with resepct to the
-        global :math:`xyz` -axis.
-    """
-    if local_axis is None:
-        local_axis = np.eye(3)
-    else:
-        local_axis = np.array(local_axis)
-
-    ex = linear_polvec(thin, phi, 0, local_axis, direction='in')
-    ey = linear_polvec(thin, phi, np.pi/2.0, local_axis, direction='in')
-    if pol_type.strip() == 'linear':
-        ei_global = linear_polvec(thin, phi, alpha, local_axis, direction='in')
-    elif pol_type.strip() == 'left':
-        ei_global = (ex + 1j * ey) / np.sqrt(2.0)
-    elif pol_type.strip() == 'right':
-        ei_global = (ex - 1j * ey) / np.sqrt(2.0)
-    else:
-        raise Exception("Unknown polarization type for incident photon: ", pol_type)
-
-    return ei_global

-
-
-
-

-[docs]
-def quadrupole_polvec(polvec, wavevec):
-    """
-    Given dipolar polarization vector and wave-vector, return quadrupolar polarization vector.
-
-    Parameters
-    ----------
-    polvec: 3 elements of complex array
-        Dipolar polarization vector of photon, :math:`\\epsilon_{x}, \\epsilon_{y}, \\epsilon_{z}`,
-        NOTE: they can be complex when the polarization is circular.
-    wavevec: 3 elements of float array
-        Wavevector of photon, :math:`k_{x}, k_{y}, k_{z}`.
-
-    Returns
-    -------
-    quad_vec: 5 elements of float array
-        Quadrupolar polarization vector.
-    """
-
-    quad_vec = np.zeros(5, dtype=complex)
-    kvec = wavevec / np.sqrt(np.dot(wavevec, wavevec))
-
-    quad_vec[0] = 0.5 * (2 * polvec[2] * kvec[2] - polvec[0] * kvec[0] - polvec[1] * kvec[1])
-    quad_vec[1] = np.sqrt(3.0)/2.0 * (polvec[2] * kvec[0] + polvec[0] * kvec[2])
-    quad_vec[2] = np.sqrt(3.0)/2.0 * (polvec[1] * kvec[2] + polvec[2] * kvec[1])
-    quad_vec[3] = np.sqrt(3.0)/2.0 * (polvec[0] * kvec[0] - polvec[1] * kvec[1])
-    quad_vec[4] = np.sqrt(3.0)/2.0 * (polvec[0] * kvec[1] + polvec[1] * kvec[0])
-
-    return quad_vec

-
-
- -
-
-
- -
- -
-

© Copyright 2019, Brookhaven National Lab.

-
- - Built with Sphinx using a - theme - provided by Read the Docs. - - -
-
-
-
-
- - - - \ No newline at end of file diff --git a/edrixs/_modules/edrixs/plot_spectrum.html b/edrixs/_modules/edrixs/plot_spectrum.html deleted file mode 100644 index 065cfe5b69..0000000000 --- a/edrixs/_modules/edrixs/plot_spectrum.html +++ /dev/null @@ -1,296 +0,0 @@ - - - - - - - - edrixs.plot_spectrum — edrixs documentation - - - - - - - - - - - - - - - - - - - - - -
- - -
- -
-
-
- -
-
-
-
- -

Source code for edrixs.plot_spectrum

-__all__ = ['get_spectra_from_poles', 'merge_pole_dicts', 'plot_spectrum', 'plot_rixs_map']
-
-import numpy as np
-import matplotlib.pyplot as plt
-from .utils import boltz_dist
-from .iostream import read_poles_from_file
-
-
-

-[docs]
-def get_spectra_from_poles(poles_dict, omega_mesh, gamma_mesh, temperature):
-    """
-    Given the dict of poles, calculate XAS or RIXS spectra using continued fraction formula,
-
-    .. math::
-        I(\\omega_{i}) =-\\frac{1}{\\pi}\\text{Im} \\left[ \\frac{1}{x - \\alpha_{0} -
-        \\frac{\\beta_{1}^2}{x-\\alpha_{1} - \\frac{\\beta_{2}^2}{x-\\alpha_{2} - ...}} }\\right],
-
-    where, :math:`x = \\omega_{i}+i\\Gamma_{i} + E_{g}`.
-
-    Parameters
-    ----------
-    poles_dict: dict
-        Dict containing information of poles, which are calculated from
-        xas_fsolver and rixs_fsolver.
-        This dict is constructed by :func:`iostream.read_poles_from_file`.
-    omega_mesh: 1d float array
-        Energy grid.
-    gamma_mesh: 1d float array
-        Life-time broadening.
-    temperature: float number
-        Temperature (K) for boltzmann distribution.
-
-    Returns
-    -------
-    spectra: 1d float array
-        The calculated XAS or RIXS spectra.
-
-    See also
-    --------
-    iostream.read_poles_from_file: read XAS or RIXS poles files.
-    """
-    nom = len(omega_mesh)
-    spectra = np.zeros(nom, dtype=np.float64)
-    gs_dist = boltz_dist(poles_dict['eigval'], temperature)
-    ngs = len(poles_dict['eigval'])
-    for i in range(ngs):
-        tmp_vec = np.zeros(nom, dtype=complex)
-        neff = poles_dict['npoles'][i]
-        alpha = poles_dict['alpha'][i]
-        beta = poles_dict['beta'][i]
-        eigval = poles_dict['eigval'][i]
-        norm = poles_dict['norm'][i]
-        for j in range(neff-1, 0, -1):
-            tmp_vec = (
-                beta[j-1]**2 / (omega_mesh + 1j * gamma_mesh + eigval - alpha[j] - tmp_vec)
-            )
-        tmp_vec = (
-            1.0 / (omega_mesh + 1j * gamma_mesh + eigval - alpha[0] - tmp_vec)
-        )
-        spectra[:] += -1.0 / np.pi * np.imag(tmp_vec) * norm * gs_dist[i]
-
-    return spectra

-
-
-
-

-[docs]
-def merge_pole_dicts(list_pole_dict):
-    """
-    Given a list of dict of poles, merge them into one dict of poles
-
-    Parameters
-    ----------
-    list_pole_dict:  list of dict
-        Dict containing information of poles, which are calculated from
-        xas_fsolver and rixs_fsolver.
-
-    Returns
-    -------
-    new_pole_dict: dict of poles
-        New dict of poles.
-    """
-    new_pole_dict = {
-        'eigval': [],
-        'npoles': [],
-        'norm': [],
-        'alpha': [],
-        'beta': []
-    }
-    for poles_dict in list(list_pole_dict):
-        new_pole_dict['eigval'].extend(poles_dict['eigval'])
-        new_pole_dict['npoles'].extend(poles_dict['npoles'])
-        new_pole_dict['norm'].extend(poles_dict['norm'])
-        new_pole_dict['alpha'].extend(poles_dict['alpha'])
-        new_pole_dict['beta'].extend(poles_dict['beta'])
-
-    return new_pole_dict

-
-
-
-

-[docs]
-def plot_spectrum(file_list, omega_mesh, gamma_mesh, T=1.0, fname='spectrum.dat',
-                  om_shift=0.0, fmt_float='{:.15f}'):
-    """
-    Reading poles :math:`\\alpha` and :math:`\\beta`, and calculate
-    the spectrum using continued fraction formula,
-
-    .. math::
-        I(\\omega_{i}) =-\\frac{1}{\\pi}\\text{Im} \\left[ \\frac{1}{x - \\alpha_{0} -
-        \\frac{\\beta_{1}^2}{x-\\alpha_{1} - \\frac{\\beta_{2}^2}{x-\\alpha_{2} - ...}} }\\right],
-
-    where, :math:`x = \\omega_{i}+i\\Gamma_{i} + E_{g}`.
-
-    Parameters
-    ----------
-    file_list: list of string
-        Name of poles file.
-    omega_mesh: 1d float array
-        The frequency mesh.
-    gamma_mesh: 1d float array
-        The broadening factor, in general, it is frequency dependent.
-    T: float (default: 1.0K)
-        Temperature (K).
-    fname: str (default: 'spectrum.dat')
-        File name to store spectrum.
-    om_shift: float (default: 0.0)
-        Energy shift.
-    fmt_float: str (default: '{:.15f}')
-        Format for printing float numbers.
-    """
-
-    pole_dict = read_poles_from_file(file_list)
-    spectrum = get_spectra_from_poles(pole_dict, omega_mesh, gamma_mesh, T)
-
-    space = "  "
-    fmt_string = (fmt_float + space) * 2 + '\n'
-    f = open(fname, 'w')
-    for i in range(len(omega_mesh)):
-        f.write(fmt_string.format(omega_mesh[i] + om_shift, spectrum[i]))
-    f.close()

-
-
-
-

-[docs]
-def plot_rixs_map(rixs_data, ominc_mesh, eloss_mesh, fname='rixsmap.pdf'):
-    """
-    Given 2d RIXS data, plot a RIXS map and save it to a pdf file.
-
-    Parameters
-    ----------
-    rixs_data: 2d float array
-        Calculated RIXS data as a function of incident energy and energy loss.
-    ominc_mesh: 1d float array
-        Incident energy mesh.
-    eloss_mesh: 1d float array
-        Energy loss mesh.
-    fname: string
-        File name to save RIXS map.
-    """
-
-    fig, ax = plt.subplots()
-    a, b, c, d = min(eloss_mesh), max(eloss_mesh), min(ominc_mesh), max(ominc_mesh)
-    m, n = np.array(rixs_data).shape
-    if len(ominc_mesh) == m and len(eloss_mesh) == n:
-        plt.imshow(
-            rixs_data, extent=[a, b, c, d], origin='lower', aspect='auto',
-            cmap='rainbow', interpolation='gaussian'
-        )
-        plt.xlabel(r'Energy loss (eV)')
-        plt.ylabel(r'Energy of incident photon (eV)')
-    elif len(eloss_mesh) == m and len(ominc_mesh) == n:
-        plt.imshow(
-            rixs_data, extent=[c, d, a, b], origin='lower', aspect='auto',
-            cmap='rainbow', interpolation='gaussian'
-        )
-        plt.ylabel(r'Energy loss (eV)')
-        plt.xlabel(r'Energy of incident photon (eV)')
-    else:
-        raise Exception(
-            "Dimension of rixs_data is not consistent with ominc_mesh or eloss_mesh"
-        )
-
-    plt.savefig(fname)

-
-
- -
-
-
- -
- -
-

© Copyright 2019, Brookhaven National Lab.

-
- - Built with Sphinx using a - theme - provided by Read the Docs. - - -
-
-
-
-
- - - - \ No newline at end of file diff --git a/edrixs/_modules/edrixs/rixs_utils.html b/edrixs/_modules/edrixs/rixs_utils.html deleted file mode 100644 index 33aa449e0d..0000000000 --- a/edrixs/_modules/edrixs/rixs_utils.html +++ /dev/null @@ -1,175 +0,0 @@ - - - - - - - - edrixs.rixs_utils — edrixs documentation - - - - - - - - - - - - - - - - - - - - - -
- - -
- -
-
-
- -
-
-
-
- -

Source code for edrixs.rixs_utils

-__all__ = ['scattering_mat']
-
-import numpy as np
-
-
-

-[docs]
-def scattering_mat(eval_i, eval_n, trans_mat_abs,
-                   trans_mat_emi, omega_inc, gamma_n):
-    """
-    Calculate X-ray scattering magnitude.
-
-    .. math::
-
-        F^{ab}_{fi} = \\sum_{n}\\frac{<f|T_{a}|n><n|T_{b}|i>}{\\omega_{in}
-                    - E_{n} + E_{i} + i\\Gamma_{n}},
-
-    where, :math:`T_{a}` and :math:`T_{b}` are components of transition
-    operators( :math:`a,b=x,y,z`).
-
-    Parameters
-    ----------
-    eval_i: 1d float array
-        Eigenvalues of the initial configuration without core-hole, :math:`E_i`.
-    eval_n: 1d float array
-        Eigenvalues of the intermediate configuration with core-hole, :math:`E_n`.
-    trans_mat_abs: 3d complex array
-        The transition operator for absorption process, :math:`<n|T_{b}|i>`.
-    trans_mat_emi: 3d complex array
-        The transition operator for emission process, :math:`<f|T_{a}|n>`.
-    omega_inc: float
-        The energy of incident photon, :math:`\\omega_{in}`.
-    gamma_n: float
-        The broadening of the core-hole (eV), :math:`\\Gamma_{n}`.
-
-    Returns
-    -------
-    Ffi: 4d complex array
-        The calculated scattering magnitude, :math:`F^{ab}_{fi}`.
-    """
-
-    num_gs = trans_mat_abs.shape[2]
-    num_ex = trans_mat_abs.shape[1]
-    num_fs = trans_mat_emi.shape[1]
-
-    npol_abs = trans_mat_abs.shape[0]
-    npol_emi = trans_mat_emi.shape[0]
-
-    Ffi = np.zeros((npol_emi, npol_abs, num_fs, num_gs), dtype=np.complex128)
-    tmp_abs = np.zeros((npol_abs, num_ex, num_gs), dtype=np.complex128)
-    denomi = np.zeros((num_ex, num_gs), dtype=np.complex128)
-
-    for i in range(num_ex):
-        for j in range(num_gs):
-            aa = omega_inc - (eval_n[i] - eval_i[j])
-            denomi[i, j] = 1.0 / (aa + 1j * gamma_n)
-
-    for i in range(npol_abs):
-        tmp_abs[i] = trans_mat_abs[i] * denomi
-
-    for i in range(npol_emi):
-        for j in range(npol_abs):
-            Ffi[i, j, :, :] = np.dot(trans_mat_emi[i], tmp_abs[j])
-
-    return Ffi

-
-
- -
-
-
- -
- -
-

© Copyright 2019, Brookhaven National Lab.

-
- - Built with Sphinx using a - theme - provided by Read the Docs. - - -
-
-
-
-
- - - - \ No newline at end of file diff --git a/edrixs/_modules/edrixs/soc.html b/edrixs/_modules/edrixs/soc.html deleted file mode 100644 index 0aa22d618e..0000000000 --- a/edrixs/_modules/edrixs/soc.html +++ /dev/null @@ -1,219 +0,0 @@ - - - - - - - - edrixs.soc — edrixs documentation - - - - - - - - - - - - - - - - - - - - - -
- - -
- -
-
-
- -
-
-
-
- -

Source code for edrixs.soc

-__all__ = ['atom_hsoc']
-
-import numpy as np
-
-
-

-[docs]
-def atom_hsoc(case, soc):
-    """
-    Return atomic spin-orbit coupling matrix :math:`\\vec{l}\\cdot\\vec{s}`
-    in complex spherical harmonics basis.
-
-    Parameters
-    ----------
-    case: str
-        String label indicating atomic shell,
-
-        - 'p': for :math:`p` -shell.
-        - 't2g': for :math:`t_{2g}` -shell.
-        - 'd': for :math:`d` -shell.
-        - 'f': for :math:`f` -shell.
-    soc: float
-        The strength of spin-orbit coupling.
-
-    Returns
-    -------
-    hsoc: 2d complex array
-        The spin-orbit coupling matrix.
-    """
-
-    sqrt2 = np.sqrt(2.0)
-    sqrt6 = np.sqrt(6.0)
-    sqrt10 = np.sqrt(10.0)
-    sqrt12 = np.sqrt(12.0)
-
-    if case.strip() == 'p':
-        hsoc = np.zeros((6, 6), dtype=np.complex128)
-        hsoc[0, 0] = -1.0
-        hsoc[3, 0] = sqrt2
-        hsoc[1, 1] = 1.0
-        hsoc[5, 2] = sqrt2
-        hsoc[0, 3] = sqrt2
-        hsoc[4, 4] = 1.0
-        hsoc[2, 5] = sqrt2
-        hsoc[5, 5] = -1.0
-        return 0.5 * soc * hsoc
-
-    elif case.strip() == 't2g':
-        hsoc = np.zeros((6, 6), dtype=np.complex128)
-        hsoc[0, 0] = -1.0
-        hsoc[3, 0] = sqrt2
-        hsoc[1, 1] = 1.0
-        hsoc[5, 2] = sqrt2
-        hsoc[0, 3] = sqrt2
-        hsoc[4, 4] = 1.0
-        hsoc[2, 5] = sqrt2
-        hsoc[5, 5] = -1.0
-        return 0.5 * -soc * hsoc
-
-    elif case.strip() == 'd':
-        hsoc = np.zeros((10, 10), dtype=np.complex128)
-        hsoc[0, 0] = -2.0
-        hsoc[3, 0] = 2.0
-        hsoc[1, 1] = 2.0
-        hsoc[2, 2] = -1.0
-        hsoc[5, 2] = sqrt6
-        hsoc[0, 3] = 2.0
-        hsoc[3, 3] = 1.0
-        hsoc[7, 4] = sqrt6
-        hsoc[2, 5] = sqrt6
-        hsoc[6, 6] = 1.0
-        hsoc[9, 6] = 2.0
-        hsoc[4, 7] = sqrt6
-        hsoc[7, 7] = -1.0
-        hsoc[8, 8] = 2.0
-        hsoc[6, 9] = 2.0
-        hsoc[9, 9] = -2.0
-        return 0.5 * soc * hsoc
-
-    elif case.strip() == 'f':
-        hsoc = np.zeros((14, 14), dtype=np.complex128)
-        hsoc[0, 0] = -3.0
-        hsoc[3, 0] = sqrt6
-        hsoc[1, 1] = 3.0
-        hsoc[2, 2] = -2.0
-        hsoc[5, 2] = sqrt10
-        hsoc[0, 3] = sqrt6
-        hsoc[3, 3] = 2.0
-        hsoc[4, 4] = -1.0
-        hsoc[7, 4] = sqrt12
-        hsoc[2, 5] = sqrt10
-        hsoc[5, 5] = 1.0
-        hsoc[9, 6] = sqrt12
-        hsoc[4, 7] = sqrt12
-        hsoc[8, 8] = 1.0
-        hsoc[11, 8] = sqrt10
-        hsoc[6, 9] = sqrt12
-        hsoc[9, 9] = -1.0
-        hsoc[10, 10] = 2.0
-        hsoc[13, 10] = sqrt6
-        hsoc[8, 11] = sqrt10
-        hsoc[11, 11] = -2.0
-        hsoc[12, 12] = 3.0
-        hsoc[10, 13] = sqrt6
-        hsoc[13, 13] = -3.0
-        return 0.5 * soc * hsoc
-
-    else:
-        raise Exception("error in atom_hsoc: DO NOT support SOC for this case: ", case)

-
-
- -
-
-
- -
- -
-

© Copyright 2019, Brookhaven National Lab.

-
- - Built with Sphinx using a - theme - provided by Read the Docs. - - -
-
-
-
-
- - - - \ No newline at end of file diff --git a/edrixs/_modules/edrixs/solvers.html b/edrixs/_modules/edrixs/solvers.html deleted file mode 100644 index d73430c521..0000000000 --- a/edrixs/_modules/edrixs/solvers.html +++ /dev/null @@ -1,2849 +0,0 @@ - - - - - - - - edrixs.solvers — edrixs documentation - - - - - - - - - - - - - - - - - - - - - -
- - -
- -
-
-
- -
-
-
-
- -

Source code for edrixs.solvers

-__all__ = ['ed_1v1c_py', 'xas_1v1c_py', 'rixs_1v1c_py',
-           'ed_1v1c_fort', 'xas_1v1c_fort', 'rixs_1v1c_fort',
-           'ed_2v1c_fort', 'xas_2v1c_fort', 'rixs_2v1c_fort',
-           'ed_siam_fort', 'xas_siam_fort', 'rixs_siam_fort']
-
-import numpy as np
-import scipy
-
-from .iostream import (
-    write_tensor, write_emat, write_umat, write_config, read_poles_from_file
-)
-from .angular_momentum import (
-    get_sx, get_sy, get_sz, get_lx, get_ly, get_lz, rmat_to_euler, get_wigner_dmat
-)
-from .photon_transition import (
-    get_trans_oper, quadrupole_polvec, dipole_polvec_xas, dipole_polvec_rixs, unit_wavevector
-)
-from .coulomb_utensor import get_umat_slater, get_umat_slater_3shells
-from .manybody_operator import two_fermion, four_fermion
-from .fock_basis import get_fock_bin_by_N, write_fock_dec_by_N
-from .basis_transform import cb_op2, tmat_r2c, cb_op
-from .utils import info_atomic_shell, slater_integrals_name, boltz_dist
-from .rixs_utils import scattering_mat
-from .plot_spectrum import get_spectra_from_poles, merge_pole_dicts
-from .soc import atom_hsoc
-
-
-

-[docs]
-def ed_1v1c_py(shell_name, *, shell_level=None, v_soc=None, c_soc=0,
-               v_noccu=1, slater=None, ext_B=None, on_which='spin',
-               v_cfmat=None, v_othermat=None, loc_axis=None, verbose=0):
-    """
-    Perform ED for the case of two atomic shells, one valence plus one Core
-    shell with pure Python solver.
-    For example, for Ni-:math:`L_3` edge RIXS, they are 3d valence and 2p core shells.
-
-    It will use scipy.linalag.eigh to exactly diagonalize both the initial and intermediate
-    Hamiltonians to get all the eigenvalues and eigenvectors, and the transition operators
-    will be built in the many-body eigenvector basis.
-
-    This solver is only suitable for small size of Hamiltonian, typically the dimension
-    of both initial and intermediate Hamiltonian are smaller than 10,000.
-
-    Parameters
-    ----------
-    shell_name: tuple of two strings
-        Names of valence and core shells. The 1st (2nd) string in the tuple is for the
-        valence (core) shell.
-
-        - The 1st string can only be 's', 'p', 't2g', 'd', 'f',
-
-        - The 2nd string can be 's', 'p', 'p12', 'p32', 'd', 'd32', 'd52',
-          'f', 'f52', 'f72'.
-
-        For example: shell_name=('d', 'p32') indicates a :math:`L_3` edge transition from
-        core :math:`p_{3/2}` shell to valence :math:`d` shell.
-    shell_level: tuple of two float numbers
-        Energy level of valence (1st element) and core (2nd element) shells.
-
-        They will be set to zero if not provided.
-    v_soc: tuple of two float numbers
-        Spin-orbit coupling strength of valence electrons, for the initial (1st element)
-        and intermediate (2nd element) Hamiltonians.
-
-        They will be set to zero if not provided.
-    c_soc: a float number
-        Spin-orbit coupling strength of core electrons.
-    v_noccu: int number
-        Number of electrons in valence shell.
-    slater: tuple of two lists
-        Slater integrals for initial (1st list) and intermediate (2nd list) Hamiltonians.
-        The order of the elements in each list should be like this:
-
-        [FX_vv, FX_vc, GX_vc, FX_cc],
-
-        where X are integers with ascending order, it can be X=0, 2, 4, 6 or X=1, 3, 5.
-        One can ignore all the continuous zeros at the end of the list.
-
-        For example, if the full list is: [F0_dd, F2_dd, F4_dd, 0, F2_dp, 0, 0, 0, 0], one can
-        just provide [F0_dd, F2_dd, F4_dd, 0, F2_dp]
-
-        All the Slater integrals will be set to zero if slater=None.
-    ext_B: tuple of three float numbers
-        Vector of external magnetic field with respect to global :math:`xyz`-axis.
-
-        They will be set to zero if not provided.
-    on_which: string
-        Apply Zeeman exchange field on which sector. Options are 'spin', 'orbital' or 'both'.
-    v_cfmat: 2d complex array
-        Crystal field splitting Hamiltonian of valence electrons. The dimension and the orbital
-        order should be consistent with the type of valence shell.
-
-        They will be zeros if not provided.
-    v_othermat: 2d complex array
-        Other possible Hamiltonian of valence electrons. The dimension and the orbital order
-        should be consistent with the type of valence shell.
-
-        They will be zeros if not provided.
-    loc_axis: 3*3 float array
-        The local axis with respect to which local orbitals are defined.
-
-        - x: local_axis[:,0],
-
-        - y: local_axis[:,1],
-
-        - z: local_axis[:,2].
-
-        It will be an identity matrix if not provided.
-    verbose: int
-        Level of writting data to files. Hopping matrices, Coulomb tensors, eigvenvalues
-        will be written if verbose > 0.
-
-    Returns
-    -------
-    eval_i:  1d float array
-        The eigenvalues of initial Hamiltonian.
-    eval_n: 1d float array
-        The eigenvalues of intermediate Hamiltonian.
-    trans_op: 3d complex array
-        The matrices of transition operators in the eigenvector basis.
-        Their components are defined with respect to the global :math:`xyz`-axis.
-    """
-    print("edrixs >>> Running ED ...")
-    v_name_options = ['s', 'p', 't2g', 'd', 'f']
-    c_name_options = ['s', 'p', 'p12', 'p32', 't2g', 'd', 'd32', 'd52', 'f', 'f52', 'f72']
-    v_name = shell_name[0].strip()
-    c_name = shell_name[1].strip()
-    if v_name not in v_name_options:
-        raise Exception("NOT supported type of valence shell: ", v_name)
-    if c_name not in c_name_options:
-        raise Exception("NOT supported type of core shell: ", c_name)
-
-    info_shell = info_atomic_shell()
-
-    # Quantum numbers of angular momentum
-    v_orbl = info_shell[v_name][0]
-
-    # number of orbitals including spin degree of freedom
-    v_norb = info_shell[v_name][1]
-    c_norb = info_shell[c_name][1]
-
-    # total number of orbitals
-    ntot = v_norb + c_norb
-
-    emat_i = np.zeros((ntot, ntot), dtype=complex)
-    emat_n = np.zeros((ntot, ntot), dtype=complex)
-
-    # Coulomb interaction
-    # Get the names of all the required slater integrals
-    slater_name = slater_integrals_name((v_name, c_name), ('v', 'c'))
-    nslat = len(slater_name)
-
-    slater_i = np.zeros(nslat, dtype=float)
-    slater_n = np.zeros(nslat, dtype=float)
-
-    if slater is not None:
-        if nslat > len(slater[0]):
-            slater_i[0:len(slater[0])] = slater[0]
-        else:
-            slater_i[:] = slater[0][0:nslat]
-        if nslat > len(slater[1]):
-            slater_n[0:len(slater[1])] = slater[1]
-        else:
-            slater_n[:] = slater[1][0:nslat]
-
-    # print summary of slater integrals
-    print()
-    print("    Summary of Slater integrals:")
-    print("    ------------------------------")
-    print("    Terms,   Initial Hamiltonian,  Intermediate Hamiltonian")
-    for i in range(nslat):
-        print("    ", slater_name[i], ":  {:20.10f}{:20.10f}".format(slater_i[i], slater_n[i]))
-    print()
-
-    case = v_name + c_name
-    umat_i = get_umat_slater(case, *slater_i)
-    umat_n = get_umat_slater(case, *slater_n)
-
-    if verbose > 0:
-        write_umat(umat_i, 'coulomb_i.in')
-        write_umat(umat_n, 'coulomb_n.in')
-
-    # SOC
-    if v_soc is not None:
-        emat_i[0:v_norb, 0:v_norb] += atom_hsoc(v_name, v_soc[0])
-        emat_n[0:v_norb, 0:v_norb] += atom_hsoc(v_name, v_soc[1])
-
-    # when the core-shell is any of p12, p32, d32, d52, f52, f72,
-    # do not need to add SOC for core shell
-    if c_name in ['p', 'd', 'f']:
-        emat_n[v_norb:ntot, v_norb:ntot] += atom_hsoc(c_name, c_soc)
-
-    # crystal field
-    if v_cfmat is not None:
-        emat_i[0:v_norb, 0:v_norb] += np.array(v_cfmat)
-        emat_n[0:v_norb, 0:v_norb] += np.array(v_cfmat)
-
-    # other hopping matrix
-    if v_othermat is not None:
-        emat_i[0:v_norb, 0:v_norb] += np.array(v_othermat)
-        emat_n[0:v_norb, 0:v_norb] += np.array(v_othermat)
-
-    # energy of shells
-    if shell_level is not None:
-        emat_i[0:v_norb, 0:v_norb] += np.eye(v_norb) * shell_level[0]
-        emat_i[v_norb:ntot, v_norb:ntot] += np.eye(c_norb) * shell_level[1]
-        emat_n[0:v_norb, 0:v_norb] += np.eye(v_norb) * shell_level[0]
-        emat_n[v_norb:ntot, v_norb:ntot] += np.eye(c_norb) * shell_level[1]
-
-    # external magnetic field
-    if v_name == 't2g':
-        lx, ly, lz = get_lx(1, True), get_ly(1, True), get_lz(1, True)
-        sx, sy, sz = get_sx(1), get_sy(1), get_sz(1)
-        lx, ly, lz = -lx, -ly, -lz
-    else:
-        lx, ly, lz = get_lx(v_orbl, True), get_ly(v_orbl, True), get_lz(v_orbl, True)
-        sx, sy, sz = get_sx(v_orbl), get_sy(v_orbl), get_sz(v_orbl)
-
-    if ext_B is not None:
-        if on_which.strip() == 'spin':
-            zeeman = ext_B[0] * (2 * sx) + ext_B[1] * (2 * sy) + ext_B[2] * (2 * sz)
-        elif on_which.strip() == 'orbital':
-            zeeman = ext_B[0] * lx + ext_B[1] * ly + ext_B[2] * lz
-        elif on_which.strip() == 'both':
-            zeeman = ext_B[0] * (lx + 2 * sx) + ext_B[1] * (ly + 2 * sy) + ext_B[2] * (lz + 2 * sz)
-        else:
-            raise Exception("Unknown value of on_which", on_which)
-        emat_i[0:v_norb, 0:v_norb] += zeeman
-        emat_n[0:v_norb, 0:v_norb] += zeeman
-
-    if verbose > 0:
-        write_emat(emat_i, 'hopping_i.in')
-        write_emat(emat_n, 'hopping_n.in')
-
-    basis_i = get_fock_bin_by_N(v_norb, v_noccu, c_norb, c_norb)
-    basis_n = get_fock_bin_by_N(v_norb, v_noccu+1, c_norb, c_norb - 1)
-    ncfg_i, ncfg_n = len(basis_i), len(basis_n)
-    print("edrixs >>> Dimension of the initial Hamiltonian: ", ncfg_i)
-    print("edrixs >>> Dimension of the intermediate Hamiltonian: ", ncfg_n)
-
-    # Build many-body Hamiltonian in Fock basis
-    print("edrixs >>> Building Many-body Hamiltonians ...")
-    hmat_i = np.zeros((ncfg_i, ncfg_i), dtype=complex)
-    hmat_n = np.zeros((ncfg_n, ncfg_n), dtype=complex)
-    hmat_i[:, :] += two_fermion(emat_i, basis_i, basis_i)
-    hmat_i[:, :] += four_fermion(umat_i, basis_i)
-    hmat_n[:, :] += two_fermion(emat_n, basis_n, basis_n)
-    hmat_n[:, :] += four_fermion(umat_n, basis_n)
-    print("edrixs >>> Done !")
-
-    # Do exact-diagonalization to get eigenvalues and eigenvectors
-    print("edrixs >>> Exact Diagonalization of Hamiltonians ...")
-    eval_i, evec_i = scipy.linalg.eigh(hmat_i)
-    eval_n, evec_n = scipy.linalg.eigh(hmat_n)
-    print("edrixs >>> Done !")
-
-    if verbose > 0:
-        write_tensor(eval_i, 'eval_i.dat')
-        write_tensor(eval_n, 'eval_n.dat')
-
-    # Build dipolar transition operators in local-xyz axis
-    if loc_axis is not None:
-        local_axis = np.array(loc_axis)
-    else:
-        local_axis = np.eye(3)
-    tmp = get_trans_oper(case)
-    npol, n, m = tmp.shape
-    tmp_g = np.zeros((npol, n, m), dtype=complex)
-    # Transform the transition operators to global-xyz axis
-    # dipolar transition
-    if npol == 3:
-        for i in range(3):
-            for j in range(3):
-                tmp_g[i] += local_axis[i, j] * tmp[j]
-
-    # quadrupolar transition
-    elif npol == 5:
-        alpha, beta, gamma = rmat_to_euler(local_axis)
-        wignerD = get_wigner_dmat(4, alpha, beta, gamma)
-        rotmat = np.dot(np.dot(tmat_r2c('d'), wignerD), np.conj(np.transpose(tmat_r2c('d'))))
-        for i in range(5):
-            for j in range(5):
-                tmp_g[i] += rotmat[i, j] * tmp[j]
-    else:
-        raise Exception("Have NOT implemented this case: ", npol)
-
-    tmp2 = np.zeros((npol, ntot, ntot), dtype=complex)
-    trans_op = np.zeros((npol, ncfg_n, ncfg_i), dtype=complex)
-    for i in range(npol):
-        tmp2[i, 0:v_norb, v_norb:ntot] = tmp_g[i]
-        trans_op[i] = two_fermion(tmp2[i], basis_n, basis_i)
-        trans_op[i] = cb_op2(trans_op[i], evec_n, evec_i)
-
-    print("edrixs >>> ED Done !")
-
-    return eval_i, eval_n, trans_op

-
-
-
-

-[docs]
-def xas_1v1c_py(eval_i, eval_n, trans_op, ominc, *, gamma_c=0.1, thin=1.0, phi=0,
-                pol_type=None, gs_list=None, temperature=1.0, scatter_axis=None):
-
-    """
-    Calculate XAS for the case of one valence shell plus one core shell with Python solver.
-
-    This solver is only suitable for small size of Hamiltonian, typically the dimension
-    of both initial and intermediate Hamiltonian are smaller than 10,000.
-
-    Parameters
-    ----------
-    eval_i: 1d float array
-        The eigenvalues of the initial Hamiltonian.
-    eval_n: 1d float array
-        The eigenvalues of the intermediate Hamiltonian.
-    trans_op: 3d complex array
-        The transition operators in the eigenstates basis.
-    ominc: 1d float array
-        Incident energy of photon.
-    gamma_c: a float number or a 1d float array with the same shape as ominc.
-        The core-hole life-time broadening factor. It can be a constant value
-        or incident energy dependent.
-    thin: float number
-        The incident angle of photon (in radian).
-    phi: float number
-        Azimuthal angle (in radian), defined with respect to the
-        :math:`x`-axis of the scattering axis: scatter_axis[:,0].
-    pol_type: list of tuples
-        Type of polarization, options can be:
-
-        - ('linear', alpha), linear polarization, where alpha is the angle between the
-          polarization vector and the scattering plane in radians.
-
-        - ('left', 0), left circular polarization.
-
-        - ('right', 0), right circular polarization.
-
-        - ('isotropic', 0). isotropic polarization.
-
-        It will set pol_type=[('isotropic', 0)] if not provided.
-    gs_list: 1d list of ints
-        The indices of initial states which will be used in XAS calculations.
-
-        It will set gs_list=[0] if not provided.
-    temperature: float number
-        Temperature (in K) for boltzmann distribution.
-    scatter_axis: 3*3 float array
-        The local axis defining the scattering plane. The scattering plane is defined in
-        the local :math:`zx`-plane.
-
-        local :math:`x`-axis: scatter_axis[:,0]
-
-        local :math:`y`-axis: scatter_axis[:,1]
-
-        local :math:`z`-axis: scatter_axis[:,2]
-
-        It will be set to an identity matrix if not provided.
-
-    Returns
-    -------
-    xas: 2d float array
-        The calculated XAS spectra. The 1st dimension is for the incident energy, and the
-        2nd dimension is for different polarizations.
-    """
-
-    print("edrixs >>> Running XAS ...")
-    n_om = len(ominc)
-    npol, ncfg_n = trans_op.shape[0], trans_op.shape[1]
-    if pol_type is None:
-        pol_type = [('isotropic', 0)]
-    if gs_list is None:
-        gs_list = [0]
-    if scatter_axis is None:
-        scatter_axis = np.eye(3)
-    else:
-        scatter_axis = np.array(scatter_axis)
-
-    xas = np.zeros((n_om, len(pol_type)), dtype=float)
-    gamma_core = np.zeros(n_om, dtype=float)
-    prob = boltz_dist([eval_i[i] for i in gs_list], temperature)
-    if np.isscalar(gamma_c):
-        gamma_core[:] = np.ones(n_om) * gamma_c
-    else:
-        gamma_core[:] = gamma_c
-
-    kvec = unit_wavevector(thin, phi, scatter_axis, 'in')
-    for i, om in enumerate(ominc):
-        for it, (pt, alpha) in enumerate(pol_type):
-            if pt.strip() not in ['left', 'right', 'linear', 'isotropic']:
-                raise Exception("Unknown polarization type: ", pt)
-            polvec = np.zeros(npol, dtype=complex)
-            if pt.strip() == 'left' or pt.strip() == 'right' or pt.strip() == 'linear':
-                pol = dipole_polvec_xas(thin, phi, alpha, scatter_axis, pt)
-                if npol == 3:  # dipolar transition
-                    polvec[:] = pol
-                if npol == 5:  # quadrupolar transition
-                    polvec[:] = quadrupole_polvec(pol, kvec)
-
-            # loop over all the initial states
-            for j, igs in enumerate(gs_list):
-                if pt.strip() == 'isotropic':
-                    for k in range(npol):
-                        xas[i, it] += (
-                            prob[j] * np.sum(np.abs(trans_op[k, :, igs])**2 * gamma_core[i] /
-                                             np.pi / ((om - (eval_n[:] - eval_i[igs]))**2 +
-                                             gamma_core[i]**2))
-                        ) / npol
-                else:
-                    F_mag = np.zeros(ncfg_n, dtype=complex)
-                    for k in range(npol):
-                        F_mag += trans_op[k, :, igs] * polvec[k]
-                    xas[i, it] += (
-                        prob[j] * np.sum(np.abs(F_mag)**2 * gamma_core[i] / np.pi /
-                                         ((om - (eval_n[:] - eval_i[igs]))**2 + gamma_core[i]**2))
-                    )
-
-    print("edrixs >>> XAS Done !")
-
-    return xas

-
-
-
-

-[docs]
-def rixs_1v1c_py(eval_i, eval_n, trans_op, ominc, eloss, *,
-                 gamma_c=0.1, gamma_f=0.01, thin=1.0, thout=1.0, phi=0.0,
-                 pol_type=None, gs_list=None, temperature=1.0, scatter_axis=None, skip_gs=False):
-    """
-    Calculate RIXS for the case of one valence shell plus one core shell with Python solver.
-
-    This solver is only suitable for small size of Hamiltonian, typically the dimension
-    of both initial and intermediate Hamiltonian are smaller than 10,000.
-
-    Parameters
-    ----------
-    eval_i: 1d float array
-        The eigenvalues of the initial Hamiltonian.
-    eval_n: 1d float array
-        The eigenvalues of the intermediate Hamiltonian.
-    trans_op: 3d complex array
-        The transition operators in the eigenstates basis.
-    ominc: 1d float array
-        Incident energy of photon.
-    eloss: 1d float array
-        Energy loss.
-    gamma_c: a float number or a 1d float array with same shape as ominc.
-        The core-hole life-time broadening factor. It can be a constant value
-        or incident energy dependent.
-    gamma_f: a float number or a 1d float array with same shape as eloss.
-        The final states life-time broadening factor. It can be a constant value
-        or energy loss dependent.
-    thin: float number
-        The incident angle of photon (in radian).
-    thout: float number
-        The scattered angle of photon (in radian).
-    phi: float number
-        Azimuthal angle (in radian), defined with respect to the
-        :math:`x`-axis of scattering axis: scatter_axis[:,0].
-    pol_type: list of 4-elements-tuples
-        Type of polarizations. It has the following form:
-
-        (str1, alpha, str2, beta)
-
-        where, str1 (str2) can be 'linear', 'left', 'right', 'isotropic' and alpha (beta) is
-        the angle (in radians) between the linear polarization vector and the scattering plane.
-
-        If str1 (or str2) is 'isotropic' then the polarization vector projects equally
-        along each axis and the other variables are ignored.
-
-        It will set pol_type=[('linear', 0, 'linear', 0)] if not provided.
-    gs_list: 1d list of ints
-        The indices of initial states which will be used in RIXS calculations.
-
-        It will set gs_list=[0] if not provided.
-    temperature: float number
-        Temperature (in K) for boltzmann distribution.
-    scatter_axis: 3*3 float array
-        The local axis defining the scattering plane. The scattering plane is defined in
-        the local :math:`zx`-plane.
-
-        - local :math:`x`-axis: scatter_axis[:,0]
-
-        - local :math:`y`-axis: scatter_axis[:,1]
-
-        - local :math:`z`-axis: scatter_axis[:,2]
-
-        It will be an identity matrix if not provided.
-    skip_gs: bool
-        If True, transitions to the ground state(s) (forming the elastic peak) are omitted from
-        the calculation.
-
-    Returns
-    -------
-    rixs: 3d float array
-        The calculated RIXS spectra. The 1st dimension is for the incident energy,
-        the 2nd dimension is for the energy loss and the 3rd dimension is for
-        different polarizations.
-    """
-
-    print("edrixs >>> Running RIXS ... ")
-    n_ominc = len(ominc)
-    n_eloss = len(eloss)
-    gamma_core = np.zeros(n_ominc, dtype=float)
-    gamma_final = np.zeros(n_eloss, dtype=float)
-    if np.isscalar(gamma_c):
-        gamma_core[:] = np.ones(n_ominc) * gamma_c
-    else:
-        gamma_core[:] = gamma_c
-
-    if np.isscalar(gamma_f):
-        gamma_final[:] = np.ones(n_eloss) * gamma_f
-    else:
-        gamma_final[:] = gamma_f
-
-    if pol_type is None:
-        pol_type = [('linear', 0, 'linear', 0)]
-    if gs_list is None:
-        gs_list = [0]
-    if scatter_axis is None:
-        scatter_axis = np.eye(3)
-    else:
-        scatter_axis = np.array(scatter_axis)
-
-    prob = boltz_dist([eval_i[i] for i in gs_list], temperature)
-    rixs = np.zeros((len(ominc), len(eloss), len(pol_type)), dtype=float)
-    npol, n, m = trans_op.shape
-    trans_emi = np.zeros((npol, m, n), dtype=np.complex128)
-    for i in range(npol):
-        trans_emi[i] = np.conj(np.transpose(trans_op[i]))
-    polvec_i = np.zeros(npol, dtype=complex)
-    polvec_f = np.zeros(npol, dtype=complex)
-
-    # Calculate RIXS
-    for i, om in enumerate(ominc):
-        F_fi = scattering_mat(eval_i, eval_n, trans_op[:, :, 0:max(gs_list)+1],
-                              trans_emi, om, gamma_core[i])
-
-        for j, (it, alpha, jt, beta) in enumerate(pol_type):
-            ei, ef = dipole_polvec_rixs(thin, thout, phi, alpha, beta,
-                                        scatter_axis, (it, jt))
-            if it.lower() == 'isotropic':
-                ei = np.ones(3)/np.sqrt(3)                        # Powder spectrum
-            if jt.lower() == 'isotropic':
-                ef = np.ones(3)/np.sqrt(3)
-            # dipolar transition
-            if npol == 3:
-                polvec_i[:] = ei
-                polvec_f[:] = ef
-            # quadrupolar transition
-            elif npol == 5:
-                ki = unit_wavevector(thin, phi, scatter_axis, direction='in')
-                kf = unit_wavevector(thout, phi, scatter_axis, direction='out')
-                polvec_i[:] = quadrupole_polvec(ei, ki)
-                polvec_f[:] = quadrupole_polvec(ef, kf)
-            else:
-                raise Exception("Have NOT implemented this type of transition operators")
-            # scattering magnitude with polarization vectors
-            F_mag = np.zeros((len(eval_i), len(gs_list)), dtype=complex)
-            for m in range(npol):
-                for n in range(npol):
-                    F_mag[:, :] += np.conj(polvec_f[m]) * F_fi[m, n] * polvec_i[n]
-
-            fs_list = np.arange(len(eval_i))
-            if skip_gs:
-                fs_list = np.delete(fs_list, gs_list)
-            for m, igs in enumerate(gs_list):
-                for n in fs_list:
-                    rixs[i, :, j] += (
-                        prob[m] * np.abs(F_mag[n, igs])**2 * gamma_final / np.pi /
-                        ((eloss - (eval_i[n] - eval_i[igs]))**2 + gamma_final**2)
-                    )
-    print("edrixs >>> RIXS Done !")
-
-    return rixs

-
-
-
-

-[docs]
-def ed_1v1c_fort(comm, shell_name, *, shell_level=None,
-                 v_soc=None, c_soc=0, v_noccu=1, slater=None,
-                 ext_B=None, on_which='spin',
-                 v_cfmat=None, v_othermat=None,
-                 do_ed=True, ed_solver=2, neval=1, nvector=1, ncv=3,
-                 idump=False, maxiter=500, eigval_tol=1e-8, min_ndim=1000):
-    """
-    Perform ED for the case of one valence shell plus one Core-shell with Fortran ED solver.
-
-    The hopping and Coulomb terms of both the initial and intermediate Hamiltonians will be
-    constructed and written to files (hopping_i.in, hopping_n.in, coulomb_i.in and coulomb_n.in).
-    Fock basis for the initial Hamiltonian will be written to file (fock_i.in).
-
-    ED will be only performed on the initial Hamiltonian to find a few lowest eigenstates
-    do_ed=True. Only input files will be written if do_ed=False.
-    Due to large Hilbert space, the ed_fsolver written in Fortran will be called.
-    mpi4py and a MPI environment (mpich or openmpi) are required to launch ed_fsolver.
-
-    If do_ed=True, it will output the eigenvalues in file (eigvals.dat) and eigenvectors in files
-    (eigvec.n), where n means the n-th eigenvectors. The eigvec.n files will be used later
-    as the inputs for XAS and RIXS calculations.
-
-    Parameters
-    ----------
-    comm: MPI_comm
-        The MPI communicator from mpi4py.
-    shell_name: tuple of two strings
-        Names of valence and core shells. The 1st (2nd) string in the tuple is for the
-        valence (core) shell.
-
-        - The 1st strings can only be 's', 'p', 't2g', 'd', 'f'
-
-        - The 2nd string can be 's', 'p', 'p12', 'p32', 'd', 'd32', 'd52',
-          'f', 'f52', 'f72'.
-
-        For example: shell_name=('d', 'p32') may indicate a :math:`L_3` edge transition from
-        core :math:`2p_{3/2}` shell to valence :math:`3d` shell for Ni.
-    shell_level: tuple of two float numbers
-        Energy level of valence (1st element) and core (2nd element) shells.
-
-        They will be set to zero if not provided.
-    v_soc: tuple of two float numbers
-        Spin-orbit coupling strength of the valence shell,
-        v1_soc[0] for the initial Hamiltonian, and
-        v1_soc[1] for the intermediate Hamiltonian.
-
-        They will be set to zero if not provided.
-    c_soc: float number
-        Spin-orbit coupling strength of core electrons.
-    v_noccu: int number
-        Total number of electrons in valence shells.
-    slater: tuple of two lists
-        Slater integrals for initial (1st list) and intermediate (2nd list) Hamiltonians.
-        The order of the elements in each list should be like this:
-
-        [FX_vv, FX_vc, GX_vc, FX_cc],
-
-        where X are integers with ascending order, it can be X=0, 2, 4, 6 or X=1, 3, 5.
-        One can ignore all the continuous zeros at the end of the list.
-
-        For example, if the full list is: [F0_dd, F2_dd, F4_dd, 0, F2_dp, 0, 0, 0, 0], one can
-        just provide [F0_dd, F2_dd, F4_dd, 0, F2_dp]
-
-        All the Slater integrals will be set to zero if slater==None.
-    ext_B: tuple of three float numbers
-        Vector of external magnetic field with respect to global :math:`xyz`-axis applied
-        on the valence shell.
-
-        It will be set to zeros if not provided.
-    on_which: string
-        Apply Zeeman exchange field on which sector. Options are 'spin', 'orbital' or 'both'.
-    v_cfmat: 2d complex array
-        Crystal field splitting Hamiltonian of the valence shell. The dimension and the orbital
-        order should be consistent with the type of the valence shell.
-
-        They will be zeros if not provided.
-    v_othermat: 2d complex array
-        Other possible Hamiltonian of the valence shell. The dimension and the orbital order
-        should be consistent with the type of the valence shell.
-
-        They will be zeros if not provided.
-    do_ed: logical
-        If do_end=True, diagonalize the Hamitlonian to find a few lowest eigenstates, return the
-        eigenvalues and density matirx, and write the eigenvectors in files eigvec.n, otherwise,
-        just write out the input files, do not perform the ED.
-    ed_solver: int
-        Type of ED solver, options can be 0, 1, 2
-
-        - 0: use Lapack to fully diagonalize Hamiltonian to get all the eigenvalues.
-
-        - 1: use standard Lanczos algorithm to find only a few lowest eigenvalues,
-          no re-orthogonalization has been applied, so it is not very accurate.
-
-        - 2: use parallel version of Arpack library to find a few lowest eigenvalues,
-          it is accurate and is the recommeded choice in real calculations of XAS and RIXS.
-    neval: int
-        Number of eigenvalues to be found. For ed_solver=2, the value should not be too small,
-        neval > 10 is usually a safe value.
-    nvector: int
-        Number of eigenvectors to be found and written into files.
-    ncv: int
-        Used for ed_solver=2, it should be at least ncv > neval + 2. Usually, set it a little
-        bit larger than neval, for example, set ncv=200 when neval=100.
-    idump: logical
-        Whether to dump the eigenvectors to files "eigvec.n", where n means the n-th vectors.
-    maxiter: int
-        Maximum number of iterations in finding all the eigenvalues, used for ed_solver=1, 2.
-    eigval_tol: float
-        The convergence criteria of eigenvalues, used for ed_solver=1, 2.
-    min_ndim: int
-        The minimum dimension of the Hamiltonian when the ed_solver=1, 2 can be used, otherwise,
-        ed_solver=1 will be used.
-
-    Returns
-    -------
-    eval_i: 1d float array, shape=(neval, )
-        The eigenvalues of initial Hamiltonian.
-    denmat: 2d complex array, shape=(nvector, v_norb, v_norb))
-        The density matrix in the eigenstates.
-    """
-    v_name_options = ['s', 'p', 't2g', 'd', 'f']
-    c_name_options = ['s', 'p', 'p12', 'p32', 't2g', 'd', 'd32', 'd52', 'f', 'f52', 'f72']
-    v_name = shell_name[0].strip()
-    c_name = shell_name[1].strip()
-    if v_name not in v_name_options:
-        raise Exception("NOT supported type of valence shell: ", v_name)
-    if c_name not in c_name_options:
-        raise Exception("NOT supported type of core shell: ", c_name)
-
-    names = (v_name, 'empty', c_name)
-    if shell_level is not None:
-        levels = (shell_level[0], 0, shell_level[1])
-    else:
-        levels = None
-
-    eval_i, denmat = _ed_1or2_valence_1core(
-        comm, names, shell_level=levels, v1_soc=v_soc, c_soc=c_soc,
-        v_tot_noccu=v_noccu, slater=slater, v1_ext_B=ext_B,
-        v1_on_which=on_which, v1_cfmat=v_cfmat, v1_othermat=v_othermat,
-        do_ed=do_ed, ed_solver=ed_solver, neval=neval, nvector=nvector,
-        ncv=ncv, idump=idump, maxiter=maxiter, eigval_tol=eigval_tol,
-        min_ndim=min_ndim
-    )
-
-    return eval_i, denmat

-
-
-
-

-[docs]
-def ed_2v1c_fort(comm, shell_name, *, shell_level=None,
-                 v1_soc=None, v2_soc=None, c_soc=0, v_tot_noccu=1, slater=None,
-                 v1_ext_B=None, v2_ext_B=None, v1_on_which='spin', v2_on_which='spin',
-                 v1_cfmat=None, v2_cfmat=None, v1_othermat=None, v2_othermat=None,
-                 hopping_v1v2=None, do_ed=True, ed_solver=2, neval=1, nvector=1, ncv=3,
-                 idump=False, maxiter=500, eigval_tol=1e-8, min_ndim=1000):
-    """
-    Perform ED for the case of two valence shell plus one core-shell with Fortran solver.
-    For example, for Ni :math:`K`-edge RIXS, :math:`1s\\rightarrow 4p` transition,
-    the valence shells involved in RIXS are :math:`3d` and :math:`4p`.
-
-    The hopping and Coulomb terms of both the initial and intermediate Hamiltonians will be
-    constructed and written to files (hopping_i.in, hopping_n.in, coulomb_i.in and coulomb_n.in).
-    Fock basis for the initial Hamiltonian will be written to file (fock_i.in).
-
-    ED will be only performed on the initial Hamiltonian to find a few lowest eigenstates
-    do_ed=True. Only input files will be written if do_ed=False.
-    Due to large Hilbert space, the ed_fsolver written in Fortran will be called.
-    mpi4py and a MPI environment (mpich or openmpi) are required to launch ed_fsolver.
-
-    If do_ed=True, it will output the eigenvalues in file (eigvals.dat) and eigenvectors in files
-    (eigvec.n), where n means the n-th eigenvectors. The eigvec.n files will be used later
-    as the inputs for XAS and RIXS calculations.
-
-    Parameters
-    ----------
-    comm: MPI_comm
-        The MPI communicator from mpi4py.
-    shell_name: tuple of three strings
-        Names of valence and core shells. The 1st (2nd) string in the tuple is for the
-        1st (2nd) valence shell, and the 3rd one is for the core shell.
-
-        - The 1st and 2nd strings can only be 's', 'p', 't2g', 'd', 'f'
-
-        - The 3nd string can be 's', 'p', 'p12', 'p32', 'd', 'd32', 'd52',
-          'f', 'f52', 'f72'.
-
-        For example: shell_name=('d', 'p', 's') may indicate a :math:`K` edge transition from
-        core :math:`1s` shell to valence :math:`3d` and :math:`4p` shell for Ni.
-    shell_level: tuple of three float numbers
-        Energy level of valence (1st and 2nd elements) and core (3nd element) shells.
-
-        They will be set to zero if not provided.
-    v1_soc: tuple of two float numbers
-        Spin-orbit coupling strength of the 1st valence shell,
-        v1_soc[0] for the initial Hamiltonian, and
-        v1_soc[1] for the intermediate Hamiltonian.
-
-        They will be set to zero if not provided.
-    v2_soc: tuple of two float numbers
-        Spin-orbit coupling strength of the 2nd valence shell,
-        v2_soc[0] for the initial Hamiltonian, and
-        v2_soc[1] for the intermediate Hamiltonian.
-
-        They will be set to zero if not provided.
-    c_soc: float number
-        Spin-orbit coupling strength of core electrons.
-    v_tot_noccu: int number
-        Total number of electrons in valence shells.
-    slater: tuple of two lists
-        Slater integrals for initial (1st list) and intermediate (2nd list) Hamiltonians.
-        The order of the elements in each list should be like this:
-
-        [FX_v1v1, FX_v1v2, GX_v1v2, FX_v2v2, FX_v1c, GX_v1c, FX_v2c, GX_v2c],
-
-        where X are integers with ascending order, it can be X=0, 2, 4, 6 or X=1, 3, 5.
-        One can ignore all the continuous zeros at the end of the list.
-
-        For example, if the full list is: [F0_dd, F2_dd, F4_dd, 0, F2_dp, 0, 0, 0, 0], one can
-        just provide [F0_dd, F2_dd, F4_dd, 0, F2_dp]
-
-        All the Slater integrals will be set to zero if slater==None.
-    v1_ext_B: tuple of three float numbers
-        Vector of external magnetic field with respect to global :math:`xyz`-axis applied
-        on the 1st valence shell.
-
-        It will be set to zeros if not provided.
-    v2_ext_B: tuple of three float numbers
-        Vector of external magnetic field with respect to global :math:`xyz`-axis applied
-        on the 2nd valence shell.
-
-        It will be set to zeros if not provided.
-    v1_on_which: string
-        Apply Zeeman exchange field on which sector. Options are 'spin', 'orbital' or 'both'.
-        For the 1st valence shell.
-    v2_on_which: string
-        Apply Zeeman exchange field on which sector. Options are 'spin', 'orbital' or 'both'.
-        For the 2nd valence shell.
-    v1_cfmat: 2d complex array
-        Crystal field splitting Hamiltonian of the 1st valence shell. The dimension and the orbital
-        order should be consistent with the type of the 1st valence shell.
-
-        They will be zeros if not provided.
-    v2_cfmat: 2d complex array
-        Crystal field splitting Hamiltonian of the 2nd valence shell. The dimension and the orbital
-        order should be consistent with the type of the 2nd valence shell.
-
-        They will be zeros if not provided.
-    v1_othermat: 2d complex array
-        Other possible Hamiltonian of the 1st valence shell. The dimension and the orbital order
-        should be consistent with the type of the 1st valence shell.
-
-        They will be zeros if not provided.
-    v2_othermat: 2d complex array
-        Other possible Hamiltonian of the 2nd valence shell. The dimension and the orbital order
-        should be consistent with the type of the 2nd valence shell.
-
-        They will be zeros if not provided.
-    hopping_v1v2: 2d complex array
-        Hopping between the two valence shells. The 1st-index (2nd-index) is the 1st (2nd)
-        valence shell.
-
-        They will be zeros if not provided.
-    do_ed: logical
-        If do_end=True, diagonalize the Hamitlonian to find a few lowest eigenstates, return the
-        eigenvalues and density matirx, and write the eigenvectors in files eigvec.n, otherwise,
-        just write out the input files, do not perform the ED.
-    ed_solver: int
-        Type of ED solver, options can be 0, 1, 2
-
-        - 0: use Lapack to fully diagonalize Hamiltonian to get all the eigenvalues.
-
-        - 1: use standard Lanczos algorithm to find only a few lowest eigenvalues,
-          no re-orthogonalization has been applied, so it is not very accurate.
-
-        - 2: use parallel version of Arpack library to find a few lowest eigenvalues,
-          it is accurate and is the recommeded choice in real calculations of XAS and RIXS.
-    neval: int
-        Number of eigenvalues to be found. For ed_solver=2, the value should not be too small,
-        neval > 10 is usually a safe value.
-    nvector: int
-        Number of eigenvectors to be found and written into files.
-    ncv: int
-        Used for ed_solver=2, it should be at least ncv > neval + 2. Usually, set it a little
-        bit larger than neval, for example, set ncv=200 when neval=100.
-    idump: logical
-        Whether to dump the eigenvectors to files "eigvec.n", where n means the n-th vectors.
-    maxiter: int
-        Maximum number of iterations in finding all the eigenvalues, used for ed_solver=1, 2.
-    eigval_tol: float
-        The convergence criteria of eigenvalues, used for ed_solver=1, 2.
-    min_ndim: int
-        The minimum dimension of the Hamiltonian when the ed_solver=1, 2 can be used, otherwise,
-        ed_solver=1 will be used.
-
-    Returns
-    -------
-    eval_i: 1d float array, shape=(neval, )
-        The eigenvalues of initial Hamiltonian.
-    denmat: 2d complex array, shape=(nvector, v1v2_norb, v1v2_norb))
-        The density matrix in the eigenstates.
-    """
-    v_name_options = ['s', 'p', 't2g', 'd', 'f']
-    c_name_options = ['s', 'p', 'p12', 'p32', 't2g', 'd', 'd32', 'd52', 'f', 'f52', 'f72']
-    v1_name = shell_name[0].strip()
-    v2_name = shell_name[1].strip()
-    c_name = shell_name[2].strip()
-    if v1_name not in v_name_options:
-        raise Exception("NOT supported type of valence shell: ", v1_name)
-    if v2_name not in v_name_options:
-        raise Exception("NOT supported type of valence shell: ", v2_name)
-    if c_name not in c_name_options:
-        raise Exception("NOT supported type of core shell: ", c_name)
-
-    eval_i, denmat = _ed_1or2_valence_1core(
-        comm, shell_name, shell_level=shell_level, v1_soc=v1_soc, v2_soc=v2_soc, c_soc=c_soc,
-        v_tot_noccu=v_tot_noccu, slater=slater, v1_ext_B=v1_ext_B, v2_ext_B=v2_ext_B,
-        v1_on_which=v1_on_which, v2_on_which=v2_on_which, v1_cfmat=v1_cfmat,
-        v2_cfmat=v2_cfmat, v1_othermat=v1_othermat, v2_othermat=v2_othermat,
-        hopping_v1v2=hopping_v1v2, do_ed=do_ed, ed_solver=ed_solver, neval=neval,
-        nvector=nvector, ncv=ncv, idump=idump, maxiter=maxiter, eigval_tol=eigval_tol,
-        min_ndim=min_ndim
-    )
-    return eval_i, denmat

-
-
-
-def _ed_1or2_valence_1core(
-        comm, shell_name, *, shell_level=None,
-        v1_soc=None, v2_soc=None, c_soc=0, v_tot_noccu=1, slater=None,
-        v1_ext_B=None, v2_ext_B=None, v1_on_which='spin', v2_on_which='spin',
-        v1_cfmat=None, v2_cfmat=None, v1_othermat=None, v2_othermat=None,
-        hopping_v1v2=None, do_ed=True, ed_solver=2, neval=1, nvector=1, ncv=3,
-        idump=False, maxiter=500, eigval_tol=1e-8, min_ndim=1000
-        ):
-    from .fedrixs import ed_fsolver
-
-    rank = comm.Get_rank()
-    size = comm.Get_size()
-    fcomm = comm.py2f()
-    if rank == 0:
-        print("edrixs >>> Running ED ...", flush=True)
-    v1_name = shell_name[0].strip()
-    v2_name = shell_name[1].strip()
-    c_name = shell_name[2].strip()
-    info_shell = info_atomic_shell()
-
-    # Quantum numbers of angular momentum
-    v1_orbl = info_shell[v1_name][0]
-    if v2_name != 'empty':
-        v2_orbl = info_shell[v2_name][0]
-    else:
-        v2_orbl = -1
-
-    # number of orbitals with spin
-    v1_norb = info_shell[v1_name][1]
-    if v2_name != 'empty':
-        v2_norb = info_shell[v2_name][1]
-    else:
-        v2_norb = 0
-    c_norb = info_shell[c_name][1]
-
-    # total number of orbitals
-    ntot = v1_norb + v2_norb + c_norb
-    v1v2_norb = v1_norb + v2_norb
-
-    # Coulomb interaction
-    if v2_name == 'empty':
-        slater_name = slater_integrals_name((v1_name, c_name), ('v', 'c'))
-    else:
-        slater_name = slater_integrals_name((v1_name, v2_name, c_name), ('v1', 'v2', 'c1'))
-    nslat = len(slater_name)
-    slater_i = np.zeros(nslat, dtype=float)
-    slater_n = np.zeros(nslat, dtype=float)
-
-    if slater is not None:
-        if nslat > len(slater[0]):
-            slater_i[0:len(slater[0])] = slater[0]
-        else:
-            slater_i[:] = slater[0][0:nslat]
-        if nslat > len(slater[1]):
-            slater_n[0:len(slater[1])] = slater[1]
-        else:
-            slater_n[:] = slater[1][0:nslat]
-
-    # print summary of slater integrals
-    if rank == 0:
-        print(flush=True)
-        print("    Summary of Slater integrals:", flush=True)
-        print("    ------------------------------", flush=True)
-        print("    Terms,  Initial Hamiltonian,  Intermediate Hamiltonian", flush=True)
-        for i in range(nslat):
-            print(
-                "    ", slater_name[i],
-                ":  {:20.10f}{:20.10f}".format(slater_i[i], slater_n[i]), flush=True
-            )
-        print(flush=True)
-
-    if v2_name == 'empty':
-        umat_i = get_umat_slater(v1_name + c_name, *slater_i)
-        umat_n = get_umat_slater(v1_name + c_name, *slater_n)
-    else:
-        umat_i = get_umat_slater_3shells((v1_name, v2_name, c_name), *slater_i)
-        umat_n = get_umat_slater_3shells((v1_name, v2_name, c_name), *slater_n)
-
-    if rank == 0:
-        write_umat(umat_i, 'coulomb_i.in')
-        write_umat(umat_n, 'coulomb_n.in')
-
-    emat_i = np.zeros((ntot, ntot), dtype=complex)
-    emat_n = np.zeros((ntot, ntot), dtype=complex)
-    # SOC
-    if v1_soc is not None and v1_name in ['p', 'd', 't2g', 'f']:
-        emat_i[0:v1_norb, 0:v1_norb] += atom_hsoc(v1_name, v1_soc[0])
-        emat_n[0:v1_norb, 0:v1_norb] += atom_hsoc(v1_name, v1_soc[1])
-
-    if v2_soc is not None and v2_name in ['p', 'd', 't2g', 'f']:
-        emat_i[v1_norb:v1v2_norb, v1_norb:v1v2_norb] += atom_hsoc(v2_name, v2_soc[0])
-        emat_n[v1_norb:v1v2_norb, v1_norb:v1v2_norb] += atom_hsoc(v2_name, v2_soc[1])
-
-    if c_name in ['p', 'd', 'f']:
-        emat_n[v1v2_norb:ntot, v1v2_norb:ntot] += atom_hsoc(c_name, c_soc)
-
-    # crystal field
-    if v1_cfmat is not None:
-        emat_i[0:v1_norb, 0:v1_norb] += np.array(v1_cfmat)
-        emat_n[0:v1_norb, 0:v1_norb] += np.array(v1_cfmat)
-
-    if v2_cfmat is not None and v2_name != 'empty':
-        emat_i[v1_norb:v1v2_norb, v1_norb:v1v2_norb] += np.array(v2_cfmat)
-        emat_n[v1_norb:v1v2_norb, v1_norb:v1v2_norb] += np.array(v2_cfmat)
-
-    # other mat
-    if v1_othermat is not None:
-        emat_i[0:v1_norb, 0:v1_norb] += np.array(v1_othermat)
-        emat_n[0:v1_norb, 0:v1_norb] += np.array(v1_othermat)
-
-    if v2_othermat is not None and v2_name != 'empty':
-        emat_i[v1_norb:v1v2_norb, v1_norb:v1v2_norb] += np.array(v2_othermat)
-        emat_n[v1_norb:v1v2_norb, v1_norb:v1v2_norb] += np.array(v2_othermat)
-
-    # energy of shell
-    if shell_level is not None:
-        eval_shift = shell_level[2] * c_norb / v_tot_noccu
-        emat_i[0:v1_norb, 0:v1_norb] += np.eye(v1_norb) * shell_level[0]
-        emat_i[0:v1_norb, 0:v1_norb] += np.eye(v1_norb) * eval_shift
-        emat_n[0:v1_norb, 0:v1_norb] += np.eye(v1_norb) * shell_level[0]
-        emat_n[v1v2_norb:ntot, v1v2_norb:ntot] += np.eye(c_norb) * shell_level[2]
-        if v2_name != 'empty':
-            emat_i[v1_norb:v1v2_norb, v1_norb:v1v2_norb] += np.eye(v2_norb) * shell_level[1]
-            emat_i[v1_norb:v1v2_norb, v1_norb:v1v2_norb] += np.eye(v2_norb) * eval_shift
-            emat_n[v1_norb:v1v2_norb, v1_norb:v1v2_norb] += np.eye(v2_norb) * shell_level[1]
-
-    # external magnetic field
-    for name, l, ext_B, which, i1, i2 in [
-        (v1_name, v1_orbl, v1_ext_B, v1_on_which, 0, v1_norb),
-        (v2_name, v2_orbl, v2_ext_B, v2_on_which, v1_norb, v1v2_norb)
-    ]:
-        if name == 'empty':
-            continue
-        if name == 't2g':
-            lx, ly, lz = get_lx(1, True), get_ly(1, True), get_lz(1, True)
-            sx, sy, sz = get_sx(1), get_sy(1), get_sz(1)
-            lx, ly, lz = -lx, -ly, -lz
-        else:
-            lx, ly, lz = get_lx(l, True), get_ly(l, True), get_lz(l, True)
-            sx, sy, sz = get_sx(l), get_sy(l), get_sz(l)
-        if ext_B is not None:
-            if which.strip() == 'spin':
-                zeeman = ext_B[0] * (2 * sx) + ext_B[1] * (2 * sy) + ext_B[2] * (2 * sz)
-            elif which.strip() == 'orbital':
-                zeeman = ext_B[0] * lx + ext_B[1] * ly + ext_B[2] * lz
-            elif which.strip() == 'both':
-                zeeman = (ext_B[0] * (lx + 2 * sx) +
-                          ext_B[1] * (ly + 2 * sy) +
-                          ext_B[2] * (lz + 2 * sz))
-            else:
-                raise Exception("Unknown value of zeeman_on_which", which)
-            emat_i[i1:i2, i1:i2] += zeeman
-            emat_n[i1:i2, i1:i2] += zeeman
-
-    # hopping between the two valence shells
-    if hopping_v1v2 is not None and v2_name != 'empty':
-        emat_i[0:v1_norb, v1_norb:v1v2_norb] += np.array(hopping_v1v2)
-        emat_i[v1_norb:v1v2_norb, 0:v1_norb] += np.conj(np.transpose(hopping_v1v2))
-        emat_n[0:v1_norb, v1_norb:v1v2_norb] += np.array(hopping_v1v2)
-        emat_n[v1_norb:v1v2_norb, 0:v1_norb] += np.conj(np.transpose(hopping_v1v2))
-
-    if rank == 0:
-        write_emat(emat_i, 'hopping_i.in')
-        write_emat(emat_n, 'hopping_n.in')
-        write_config(
-            './', ed_solver, v1v2_norb, c_norb, neval, nvector, ncv, idump,
-            maxiter=maxiter, min_ndim=min_ndim, eigval_tol=eigval_tol
-        )
-        write_fock_dec_by_N(v1v2_norb, v_tot_noccu, "fock_i.in")
-
-    if do_ed:
-        # now, call ed solver
-        comm.Barrier()
-        ed_fsolver(fcomm, rank, size)
-        comm.Barrier()
-
-        # read eigvals.dat and denmat.dat
-        data = np.loadtxt('eigvals.dat', ndmin=2)
-        eval_i = np.zeros(neval, dtype=float)
-        eval_i[0:neval] = data[0:neval, 1]
-        data = np.loadtxt('denmat.dat', ndmin=2)
-        tmp = (nvector, v1v2_norb, v1v2_norb)
-        denmat = data[:, 3].reshape(tmp) + 1j * data[:, 4].reshape(tmp)
-
-        return eval_i, denmat
-    else:
-        return None, None
-
-
-

-[docs]
-def xas_1v1c_fort(comm, shell_name, ominc, *, gamma_c=0.1,
-                  v_noccu=1, thin=1.0, phi=0, pol_type=None,
-                  num_gs=1, nkryl=200, temperature=1.0,
-                  loc_axis=None, scatter_axis=None):
-    """
-    Calculate XAS for the case with one valence shells plus one core shell with Fortran solver.
-
-    Parameters
-    ----------
-    comm: MPI_comm
-        MPI communicator.
-    shell_name: tuple of two strings
-        Names of valence and core shells. The 1st (2nd) string in the tuple is for the
-        valence (core) shell.
-
-        - The 1st string can only be 's', 'p', 't2g', 'd', 'f',
-
-        - The 2nd string can be 's', 'p', 'p12', 'p32', 'd', 'd32', 'd52',
-          'f', 'f52', 'f72'.
-
-        For example: shell_name=('d', 'p32') may indicate a :math:`L_3` edge transition from
-        core :math:`2p_{3/2}` shell to valence :math:`3d` shell for Ni.
-    ominc: 1d float array
-        Incident energy of photon.
-    gamma_c: a float number or a 1d float array with the same shape as ominc.
-        The core-hole life-time broadening factor. It can be a constant value
-        or incident energy dependent.
-    v_noccu: int
-        Total occupancy of valence shells.
-    thin: float number
-        The incident angle of photon (in radian).
-    phi: float number
-        Azimuthal angle (in radian), defined with respect to the
-        :math:`x`-axis of the local scattering axis: scatter_axis[:,0].
-    pol_type: list of tuples
-        Type of polarization, options can be:
-
-        - ('linear', alpha), linear polarization, where alpha is the angle between the
-          polarization vector and the scattering plane in radians.
-
-        - ('left', 0), left circular polarization.
-
-        - ('right', 0), right circular polarization.
-
-        - ('isotropic', 0). isotropic polarization.
-
-        It will set pol_type=[('isotropic', 0)] if not provided.
-    num_gs: int
-        Number of initial states used in XAS calculations.
-    nkryl: int
-        Maximum number of poles obtained.
-    temperature: float number
-        Temperature (in K) for boltzmann distribution.
-    loc_axis: 3*3 float array
-        The local axis with respect to which local orbitals are defined.
-
-        - x: local_axis[:,0],
-
-        - y: local_axis[:,1],
-
-        - z: local_axis[:,2].
-
-        It will be an identity matrix if not provided.
-    scatter_axis: 3*3 float array
-        The local axis defining the scattering geometry. The scattering plane is defined in
-        the local :math:`zx`-plane.
-
-        - local :math:`x`-axis: scatter_axis[:,0]
-
-        - local :math:`y`-axis: scatter_axis[:,1]
-
-        - local :math:`z`-axis: scatter_axis[:,2]
-
-        It will be set to an identity matrix if not provided.
-
-    Returns
-    -------
-    xas: 2d array, shape=(len(ominc), len(pol_type))
-        The calculated XAS spectra. The first dimension is for ominc, and the second dimension
-        if for different polarizations.
-    poles: list of dict, shape=(len(pol_type), )
-        The calculated XAS poles for different polarizations.
-    """
-    v_name_options = ['s', 'p', 't2g', 'd', 'f']
-    c_name_options = ['s', 'p', 'p12', 'p32', 't2g', 'd', 'd32', 'd52', 'f', 'f52', 'f72']
-
-    v_name = shell_name[0].strip()
-    c_name = shell_name[1].strip()
-    if v_name not in v_name_options:
-        raise Exception("NOT supported type of valence shell: ", v_name)
-    if c_name not in c_name_options:
-        raise Exception("NOT supported type of core shell: ", c_name)
-
-    names = (v_name, 'empty', c_name)
-
-    xas, poles = _xas_1or2_valence_1core(
-        comm, names, ominc, gamma_c=gamma_c, v_tot_noccu=v_noccu,
-        trans_to_which=1, thin=thin, phi=phi, pol_type=pol_type,
-        num_gs=num_gs, nkryl=nkryl, temperature=temperature,
-        loc_axis=loc_axis, scatter_axis=scatter_axis
-    )
-
-    return xas, poles

-
-
-
-

-[docs]
-def xas_2v1c_fort(comm, shell_name, ominc, *, gamma_c=0.1,
-                  v_tot_noccu=1, trans_to_which=1, thin=1.0, phi=0,
-                  pol_type=None, num_gs=1, nkryl=200, temperature=1.0,
-                  loc_axis=None, scatter_axis=None):
-    """
-    Calculate XAS for the case with two valence shells plus one core shell with Fortran solver.
-
-    Parameters
-    ----------
-    comm: MPI_comm
-        MPI communicator.
-    shell_name: tuple of three strings
-        Names of valence and core shells. The 1st (2nd) string in the tuple is for the
-        1st (2nd) valence shell, and the 3rd one is for the core shell.
-
-        - The 1st and 2nd strings can only be 's', 'p', 't2g', 'd', 'f',
-
-        - The 3nd string can be 's', 'p', 'p12', 'p32', 'd', 'd32', 'd52',
-          'f', 'f52', 'f72'.
-
-        For example: shell_name=('d', 'p', 's') may indicate a :math:`K` edge transition from
-        core :math:`1s` shell to valence :math:`3d` and :math:`4p` shell for Ni.
-    ominc: 1d float array
-        Incident energy of photon.
-    gamma_c: a float number or a 1d float array with the same shape as ominc.
-        The core-hole life-time broadening factor. It can be a constant value
-        or incident energy dependent.
-    v_tot_noccu: int
-        Total occupancy of valence shells.
-    trans_to_which: int
-        Photon transition to which valence shell.
-
-        - 1: to 1st valence shell,
-
-        - 2: to 2nd valence shell.
-    thin: float number
-        The incident angle of photon (in radian).
-    phi: float number
-        Azimuthal angle (in radian), defined with respect to the
-        :math:`x`-axis of the local scattering axis: scatter_axis[:,0].
-    pol_type: list of tuples
-        Type of polarization, options can be:
-
-        - ('linear', alpha), linear polarization, where alpha is the angle between the
-          polarization vector and the scattering plane in radians.
-
-        - ('left', 0), left circular polarization.
-
-        - ('right', 0), right circular polarization.
-
-        - ('isotropic', 0). isotropic polarization.
-
-        It will set pol_type=[('isotropic', 0)] if not provided.
-    num_gs: int
-        Number of initial states used in XAS calculations.
-    nkryl: int
-        Maximum number of poles obtained.
-    temperature: float number
-        Temperature (in K) for boltzmann distribution.
-    loc_axis: 3*3 float array
-        The local axis with respect to which local orbitals are defined.
-
-        - x: local_axis[:,0],
-
-        - y: local_axis[:,1],
-
-        - z: local_axis[:,2].
-
-        It will be an identity matrix if not provided.
-    scatter_axis: 3*3 float array
-        The local axis defining the scattering geometry. The scattering plane is defined in
-        the local :math:`zx`-plane.
-
-        - local :math:`x`-axis: scatter_axis[:,0]
-
-        - local :math:`y`-axis: scatter_axis[:,1]
-
-        - local :math:`z`-axis: scatter_axis[:,2]
-
-        It will be set to an identity matrix if not provided.
-
-    Returns
-    -------
-    xas: 2d array, shape=(len(ominc), len(pol_type))
-        The calculated XAS spectra. The first dimension is for ominc, and the second dimension
-        if for different polarizations.
-    poles: list of dict, shape=(len(pol_type), )
-        The calculated XAS poles for different polarizations.
-    """
-    v_name_options = ['s', 'p', 't2g', 'd', 'f']
-    c_name_options = ['s', 'p', 'p12', 'p32', 't2g', 'd', 'd32', 'd52', 'f', 'f52', 'f72']
-
-    v1_name = shell_name[0].strip()
-    v2_name = shell_name[1].strip()
-    c_name = shell_name[2].strip()
-    if v1_name not in v_name_options:
-        raise Exception("NOT supported type of valence shell: ", v1_name)
-    if v2_name not in v_name_options:
-        raise Exception("NOT supported type of valence shell: ", v2_name)
-    if c_name not in c_name_options:
-        raise Exception("NOT supported type of core shell: ", c_name)
-
-    xas, poles = _xas_1or2_valence_1core(
-        comm, shell_name, ominc, gamma_c=gamma_c, v_tot_noccu=v_tot_noccu,
-        trans_to_which=trans_to_which, thin=thin, phi=phi, pol_type=pol_type,
-        num_gs=num_gs, nkryl=nkryl, temperature=temperature,
-        loc_axis=loc_axis, scatter_axis=scatter_axis
-    )
-
-    return xas, poles

-
-
-
-def _xas_1or2_valence_1core(
-        comm, shell_name, ominc, *, gamma_c=0.1,
-        v_tot_noccu=1, trans_to_which=1, thin=1.0, phi=0,
-        pol_type=None, num_gs=1, nkryl=200, temperature=1.0,
-        loc_axis=None, scatter_axis=None
-        ):
-    from .fedrixs import xas_fsolver
-
-    rank = comm.Get_rank()
-    size = comm.Get_size()
-    fcomm = comm.py2f()
-
-    v1_name = shell_name[0].strip()
-    v2_name = shell_name[1].strip()
-    c_name = shell_name[2].strip()
-
-    info_shell = info_atomic_shell()
-    v1_norb = info_shell[v1_name][1]
-    if v2_name != 'empty':
-        v2_norb = info_shell[v2_name][1]
-    else:
-        v2_norb = 0
-
-    c_norb = info_shell[c_name][1]
-    ntot = v1_norb + v2_norb + c_norb
-    v1v2_norb = v1_norb + v2_norb
-    if pol_type is None:
-        pol_type = [('isotropic', 0)]
-    if loc_axis is None:
-        loc_axis = np.eye(3)
-    else:
-        loc_axis = np.array(loc_axis)
-    if scatter_axis is None:
-        scatter_axis = np.eye(3)
-    else:
-        scatter_axis = np.array(scatter_axis)
-
-    if rank == 0:
-        print("edrixs >>> Running XAS ...", flush=True)
-        write_config(num_val_orbs=v1v2_norb, num_core_orbs=c_norb,
-                     num_gs=num_gs, nkryl=nkryl)
-        write_fock_dec_by_N(v1v2_norb, v_tot_noccu, "fock_i.in")
-        write_fock_dec_by_N(v1v2_norb, v_tot_noccu + 1, "fock_n.in")
-
-    # Build transition operators in local-xyz axis
-    if trans_to_which == 1:
-        case = v1_name + c_name
-    elif trans_to_which == 2 and v2_name != 'empty':
-        case = v2_name + c_name
-    else:
-        raise Exception('Unkonwn trans_to_which: ', trans_to_which)
-    tmp = get_trans_oper(case)
-    npol, n, m = tmp.shape
-    tmp_g = np.zeros((npol, n, m), dtype=complex)
-    trans_mat = np.zeros((npol, ntot, ntot), dtype=complex)
-    # Transform the transition operators to global-xyz axis
-    # dipolar transition
-    if npol == 3:
-        for i in range(3):
-            for j in range(3):
-                tmp_g[i] += loc_axis[i, j] * tmp[j]
-    # quadrupolar transition
-    elif npol == 5:
-        alpha, beta, gamma = rmat_to_euler(loc_axis)
-        wignerD = get_wigner_dmat(4, alpha, beta, gamma)
-        rotmat = np.dot(np.dot(tmat_r2c('d'), wignerD), np.conj(np.transpose(tmat_r2c('d'))))
-        for i in range(5):
-            for j in range(5):
-                tmp_g[i] += rotmat[i, j] * tmp[j]
-    else:
-        raise Exception("Have NOT implemented this case: ", npol)
-    if trans_to_which == 1:
-        trans_mat[:, 0:v1_norb, v1v2_norb:ntot] = tmp_g
-    else:
-        trans_mat[:, v1_norb:v1v2_norb, v1v2_norb:ntot] = tmp_g
-
-    n_om = len(ominc)
-    gamma_core = np.zeros(n_om, dtype=float)
-    if np.isscalar(gamma_c):
-        gamma_core[:] = np.ones(n_om) * gamma_c
-    else:
-        gamma_core[:] = gamma_c
-
-    # loop over different polarization
-    xas = np.zeros((n_om, len(pol_type)), dtype=float)
-    poles = []
-    comm.Barrier()
-    for it, (pt, alpha) in enumerate(pol_type):
-        if pt.strip() == 'left' or pt.strip() == 'right' or pt.strip() == 'linear':
-            if rank == 0:
-                print("edrixs >>> Loop over for polarization: ", it, pt, flush=True)
-                kvec = unit_wavevector(thin, phi, scatter_axis, 'in')
-                polvec = np.zeros(npol, dtype=complex)
-                pol = dipole_polvec_xas(thin, phi, alpha, scatter_axis, pt)
-                if npol == 3:  # Dipolar transition
-                    polvec[:] = pol
-                if npol == 5:  # Quadrupolar transition
-                    polvec[:] = quadrupole_polvec(pol, kvec)
-                trans = np.zeros((ntot, ntot), dtype=complex)
-                for i in range(npol):
-                    trans[:, :] += trans_mat[i] * polvec[i]
-                write_emat(trans, 'transop_xas.in')
-
-            # call XAS solver in fedrixs
-            comm.Barrier()
-            xas_fsolver(fcomm, rank, size)
-            comm.Barrier()
-
-            file_list = ['xas_poles.' + str(i+1) for i in range(num_gs)]
-            pole_dict = read_poles_from_file(file_list)
-            poles.append(pole_dict)
-            xas[:, it] = get_spectra_from_poles(pole_dict, ominc, gamma_core, temperature)
-        elif pt.strip() == 'isotropic':
-            pole_dicts = []
-            for k in range(npol):
-                if rank == 0:
-                    print("edrixs >>> Loop over for polarization: ", it, pt, flush=True)
-                    print("edrixs >>> Isotropic, component: ", k, flush=True)
-                    write_emat(trans_mat[k], 'transop_xas.in')
-                # call XAS solver in fedrixs
-                comm.Barrier()
-                xas_fsolver(fcomm, rank, size)
-                comm.Barrier()
-
-                file_list = ['xas_poles.' + str(i+1) for i in range(num_gs)]
-                pole_tmp = read_poles_from_file(file_list)
-                xas[:, it] += get_spectra_from_poles(pole_tmp, ominc, gamma_core, temperature)
-                pole_dicts.append(pole_tmp)
-            xas[:, it] = xas[:, it] / npol
-            poles.append(merge_pole_dicts(pole_dicts))
-        else:
-            raise Exception("Unknown polarization type: ", pt)
-
-    return xas, poles
-
-
-

-[docs]
-def rixs_1v1c_fort(comm, shell_name, ominc, eloss, *, gamma_c=0.1, gamma_f=0.1,
-                   v_noccu=1, thin=1.0, thout=1.0, phi=0, pol_type=None,
-                   num_gs=1, nkryl=200, linsys_max=500, linsys_tol=1e-8,
-                   temperature=1.0, loc_axis=None, scatter_axis=None):
-    """
-    Calculate RIXS for the case with one valence shell plus one core shell with Fortran solver.
-
-    Parameters
-    ----------
-    comm: MPI_comm
-        MPI communicator.
-    shell_name: tuple of two strings
-        Names of valence and core shells. The 1st (2nd) string in the tuple is for the
-        valence (core) shell.
-
-        - The 1st string can only be 's', 'p', 't2g', 'd', 'f',
-
-        - The 2nd string can be 's', 'p', 'p12', 'p32', 'd', 'd32', 'd52',
-          'f', 'f52', 'f72'.
-
-        For example: shell_name=('d', 'p32') may indicate a :math:`L_3` edge transition from
-        core :math:`2p_{3/2}` shell to valence :math:`3d` shell for Ni.
-    ominc: 1d float array
-        Incident energy of photon.
-    eloss: 1d float array
-        Energy loss.
-    gamma_c: a float number or a 1d float array with same shape as ominc.
-        The core-hole life-time broadening factor. It can be a constant value
-        or incident energy dependent.
-    gamma_f: a float number or a 1d float array with same shape as eloss.
-        The final states life-time broadening factor. It can be a constant value
-        or energy loss dependent.
-    v_noccu: int
-        Total occupancy of valence shells.
-    thin: float number
-        The incident angle of photon (in radian).
-    thout: float number
-        The scattered angle of photon (in radian).
-    phi: float number
-        Azimuthal angle (in radian), defined with respect to the
-        :math:`x`-axis of scattering axis: scatter_axis[:,0].
-    pol_type: list of 4-elements-tuples
-        Type of polarizations. It has the following form:
-
-        (str1, alpha, str2, beta)
-
-        where, str1 (str2) can be 'linear', 'left', 'right', and alpha (beta) is
-        the angle (in radians) between the linear polarization vector and the scattering plane.
-
-        It will set pol_type=[('linear', 0, 'linear', 0)] if not provided.
-    num_gs: int
-        Number of initial states used in RIXS calculations.
-    nkryl: int
-        Maximum number of poles obtained.
-    linsys_max: int
-        Maximum iterations of solving linear equations.
-    linsys_tol: float
-        Convergence for solving linear equations.
-    temperature: float number
-        Temperature (in K) for boltzmann distribution.
-    loc_axis: 3*3 float array
-        The local axis with respect to which local orbitals are defined.
-
-        - x: local_axis[:,0],
-
-        - y: local_axis[:,1],
-
-        - z: local_axis[:,2].
-
-        It will be an identity matrix if not provided.
-    scatter_axis: 3*3 float array
-        The local axis defining the scattering geometry. The scattering plane is defined in
-        the local :math:`zx`-plane.
-
-        - local :math:`x`-axis: scatter_axis[:,0]
-
-        - local :math:`y`-axis: scatter_axis[:,1]
-
-        - local :math:`z`-axis: scatter_axis[:,2]
-
-        It will be set to an identity matrix if not provided.
-
-    Returns
-    -------
-    rixs: 3d float array, shape=(len(ominc), len(eloss), len(pol_type))
-        The calculated RIXS spectra. The 1st dimension is for the incident energy,
-        the 2nd dimension is for the energy loss and the 3rd dimension is for
-        different polarizations.
-    poles: 2d list of dict, shape=(len(ominc), len(pol_type))
-        The calculated RIXS poles. The 1st dimension is for incident energy, and the
-        2nd dimension is for different polarizations.
-    """
-    v_name_options = ['s', 'p', 't2g', 'd', 'f']
-    c_name_options = ['s', 'p', 'p12', 'p32', 't2g', 'd', 'd32', 'd52', 'f', 'f52', 'f72']
-    v_name = shell_name[0].strip()
-    c_name = shell_name[1].strip()
-    if v_name not in v_name_options:
-        raise Exception("NOT supported type of valence shell: ", v_name)
-    if c_name not in c_name_options:
-        raise Exception("NOT supported type of core shell: ", c_name)
-
-    names = (v_name, 'empty', c_name)
-    rixs, poles = _rixs_1or2_valence_1core(
-        comm, names, ominc, eloss, gamma_c=gamma_c, gamma_f=gamma_f,
-        v_tot_noccu=v_noccu, trans_to_which=1, thin=thin,
-        thout=thout, phi=phi, pol_type=pol_type, num_gs=num_gs, nkryl=nkryl,
-        linsys_max=linsys_max, linsys_tol=linsys_tol, temperature=temperature,
-        loc_axis=loc_axis, scatter_axis=loc_axis
-    )
-
-    return rixs, poles

-
-
-
-

-[docs]
-def rixs_2v1c_fort(comm, shell_name, ominc, eloss, *, gamma_c=0.1, gamma_f=0.1,
-                   v_tot_noccu=1, trans_to_which=1, thin=1.0, thout=1.0, phi=0,
-                   pol_type=None, num_gs=1, nkryl=200, linsys_max=500, linsys_tol=1e-8,
-                   temperature=1.0, loc_axis=None, scatter_axis=None):
-    """
-    Calculate RIXS for the case with 2 valence shells plus 1 core shell.
-
-    Parameters
-    ----------
-    comm: MPI_comm
-        MPI communicator.
-    shell_name: tuple of three strings
-        Names of valence and core shells. The 1st (2nd) string in the tuple is for the
-        1st (2nd) valence shell, and the 3rd one is for the core shell.
-
-        - The 1st and 2nd strings can only be 's', 'p', 't2g', 'd', 'f',
-
-        - The 3nd string can be 's', 'p', 'p12', 'p32', 'd', 'd32', 'd52',
-          'f', 'f52', 'f72'.
-
-        For example: shell_name=('d', 'p', 's') may indicate a :math:`K` edge transition from
-        core :math:`1s` shell to valence :math:`3d` and :math:`4p` shell for Ni.
-    ominc: 1d float array
-        Incident energy of photon.
-    eloss: 1d float array
-        Energy loss.
-    gamma_c: a float number or a 1d float array with same shape as ominc.
-        The core-hole life-time broadening factor. It can be a constant value
-        or incident energy dependent.
-    gamma_f: a float number or a 1d float array with same shape as eloss.
-        The final states life-time broadening factor. It can be a constant value
-        or energy loss dependent.
-    v_tot_noccu: int
-        Total occupancy of valence shells.
-    trans_to_which: int
-        Photon transition to which valence shell.
-
-        - 1: to 1st valence shell,
-
-        - 2: to 2nd valence shell.
-    thin: float number
-        The incident angle of photon (in radian).
-    thout: float number
-        The scattered angle of photon (in radian).
-    phi: float number
-        Azimuthal angle (in radian), defined with respect to the
-        :math:`x`-axis of scattering axis: scatter_axis[:,0].
-    pol_type: list of 4-elements-tuples
-        Type of polarizations. It has the following form:
-
-        (str1, alpha, str2, beta)
-
-        where, str1 (str2) can be 'linear', 'left', 'right', and alpha (beta) is
-        the angle (in radians) between the linear polarization vector and the scattering plane.
-
-        It will set pol_type=[('linear', 0, 'linear', 0)] if not provided.
-    num_gs: int
-        Number of initial states used in RIXS calculations.
-    nkryl: int
-        Maximum number of poles obtained.
-    linsys_max: int
-        Maximum iterations of solving linear equations.
-    linsys_tol: float
-        Convergence for solving linear equations.
-    temperature: float number
-        Temperature (in K) for boltzmann distribution.
-    loc_axis: 3*3 float array
-        The local axis with respect to which local orbitals are defined.
-
-        - x: local_axis[:,0],
-
-        - y: local_axis[:,1],
-
-        - z: local_axis[:,2].
-
-        It will be an identity matrix if not provided.
-    scatter_axis: 3*3 float array
-        The local axis defining the scattering geometry. The scattering plane is defined in
-        the local :math:`zx`-plane.
-
-        - local :math:`x`-axis: scatter_axis[:,0]
-
-        - local :math:`y`-axis: scatter_axis[:,1]
-
-        - local :math:`z`-axis: scatter_axis[:,2]
-
-        It will be set to an identity matrix if not provided.
-
-    Returns
-    -------
-    rixs: 3d float array, shape=(len(ominc), len(eloss), len(pol_type))
-        The calculated RIXS spectra. The 1st dimension is for the incident energy,
-        the 2nd dimension is for the energy loss and the 3rd dimension is for
-        different polarizations.
-    poles: 2d list of dict, shape=(len(ominc), len(pol_type))
-        The calculated RIXS poles. The 1st dimension is for incident energy, and the
-        2nd dimension is for different polarizations.
-    """
-    v_name_options = ['s', 'p', 't2g', 'd', 'f']
-    c_name_options = ['s', 'p', 'p12', 'p32', 't2g', 'd', 'd32', 'd52', 'f', 'f52', 'f72']
-    v1_name = shell_name[0].strip()
-    v2_name = shell_name[1].strip()
-    c_name = shell_name[2].strip()
-    if v1_name not in v_name_options:
-        raise Exception("NOT supported type of valence shell: ", v1_name)
-    if v2_name not in v_name_options:
-        raise Exception("NOT supported type of valence shell: ", v2_name)
-    if c_name not in c_name_options:
-        raise Exception("NOT supported type of core shell: ", c_name)
-
-    rixs, poles = _rixs_1or2_valence_1core(
-        comm, shell_name, ominc, eloss, gamma_c=gamma_c, gamma_f=gamma_f,
-        v_tot_noccu=v_tot_noccu, trans_to_which=trans_to_which, thin=thin,
-        thout=thout, phi=phi, pol_type=pol_type, num_gs=num_gs, nkryl=nkryl,
-        linsys_max=linsys_max, linsys_tol=linsys_tol, temperature=temperature,
-        loc_axis=loc_axis, scatter_axis=loc_axis
-    )
-
-    return rixs, poles

-
-
-
-def _rixs_1or2_valence_1core(
-        comm, shell_name, ominc, eloss, *, gamma_c=0.1, gamma_f=0.1,
-        v_tot_noccu=1, trans_to_which=1, thin=1.0, thout=1.0, phi=0,
-        pol_type=None, num_gs=1, nkryl=200, linsys_max=500, linsys_tol=1e-8,
-        temperature=1.0, loc_axis=None, scatter_axis=None
-        ):
-    from .fedrixs import rixs_fsolver
-
-    rank = comm.Get_rank()
-    size = comm.Get_size()
-    fcomm = comm.py2f()
-
-    v1_name = shell_name[0].strip()
-    v2_name = shell_name[1].strip()
-    c_name = shell_name[2].strip()
-
-    info_shell = info_atomic_shell()
-
-    v1_norb = info_shell[v1_name][1]
-    if v2_name != 'empty':
-        v2_norb = info_shell[v2_name][1]
-    else:
-        v2_norb = 0
-    c_norb = info_shell[c_name][1]
-    ntot = v1_norb + v2_norb + c_norb
-    v1v2_norb = v1_norb + v2_norb
-    if pol_type is None:
-        pol_type = [('linear', 0, 'linear', 0)]
-    if loc_axis is None:
-        loc_axis = np.eye(3)
-    else:
-        loc_axis = np.array(loc_axis)
-    if scatter_axis is None:
-        scatter_axis = np.eye(3)
-    else:
-        scatter_axis = np.array(scatter_axis)
-
-    if rank == 0:
-        print("edrixs >>> Running RIXS ...", flush=True)
-        write_fock_dec_by_N(v1v2_norb, v_tot_noccu, "fock_i.in")
-        write_fock_dec_by_N(v1v2_norb, v_tot_noccu + 1, "fock_n.in")
-        write_fock_dec_by_N(v1v2_norb, v_tot_noccu, "fock_f.in")
-
-        # Build transition operators in local-xyz axis
-        if trans_to_which == 1:
-            case = v1_name + c_name
-        elif trans_to_which == 2:
-            case = v2_name + c_name
-        else:
-            raise Exception('Unkonwn trans_to_which: ', trans_to_which)
-        tmp = get_trans_oper(case)
-        npol, n, m = tmp.shape
-        tmp_g = np.zeros((npol, n, m), dtype=complex)
-        trans_mat = np.zeros((npol, ntot, ntot), dtype=complex)
-        # Transform the transition operators to global-xyz axis
-        # dipolar transition
-        if npol == 3:
-            for i in range(3):
-                for j in range(3):
-                    tmp_g[i] += loc_axis[i, j] * tmp[j]
-        # quadrupolar transition
-        elif npol == 5:
-            alpha, beta, gamma = rmat_to_euler(loc_axis)
-            wignerD = get_wigner_dmat(4, alpha, beta, gamma)
-            rotmat = np.dot(np.dot(tmat_r2c('d'), wignerD), np.conj(np.transpose(tmat_r2c('d'))))
-            for i in range(5):
-                for j in range(5):
-                    tmp_g[i] += rotmat[i, j] * tmp[j]
-        else:
-            raise Exception("Have NOT implemented this case: ", npol)
-        if trans_to_which == 1:
-            trans_mat[:, 0:v1_norb, v1v2_norb:ntot] = tmp_g
-        else:
-            trans_mat[:, v1_norb:v1v2_norb, v1v2_norb:ntot] = tmp_g
-
-    n_om = len(ominc)
-    neloss = len(eloss)
-    gamma_core = np.zeros(n_om, dtype=float)
-    if np.isscalar(gamma_c):
-        gamma_core[:] = np.ones(n_om) * gamma_c
-    else:
-        gamma_core[:] = gamma_c
-    gamma_final = np.zeros(neloss, dtype=float)
-    if np.isscalar(gamma_f):
-        gamma_final[:] = np.ones(neloss) * gamma_f
-    else:
-        gamma_final[:] = gamma_f
-
-    # loop over different polarization
-    rixs = np.zeros((n_om, neloss, len(pol_type)), dtype=float)
-    poles = []
-    comm.Barrier()
-    # loop over different polarization
-    for iom, omega in enumerate(ominc):
-        if rank == 0:
-            write_config(
-                num_val_orbs=v1v2_norb, num_core_orbs=c_norb,
-                omega_in=omega, gamma_in=gamma_core[iom],
-                num_gs=num_gs, nkryl=nkryl, linsys_max=linsys_max,
-                linsys_tol=linsys_tol
-            )
-        poles_per_om = []
-        # loop over polarization
-        for ip, (it, alpha, jt, beta) in enumerate(pol_type):
-            if rank == 0:
-                print(flush=True)
-                print("edrixs >>> Calculate RIXS for incident energy: ", omega, flush=True)
-                print("edrixs >>> Polarization: ", ip, flush=True)
-                polvec_i = np.zeros(npol, dtype=complex)
-                polvec_f = np.zeros(npol, dtype=complex)
-                ei, ef = dipole_polvec_rixs(thin, thout, phi, alpha, beta,
-                                            scatter_axis, (it, jt))
-                # dipolar transition
-                if npol == 3:
-                    polvec_i[:] = ei
-                    polvec_f[:] = ef
-                # quadrupolar transition
-                elif npol == 5:
-                    ki = unit_wavevector(thin, phi, scatter_axis, direction='in')
-                    kf = unit_wavevector(thout, phi, scatter_axis, direction='out')
-                    polvec_i[:] = quadrupole_polvec(ei, ki)
-                    polvec_f[:] = quadrupole_polvec(ef, kf)
-                else:
-                    raise Exception("Have NOT implemented this type of transition operators")
-                trans_i = np.zeros((ntot, ntot), dtype=complex)
-                trans_f = np.zeros((ntot, ntot), dtype=complex)
-                for i in range(npol):
-                    trans_i[:, :] += trans_mat[i] * polvec_i[i]
-                write_emat(trans_i, 'transop_rixs_i.in')
-                for i in range(npol):
-                    trans_f[:, :] += trans_mat[i] * polvec_f[i]
-                write_emat(np.conj(np.transpose(trans_f)), 'transop_rixs_f.in')
-
-            # call RIXS solver in fedrixs
-            comm.Barrier()
-            rixs_fsolver(fcomm, rank, size)
-            comm.Barrier()
-
-            file_list = ['rixs_poles.' + str(i+1) for i in range(num_gs)]
-            pole_dict = read_poles_from_file(file_list)
-            poles_per_om.append(pole_dict)
-            rixs[iom, :, ip] = get_spectra_from_poles(pole_dict, eloss,
-                                                      gamma_final, temperature)
-
-        poles.append(poles_per_om)
-
-    return rixs, poles
-
-
-

-[docs]
-def ed_siam_fort(comm, shell_name, nbath, *, siam_type=0, v_noccu=1, static_core_pot=0, c_level=0,
-                 c_soc=0, trans_c2n=None, imp_mat=None, imp_mat_n=None, bath_level=None,
-                 bath_level_n=None, hyb=None, hyb_n=None, hopping=None, hopping_n=None,
-                 slater=None, ext_B=None, on_which='spin', do_ed=0, ed_solver=2, neval=1,
-                 nvector=1, ncv=3, idump=False, maxiter=1000, eigval_tol=1e-8, min_ndim=1000):
-    """
-    Find the ground state of the initial Hamiltonian of a Single Impuirty Anderson Model (SIAM),
-    and also prepare input files, *hopping_i.in*, *hopping_n.in*, *coulomb_i.in*, *coulomb_n.in*
-    for following XAS and RIXS calculations.
-
-    Parameters
-    ----------
-    comm: MPI_Comm
-        MPI Communicator
-    shell_name: tuple of two strings
-        Names of valence and core shells. The 1st (2nd) string in the tuple is for the
-        valence (core) shell.
-
-        - The 1st string can only be 's', 'p', 't2g', 'd', 'f',
-
-        - The 2nd string can be 's', 'p', 'p12', 'p32', 'd', 'd32', 'd52',
-          'f', 'f52', 'f72'.
-
-        For example: shell_name=('d', 'p32') indicates a :math:`L_3` edge transition from
-        core :math:`p_{3/2}` shell to valence :math:`d` shell.
-    nbath: int
-        Number of bath sites.
-    siam_type: int
-        Type of SIAM Hamiltonian,
-
-        - 0: diagonal hybridization function, parameterized by *imp_mat*, *bath_level* and *hyb*
-
-        - 1: general hybridization function, parameterized by matrix *hopping*
-
-        if *siam_type=0*, only *imp_mat*, *bath_level* and *hyb* are required,
-        if *siam_type=1*, only *hopping* is required.
-    v_noccu: int
-        Number of total occupancy of impurity and baths orbitals, required when do_ed=1, 2
-    static_core_pot: float
-        Static core hole potential.
-    c_level: float
-        Energy level of core shell.
-    c_soc: float
-        Spin-orbit coupling strength of core electrons.
-    trans_c2n: 2d complex array
-        The transformation matrix from the spherical harmonics basis to the basis on which
-        the `imp_mat` and hybridization function (`bath_level`, `hyb`, `hopping`) are defined.
-    imp_mat: 2d complex array
-        Impurity matrix for the impurity site, including CF or SOC, for siam_type=0
-        and the initial configurations.
-    imp_mat_n: 2d complex array
-        Impurity matrix for the impurity site, including CF or SOC, for siam_type=0
-        and the intermediate configurations. If imp_mat_n=None, imp_mat will be used.
-    bath_level: 2d complex array
-        Energy level of bath sites, 1st (2nd) dimension is for different bath sites (orbitals),
-        for siam_type=0 and the initial configurations.
-    bath_level_n: 2d complex array
-        Energy level of bath sites, 1st (2nd) dimension is for different bath sites (orbitals),
-        for siam_type=0 and the intermediate configurations. If bath_level_n=None,
-        bath_level will be used.
-    hyb: 2d complex array
-        Hybridization strength of bath sites, 1st (2nd) dimension is for different bath
-        sites (orbitals), for siam_type=0 and the initial configurations.
-    hyb_n: 2d complex array
-        Hybridization strength of bath sites, 1st (2nd) dimension is for different bath
-        sites (orbitals), for siam_type=0 and the intermediate configurations.
-        If hyb_n=None, hyb will be used.
-    hopping: 2d complex array
-        General hopping matrix when siam_type=1, including imp_mat and hybridization functions,
-        for siam_type=1 and the initial configurations.
-    hopping_n: 2d complex array
-        General hopping matrix when siam_type=1, including imp_mat and hybridization functions,
-        for siam_type=1 and the intermediate configurations. If hopping_n=None,
-        hopping will be used.
-    slater: tuple of two lists
-        Slater integrals for initial (1st list) and intermediate (2nd list) Hamiltonians.
-        The order of the elements in each list should be like this:
-
-        [FX_vv, FX_vc, GX_vc, FX_cc],
-
-        where X are integers with ascending order, it can be X=0, 2, 4, 6 or X=1, 3, 5.
-        One can ignore all the continuous zeros at the end of the list.
-
-        For example, if the full list is: [F0_dd, F2_dd, F4_dd, 0, F2_dp, 0, 0, 0, 0], one can
-        just provide [F0_dd, F2_dd, F4_dd, 0, F2_dp]
-
-        All the Slater integrals will be set to zero if slater=None.
-    ext_B: tuple of three float numbers
-        Vector of external magnetic field with respect to global :math:`xyz`-axis.
-
-        They will be set to zero if not provided.
-    on_which: string
-        Apply Zeeman exchange field on which sector. Options are 'spin', 'orbital' or 'both'.
-    do_ed: int
-        - 0: First, search the ground state in different subspaces of total occupancy
-          :math:`N` with ed_solver=1, and then do a more accurate ED in the subspace
-          :math:`N` where the ground state lies to find a few lowest eigenstates, return
-          the eigenvalues and density matirx, and write the eigenvectors in files eigvec.n
-
-        - 1: Only do ED for given occupancy number *v_noccu*, return eigenvalues and
-          density matrix, write eigenvectors to files eigvec.n
-
-        - 2: Do not do ED, only write parameters into files: *hopping_i.in*, *hopping_n.in*,
-          *coulomb_i.in*, *coulomb_n.in* for later XAS or RIXS calculations.
-    ed_solver: int
-        Type of ED solver, options can be 0, 1, 2
-
-        - 0: use Lapack to fully diagonalize Hamiltonian to get all the eigenvalues.
-
-        - 1: use standard Lanczos algorithm to find only a few lowest eigenvalues,
-          no re-orthogonalization has been applied, so it is not very accurate.
-
-        - 2: use parallel version of Arpack library to find a few lowest eigenvalues,
-          it is accurate and is the recommeded choice in real calculations of XAS and RIXS.
-    neval: int
-        Number of eigenvalues to be found. For ed_solver=2, the value should not be too small,
-        neval > 10 is usually a safe value.
-    nvector: int
-        Number of eigenvectors to be found and written into files.
-    ncv: int
-        Used for ed_solver=2, it should be at least ncv > neval + 2. Usually, set it a little
-        bit larger than neval, for example, set ncv=200 when neval=100.
-    idump: logical
-        Whether to dump the eigenvectors to files "eigvec.n", where n means the n-th vectors.
-    maxiter: int
-        Maximum number of iterations in finding all the eigenvalues, used for ed_solver=1, 2.
-    eigval_tol: float
-        The convergence criteria of eigenvalues, used for ed_solver=1, 2.
-    min_ndim: int
-        The minimum dimension of the Hamiltonian when the ed_solver=1, 2 can be used, otherwise,
-        ed_solver=1 will be used.
-
-    Returns
-    -------
-    eval_i: 1d float array
-        Eigenvalues of initial Hamiltonian.
-    denmat: 2d complex array
-        Density matrix.
-    noccu_gs: int
-        Occupancy of the ground state.
-    """
-    from .fedrixs import ed_fsolver
-
-    rank = comm.Get_rank()
-    size = comm.Get_size()
-    fcomm = comm.py2f()
-    if rank == 0:
-        print("edrixs >>> Running ED ...", flush=True)
-
-    v_name_options = ['s', 'p', 't2g', 'd', 'f']
-    c_name_options = ['s', 'p', 'p12', 'p32', 't2g', 'd', 'd32', 'd52', 'f', 'f52', 'f72']
-    v_name = shell_name[0].strip()
-    c_name = shell_name[1].strip()
-    if v_name not in v_name_options:
-        raise Exception("NOT supported type of valence shell: ", v_name)
-    if c_name not in c_name_options:
-        raise Exception("NOT supported type of core shell: ", c_name)
-
-    info_shell = info_atomic_shell()
-    v_orbl = info_shell[v_name][0]
-    v_norb = info_shell[v_name][1]
-    c_norb = info_shell[c_name][1]
-    ntot_v = v_norb * (nbath + 1)
-    ntot = ntot_v + c_norb
-
-    slater_name = slater_integrals_name((v_name, c_name), ('v', 'c'))
-    nslat = len(slater_name)
-    slater_i = np.zeros(nslat, dtype=float)
-    slater_n = np.zeros(nslat, dtype=float)
-
-    if slater is not None:
-        if nslat > len(slater[0]):
-            slater_i[0:len(slater[0])] = slater[0]
-        else:
-            slater_i[:] = slater[0][0:nslat]
-        if nslat > len(slater[1]):
-            slater_n[0:len(slater[1])] = slater[1]
-        else:
-            slater_n[:] = slater[1][0:nslat]
-
-    # print summary of slater integrals
-    if rank == 0:
-        print(flush=True)
-        print("    Summary of Slater integrals:", flush=True)
-        print("    ------------------------------", flush=True)
-        print("    Terms,  Initial Hamiltonian,  Intermediate Hamiltonian", flush=True)
-        for i in range(nslat):
-            print(
-                "    ", slater_name[i],
-                ":  {:20.10f}{:20.10f}".format(slater_i[i], slater_n[i]), flush=True
-            )
-        print(flush=True)
-
-    umat_tmp_i = get_umat_slater(v_name + c_name, *slater_i)
-    umat_tmp_n = get_umat_slater(v_name + c_name, *slater_n)
-
-    umat_i = np.zeros((ntot, ntot, ntot, ntot), dtype=complex)
-    umat_n = np.zeros((ntot, ntot, ntot, ntot), dtype=complex)
-
-    indx = list(range(0, v_norb)) + [ntot_v + i for i in range(0, c_norb)]
-    for i in range(v_norb+c_norb):
-        for j in range(v_norb+c_norb):
-            for k in range(v_norb+c_norb):
-                for m in range(v_norb+c_norb):
-                    umat_i[indx[i], indx[j], indx[k], indx[m]] = umat_tmp_i[i, j, k, m]
-                    umat_n[indx[i], indx[j], indx[k], indx[m]] = umat_tmp_n[i, j, k, m]
-    if rank == 0:
-        write_umat(umat_i, 'coulomb_i.in')
-        write_umat(umat_n, 'coulomb_n.in')
-
-    emat_i = np.zeros((ntot, ntot), dtype=complex)
-    emat_n = np.zeros((ntot, ntot), dtype=complex)
-    # General hybridization function, including off-diagonal terms
-    if siam_type == 1:
-        if hopping is not None:
-            emat_i[0:ntot_v, 0:ntot_v] += hopping
-        if hopping_n is not None:
-            emat_n[0:ntot_v, 0:ntot_v] += hopping_n
-        elif hopping is not None:
-            emat_n[0:ntot_v, 0:ntot_v] += hopping
-    # Diagonal hybridization function
-    elif siam_type == 0:
-        # matrix (CF or SOC) for impuirty site
-        if imp_mat is not None:
-            emat_i[0:v_norb, 0:v_norb] += imp_mat
-        if imp_mat_n is not None:
-            emat_n[0:v_norb, 0:v_norb] += imp_mat_n
-        elif imp_mat is not None:
-            emat_n[0:v_norb, 0:v_norb] += imp_mat
-        # bath levels
-        if bath_level is not None:
-            for i in range(nbath):
-                for j in range(v_norb):
-                    indx = (i + 1) * v_norb + j
-                    emat_i[indx, indx] += bath_level[i, j]
-        if bath_level_n is not None:
-            for i in range(nbath):
-                for j in range(v_norb):
-                    indx = (i + 1) * v_norb + j
-                    emat_n[indx, indx] += bath_level_n[i, j]
-        elif bath_level is not None:
-            for i in range(nbath):
-                for j in range(v_norb):
-                    indx = (i + 1) * v_norb + j
-                    emat_n[indx, indx] += bath_level[i, j]
-        if hyb is not None:
-            for i in range(nbath):
-                for j in range(v_norb):
-                    indx1, indx2 = j, (i + 1) * v_norb + j
-                    emat_i[indx1, indx2] += hyb[i, j]
-                    emat_i[indx2, indx1] += np.conj(hyb[i, j])
-        if hyb_n is not None:
-            for i in range(nbath):
-                for j in range(v_norb):
-                    indx1, indx2 = j, (i + 1) * v_norb + j
-                    emat_n[indx1, indx2] += hyb_n[i, j]
-                    emat_n[indx2, indx1] += np.conj(hyb_n[i, j])
-        elif hyb is not None:
-            for i in range(nbath):
-                for j in range(v_norb):
-                    indx1, indx2 = j, (i + 1) * v_norb + j
-                    emat_n[indx1, indx2] += hyb[i, j]
-                    emat_n[indx2, indx1] += np.conj(hyb[i, j])
-    else:
-        raise Exception("Unknown siam_type: ", siam_type)
-
-    if c_name in ['p', 'd', 'f']:
-        emat_n[ntot_v:ntot, ntot_v:ntot] += atom_hsoc(c_name, c_soc)
-
-    # static core potential
-    emat_n[0:v_norb, 0:v_norb] -= np.eye(v_norb) * static_core_pot
-
-    if trans_c2n is None:
-        trans_c2n = np.eye(v_norb, dtype=complex)
-    else:
-        trans_c2n = np.array(trans_c2n)
-
-    tmat = np.eye(ntot, dtype=complex)
-    for i in range(nbath+1):
-        off = i * v_norb
-        tmat[off:off+v_norb, off:off+v_norb] = np.conj(np.transpose(trans_c2n))
-    emat_i[:, :] = cb_op(emat_i, tmat)
-    emat_n[:, :] = cb_op(emat_n, tmat)
-
-    # zeeman field
-    if v_name == 't2g':
-        lx, ly, lz = get_lx(1, True), get_ly(1, True), get_lz(1, True)
-        sx, sy, sz = get_sx(1), get_sy(1), get_sz(1)
-        lx, ly, lz = -lx, -ly, -lz
-    else:
-        lx, ly, lz = get_lx(v_orbl, True), get_ly(v_orbl, True), get_lz(v_orbl, True)
-        sx, sy, sz = get_sx(v_orbl), get_sy(v_orbl), get_sz(v_orbl)
-
-    if ext_B is not None:
-        if on_which.strip() == 'spin':
-            zeeman = ext_B[0] * (2 * sx) + ext_B[1] * (2 * sy) + ext_B[2] * (2 * sz)
-        elif on_which.strip() == 'orbital':
-            zeeman = ext_B[0] * lx + ext_B[1] * ly + ext_B[2] * lz
-        elif on_which.strip() == 'both':
-            zeeman = ext_B[0] * (lx + 2 * sx) + ext_B[1] * (ly + 2 * sy) + ext_B[2] * (lz + 2 * sz)
-        else:
-            raise Exception("Unknown value of on_which", on_which)
-        emat_i[0:v_norb, 0:v_norb] += zeeman
-        emat_n[0:v_norb, 0:v_norb] += zeeman
-
-    # Perform ED if necessary
-    if do_ed == 1 or do_ed == 2:
-        eval_shift = c_level * c_norb / v_noccu
-        emat_i[0:ntot_v, 0:ntot_v] += np.eye(ntot_v) * eval_shift
-        emat_n[ntot_v:ntot, ntot_v:ntot] += np.eye(c_norb) * c_level
-        if rank == 0:
-            write_emat(emat_i, 'hopping_i.in')
-            write_emat(emat_n, 'hopping_n.in')
-            write_config(
-                ed_solver=ed_solver, num_val_orbs=ntot_v, neval=neval, nvector=nvector, ncv=ncv,
-                idump=idump, maxiter=maxiter, min_ndim=min_ndim, eigval_tol=eigval_tol
-            )
-            write_fock_dec_by_N(ntot_v, v_noccu, "fock_i.in")
-        if do_ed == 1:
-            if rank == 0:
-                print("edrixs >>> do_ed=1, perform ED at noccu: ", v_noccu, flush=True)
-            comm.Barrier()
-            ed_fsolver(fcomm, rank, size)
-            comm.Barrier()
-            data = np.loadtxt('eigvals.dat', ndmin=2)
-            eval_i = np.zeros(neval, dtype=float)
-            eval_i[0:neval] = data[0:neval, 1]
-            data = np.loadtxt('denmat.dat', ndmin=2)
-            tmp = (nvector, ntot_v, ntot_v)
-            denmat = data[:, 3].reshape(tmp) + 1j * data[:, 4].reshape(tmp)
-            return eval_i, denmat, v_noccu
-        else:
-            if rank == 0:
-                print("edrixs >>> do_ed=2, Do not perform ED, only write files", flush=True)
-            return None, None, None
-
-    # Find the ground states by total occupancy N
-    elif do_ed == 0:
-        if rank == 0:
-            print("edrixs >>> do_ed=0, serach ground state by total occupancy N", flush=True)
-            flog = open('search_gs.log', 'w')
-        write_emat(emat_i, 'hopping_i.in')
-        res = []
-        num_electron = ntot_v // 2
-        noccu_gs = num_electron
-        if rank == 0:
-            write_config(
-                ed_solver=1, num_val_orbs=ntot_v, neval=1, nvector=1, idump=False,
-                maxiter=maxiter, min_ndim=min_ndim, eigval_tol=eigval_tol
-            )
-            write_fock_dec_by_N(ntot_v, num_electron, "fock_i.in")
-        comm.Barrier()
-        ed_fsolver(fcomm, rank, size)
-        comm.Barrier()
-        data = np.loadtxt('eigvals.dat', ndmin=2)
-        eval_gs = data[0, 1]
-        data = np.loadtxt('denmat.dat', ndmin=2)
-        tmp = (1, ntot_v, ntot_v)
-        denmat = data[:, 3].reshape(tmp) + 1j * data[:, 4].reshape(tmp)
-        imp_occu = np.sum(denmat[0].diagonal()[0:v_norb]).real
-        res.append((num_electron, eval_gs, imp_occu))
-        if rank == 0:
-            print(num_electron, eval_gs, imp_occu, file=flog, flush=True)
-
-        nplus_list = [num_electron + i + 1 for i in range(ntot_v // 2)]
-        nminus_list = [num_electron - i - 1 for i in range(ntot_v // 2)]
-        nplus_direction = True
-        nminus_direction = True
-        for i in range(ntot_v // 2):
-            if nplus_direction:
-                num_electron = nplus_list[i]
-                if rank == 0:
-                    write_fock_dec_by_N(ntot_v, num_electron, "fock_i.in")
-                comm.Barrier()
-                ed_fsolver(fcomm, rank, size)
-                comm.Barrier()
-                data = np.loadtxt('eigvals.dat', ndmin=2)
-                eigval = data[0, 1]
-                if eigval > eval_gs:
-                    nplus_direction = False
-                else:
-                    nminus_direction = False
-                    eval_gs = eigval
-                    noccu_gs = num_electron
-                data = np.loadtxt('denmat.dat', ndmin=2)
-                tmp = (1, ntot_v, ntot_v)
-                denmat = data[:, 3].reshape(tmp) + 1j * data[:, 4].reshape(tmp)
-                imp_occu = np.sum(denmat[0].diagonal()[0:v_norb]).real
-                res.append((num_electron, eigval, imp_occu))
-                if rank == 0:
-                    print(num_electron, eigval, imp_occu, file=flog, flush=True)
-
-            if nminus_direction:
-                num_electron = nminus_list[i]
-                if rank == 0:
-                    write_fock_dec_by_N(ntot_v, num_electron, "fock_i.in")
-                comm.Barrier()
-                ed_fsolver(fcomm, rank, size)
-                comm.Barrier()
-                data = np.loadtxt('eigvals.dat', ndmin=2)
-                eigval = data[0, 1]
-                if eigval > eval_gs:
-                    nminus_direction = False
-                else:
-                    nplus_direction = False
-                    eval_gs = eigval
-                    noccu_gs = num_electron
-                data = np.loadtxt('denmat.dat', ndmin=2)
-                tmp = (1, ntot_v, ntot_v)
-                denmat = data[:, 3].reshape(tmp) + 1j * data[:, 4].reshape(tmp)
-                imp_occu = np.sum(denmat[0].diagonal()[0:v_norb]).real
-                res.append((num_electron, eigval, imp_occu))
-                if rank == 0:
-                    print(num_electron, eigval, imp_occu, file=flog, flush=True)
-        if rank == 0:
-            flog.close()
-            res.sort(key=lambda x: x[1])
-            f = open('search_result.dat', 'w')
-            for item in res:
-                f.write("{:10d}{:20.10f}{:20.10f}\n".format(item[0], item[1], item[2]))
-            f.close()
-            print("edrixs >>> do_ed=0, Perform ED at occupancy: ", noccu_gs,
-                  "with more accuracy", flush=True)
-        # Do ED for the occupancy of ground state with more accuracy
-        eval_shift = c_level * c_norb / noccu_gs
-        emat_i[0:ntot_v, 0:ntot_v] += np.eye(ntot_v) * eval_shift
-        emat_n[ntot_v:ntot, ntot_v:ntot] += np.eye(c_norb) * c_level
-        if rank == 0:
-            write_emat(emat_i, 'hopping_i.in')
-            write_emat(emat_n, 'hopping_n.in')
-            write_config(
-                ed_solver=ed_solver, num_val_orbs=ntot_v, neval=neval, nvector=nvector, ncv=ncv,
-                idump=idump, maxiter=maxiter, min_ndim=min_ndim, eigval_tol=eigval_tol
-            )
-            write_fock_dec_by_N(ntot_v, noccu_gs, "fock_i.in")
-        comm.Barrier()
-        ed_fsolver(fcomm, rank, size)
-        comm.Barrier()
-        data = np.loadtxt('eigvals.dat', ndmin=2)
-        eval_i = np.zeros(neval, dtype=float)
-        eval_i[0:neval] = data[0:neval, 1]
-        data = np.loadtxt('denmat.dat', ndmin=2)
-        tmp = (nvector, ntot_v, ntot_v)
-        denmat = data[:, 3].reshape(tmp) + 1j * data[:, 4].reshape(tmp)
-        return eval_i, denmat, noccu_gs
-    else:
-        raise Exception("Unknown case of do_ed ", do_ed)

-
-
-
-

-[docs]
-def xas_siam_fort(comm, shell_name, nbath, ominc, *, gamma_c=0.1,
-                  v_noccu=1, thin=1.0, phi=0, pol_type=None,
-                  num_gs=1, nkryl=200, temperature=1.0,
-                  loc_axis=None, scatter_axis=None):
-    """
-    Calculate XAS for single impurity Anderson model (SIAM) with Fortran solver.
-
-    Parameters
-    ----------
-    comm: MPI_comm
-        MPI communicator.
-    shell_name: tuple of two strings
-        Names of valence and core shells. The 1st (2nd) string in the tuple is for the
-        valence (core) shell.
-
-        - The 1st string can only be 's', 'p', 't2g', 'd', 'f',
-
-        - The 2nd string can be 's', 'p', 'p12', 'p32', 'd', 'd32', 'd52',
-          'f', 'f52', 'f72'.
-
-        For example: shell_name=('d', 'p32') may indicate a :math:`L_3` edge transition from
-        core :math:`2p_{3/2}` shell to valence :math:`3d` shell for Ni.
-    nbath: int
-        Number of bath sites.
-    ominc: 1d float array
-        Incident energy of photon.
-    gamma_c: a float number or a 1d float array with the same shape as ominc.
-        The core-hole life-time broadening factor. It can be a constant value
-        or incident energy dependent.
-    v_noccu: int
-        Total occupancy of valence shells.
-    thin: float number
-        The incident angle of photon (in radian).
-    phi: float number
-        Azimuthal angle (in radian), defined with respect to the
-        :math:`x`-axis of the local scattering axis: scatter_axis[:,0].
-    pol_type: list of tuples
-        Type of polarization, options can be:
-
-        - ('linear', alpha), linear polarization, where alpha is the angle between the
-          polarization vector and the scattering plane in radians.
-
-        - ('left', 0), left circular polarization.
-
-        - ('right', 0), right circular polarization.
-
-        - ('isotropic', 0). isotropic polarization.
-
-        It will set pol_type=[('isotropic', 0)] if not provided.
-    num_gs: int
-        Number of initial states used in XAS calculations.
-    nkryl: int
-        Maximum number of poles obtained.
-    temperature: float number
-        Temperature (in K) for boltzmann distribution.
-    loc_axis: 3*3 float array
-        The local axis with respect to which local orbitals are defined.
-
-        - x: local_axis[:,0],
-
-        - y: local_axis[:,1],
-
-        - z: local_axis[:,2].
-
-        It will be an identity matrix if not provided.
-    scatter_axis: 3*3 float array
-        The local axis defining the scattering geometry. The scattering plane is defined in
-        the local :math:`zx`-plane.
-
-        - local :math:`x`-axis: scatter_axis[:,0]
-
-        - local :math:`y`-axis: scatter_axis[:,1]
-
-        - local :math:`z`-axis: scatter_axis[:,2]
-
-        It will be set to an identity matrix if not provided.
-
-    Returns
-    -------
-    xas: 2d array, shape=(len(ominc), len(pol_type))
-        The calculated XAS spectra. The first dimension is for ominc, and the second dimension
-        if for different polarizations.
-    poles: list of dict, shape=(len(pol_type), )
-        The calculated XAS poles for different polarizations.
-    """
-    from .fedrixs import xas_fsolver
-
-    rank = comm.Get_rank()
-    size = comm.Get_size()
-    fcomm = comm.py2f()
-
-    v_name_options = ['s', 'p', 't2g', 'd', 'f']
-    c_name_options = ['s', 'p', 'p12', 'p32', 't2g', 'd', 'd32', 'd52', 'f', 'f52', 'f72']
-
-    v_name = shell_name[0].strip()
-    c_name = shell_name[1].strip()
-    if v_name not in v_name_options:
-        raise Exception("NOT supported type of valence shell: ", v_name)
-    if c_name not in c_name_options:
-        raise Exception("NOT supported type of core shell: ", c_name)
-
-    info_shell = info_atomic_shell()
-    v_norb = info_shell[v_name][1]
-    c_norb = info_shell[c_name][1]
-
-    ntot_v = v_norb * (nbath + 1)
-    ntot = ntot_v + c_norb
-    if pol_type is None:
-        pol_type = [('isotropic', 0)]
-    if loc_axis is None:
-        loc_axis = np.eye(3)
-    else:
-        loc_axis = np.array(loc_axis)
-    if scatter_axis is None:
-        scatter_axis = np.eye(3)
-    else:
-        scatter_axis = np.array(scatter_axis)
-
-    if rank == 0:
-        print("edrixs >>> Running XAS ...", flush=True)
-        write_config(num_val_orbs=ntot_v, num_core_orbs=c_norb,
-                     num_gs=num_gs, nkryl=nkryl)
-        write_fock_dec_by_N(ntot_v, v_noccu, "fock_i.in")
-        write_fock_dec_by_N(ntot_v, v_noccu + 1, "fock_n.in")
-
-    case = v_name + c_name
-    tmp = get_trans_oper(case)
-    npol, n, m = tmp.shape
-    tmp_g = np.zeros((npol, n, m), dtype=complex)
-    trans_mat = np.zeros((npol, ntot, ntot), dtype=complex)
-    # Transform the transition operators to global-xyz axis
-    # dipolar transition
-    if npol == 3:
-        for i in range(3):
-            for j in range(3):
-                tmp_g[i] += loc_axis[i, j] * tmp[j]
-    # quadrupolar transition
-    elif npol == 5:
-        alpha, beta, gamma = rmat_to_euler(loc_axis)
-        wignerD = get_wigner_dmat(4, alpha, beta, gamma)
-        rotmat = np.dot(np.dot(tmat_r2c('d'), wignerD), np.conj(np.transpose(tmat_r2c('d'))))
-        for i in range(5):
-            for j in range(5):
-                tmp_g[i] += rotmat[i, j] * tmp[j]
-    else:
-        raise Exception("Have NOT implemented this case: ", npol)
-    trans_mat[:, 0:v_norb, ntot_v:ntot] = tmp_g
-
-    n_om = len(ominc)
-    gamma_core = np.zeros(n_om, dtype=float)
-    if np.isscalar(gamma_c):
-        gamma_core[:] = np.ones(n_om) * gamma_c
-    else:
-        gamma_core[:] = gamma_c
-
-    # loop over different polarization
-    xas = np.zeros((n_om, len(pol_type)), dtype=float)
-    poles = []
-    comm.Barrier()
-    for it, (pt, alpha) in enumerate(pol_type):
-        if pt.strip() == 'left' or pt.strip() == 'right' or pt.strip() == 'linear':
-            if rank == 0:
-                print("edrixs >>> Loop over for polarization: ", it, pt, flush=True)
-                kvec = unit_wavevector(thin, phi, scatter_axis, 'in')
-                polvec = np.zeros(npol, dtype=complex)
-                pol = dipole_polvec_xas(thin, phi, alpha, scatter_axis, pt)
-                if npol == 3:  # Dipolar transition
-                    polvec[:] = pol
-                if npol == 5:  # Quadrupolar transition
-                    polvec[:] = quadrupole_polvec(pol, kvec)
-
-                trans = np.zeros((ntot, ntot), dtype=complex)
-                for i in range(npol):
-                    trans[:, :] += trans_mat[i] * polvec[i]
-                write_emat(trans, 'transop_xas.in')
-
-            # call XAS solver in fedrixs
-            comm.Barrier()
-            xas_fsolver(fcomm, rank, size)
-            comm.Barrier()
-
-            file_list = ['xas_poles.' + str(i+1) for i in range(num_gs)]
-            pole_dict = read_poles_from_file(file_list)
-            poles.append(pole_dict)
-            xas[:, it] = get_spectra_from_poles(pole_dict, ominc, gamma_core, temperature)
-        elif pt.strip() == 'isotropic':
-            pole_dicts = []
-            for k in range(npol):
-                if rank == 0:
-                    print("edrixs >>> Loop over for polarization: ", it, pt, flush=True)
-                    print("edrixs >>> Isotropic, component: ", k, flush=True)
-                    write_emat(trans_mat[k], 'transop_xas.in')
-
-                # call XAS solver in fedrixs
-                comm.Barrier()
-                xas_fsolver(fcomm, rank, size)
-                comm.Barrier()
-
-                file_list = ['xas_poles.' + str(i+1) for i in range(num_gs)]
-                pole_tmp = read_poles_from_file(file_list)
-                xas[:, it] += get_spectra_from_poles(pole_tmp, ominc, gamma_core, temperature)
-                pole_dicts.append(pole_tmp)
-            xas[:, it] = xas[:, it] / npol
-            poles.append(merge_pole_dicts(pole_dicts))
-        else:
-            raise Exception("Unknown polarization type: ", pt)
-
-    return xas, poles

-
-
-
-

-[docs]
-def rixs_siam_fort(comm, shell_name, nbath, ominc, eloss, *, gamma_c=0.1, gamma_f=0.1,
-                   v_noccu=1, thin=1.0, thout=1.0, phi=0, pol_type=None, num_gs=1,
-                   nkryl=200, linsys_max=1000, linsys_tol=1e-10, temperature=1.0,
-                   loc_axis=None, scatter_axis=None):
-    """
-    Calculate RIXS for single impurity Anderson model with Fortran solver.
-
-    Parameters
-    ----------
-    comm: MPI_comm
-        MPI communicator.
-    shell_name: tuple of two strings
-        Names of valence and core shells. The 1st (2nd) string in the tuple is for the
-        valence (core) shell.
-
-        - The 1st string can only be 's', 'p', 't2g', 'd', 'f',
-
-        - The 2nd string can be 's', 'p', 'p12', 'p32', 'd', 'd32', 'd52',
-          'f', 'f52', 'f72'.
-
-        For example: shell_name=('d', 'p32') may indicate a :math:`L_3` edge transition from
-        core :math:`2p_{3/2}` shell to valence :math:`3d` shell for Ni.
-    nbath: int
-        Number of bath sites.
-    ominc: 1d float array
-        Incident energy of photon.
-    eloss: 1d float array
-        Energy loss.
-    gamma_c: a float number or a 1d float array with same shape as ominc.
-        The core-hole life-time broadening factor. It can be a constant value
-        or incident energy dependent.
-    gamma_f: a float number or a 1d float array with same shape as eloss.
-        The final states life-time broadening factor. It can be a constant value
-        or energy loss dependent.
-    v_noccu: int
-        Total occupancy of valence shells.
-    thin: float number
-        The incident angle of photon (in radian).
-    thout: float number
-        The scattered angle of photon (in radian).
-    phi: float number
-        Azimuthal angle (in radian), defined with respect to the
-        :math:`x`-axis of scattering axis: scatter_axis[:,0].
-    pol_type: list of 4-elements-tuples
-        Type of polarizations. It has the following form:
-
-        (str1, alpha, str2, beta)
-
-        where, str1 (str2) can be 'linear', 'left', 'right', and alpha (beta) is
-        the angle (in radians) between the linear polarization vector and the scattering plane.
-
-        It will set pol_type=[('linear', 0, 'linear', 0)] if not provided.
-    num_gs: int
-        Number of initial states used in RIXS calculations.
-    nkryl: int
-        Maximum number of poles obtained.
-    linsys_max: int
-        Maximum iterations of solving linear equations.
-    linsys_tol: float
-        Convergence for solving linear equations.
-    temperature: float number
-        Temperature (in K) for boltzmann distribution.
-    loc_axis: 3*3 float array
-        The local axis with respect to which local orbitals are defined.
-
-        - x: local_axis[:,0],
-
-        - y: local_axis[:,1],
-
-        - z: local_axis[:,2].
-
-        It will be an identity matrix if not provided.
-    scatter_axis: 3*3 float array
-        The local axis defining the scattering geometry. The scattering plane is defined in
-        the local :math:`zx`-plane.
-
-        - local :math:`x`-axis: scatter_axis[:,0]
-
-        - local :math:`y`-axis: scatter_axis[:,1]
-
-        - local :math:`z`-axis: scatter_axis[:,2]
-
-        It will be set to an identity matrix if not provided.
-
-    Returns
-    -------
-    rixs: 3d float array, shape=(len(ominc), len(eloss), len(pol_type))
-        The calculated RIXS spectra. The 1st dimension is for the incident energy,
-        the 2nd dimension is for the energy loss and the 3rd dimension is for
-        different polarizations.
-    poles: 2d list of dict, shape=(len(ominc), len(pol_type))
-        The calculated RIXS poles. The 1st dimension is for incident energy, and the
-        2nd dimension is for different polarizations.
-    """
-    from .fedrixs import rixs_fsolver
-
-    rank = comm.Get_rank()
-    size = comm.Get_size()
-    fcomm = comm.py2f()
-
-    v_name_options = ['s', 'p', 't2g', 'd', 'f']
-    c_name_options = ['s', 'p', 'p12', 'p32', 't2g', 'd', 'd32', 'd52', 'f', 'f52', 'f72']
-    v_name = shell_name[0].strip()
-    c_name = shell_name[1].strip()
-    if v_name not in v_name_options:
-        raise Exception("NOT supported type of valence shell: ", v_name)
-    if c_name not in c_name_options:
-        raise Exception("NOT supported type of core shell: ", c_name)
-
-    info_shell = info_atomic_shell()
-    v_norb = info_shell[v_name][1]
-    c_norb = info_shell[c_name][1]
-    ntot_v = v_norb * (nbath + 1)
-    ntot = ntot_v + c_norb
-
-    if pol_type is None:
-        pol_type = [('linear', 0, 'linear', 0)]
-    if loc_axis is None:
-        loc_axis = np.eye(3)
-    else:
-        loc_axis = np.array(loc_axis)
-    if scatter_axis is None:
-        scatter_axis = np.eye(3)
-    else:
-        scatter_axis = np.array(scatter_axis)
-
-    if rank == 0:
-        print("edrixs >>> Running RIXS ...", flush=True)
-        write_fock_dec_by_N(ntot_v, v_noccu, "fock_i.in")
-        write_fock_dec_by_N(ntot_v, v_noccu + 1, "fock_n.in")
-        write_fock_dec_by_N(ntot_v, v_noccu, "fock_f.in")
-
-        case = v_name + c_name
-        tmp = get_trans_oper(case)
-        npol, n, m = tmp.shape
-        tmp_g = np.zeros((npol, n, m), dtype=complex)
-        trans_mat = np.zeros((npol, ntot, ntot), dtype=complex)
-        # Transform the transition operators to global-xyz axis
-        # dipolar transition
-        if npol == 3:
-            for i in range(3):
-                for j in range(3):
-                    tmp_g[i] += loc_axis[i, j] * tmp[j]
-        # quadrupolar transition
-        elif npol == 5:
-            alpha, beta, gamma = rmat_to_euler(loc_axis)
-            wignerD = get_wigner_dmat(4, alpha, beta, gamma)
-            rotmat = np.dot(np.dot(tmat_r2c('d'), wignerD), np.conj(np.transpose(tmat_r2c('d'))))
-            for i in range(5):
-                for j in range(5):
-                    tmp_g[i] += rotmat[i, j] * tmp[j]
-        else:
-            raise Exception("Have NOT implemented this case: ", npol)
-        trans_mat[:, 0:v_norb, ntot_v:ntot] = tmp_g
-
-    n_om = len(ominc)
-    neloss = len(eloss)
-    gamma_core = np.zeros(n_om, dtype=float)
-    if np.isscalar(gamma_c):
-        gamma_core[:] = np.ones(n_om) * gamma_c
-    else:
-        gamma_core[:] = gamma_c
-    gamma_final = np.zeros(neloss, dtype=float)
-    if np.isscalar(gamma_f):
-        gamma_final[:] = np.ones(neloss) * gamma_f
-    else:
-        gamma_final[:] = gamma_f
-
-    # loop over different polarization
-    rixs = np.zeros((n_om, neloss, len(pol_type)), dtype=float)
-    poles = []
-    comm.Barrier()
-    # loop over different polarization
-    for iom, omega in enumerate(ominc):
-        if rank == 0:
-            write_config(
-                num_val_orbs=ntot_v, num_core_orbs=c_norb,
-                omega_in=omega, gamma_in=gamma_core[iom],
-                num_gs=num_gs, nkryl=nkryl, linsys_max=linsys_max,
-                linsys_tol=linsys_tol
-            )
-        poles_per_om = []
-        # loop over polarization
-        for ip, (it, alpha, jt, beta) in enumerate(pol_type):
-            if rank == 0:
-                print(flush=True)
-                print("edrixs >>> Calculate RIXS for incident energy: ", omega, flush=True)
-                print("edrixs >>> Polarization: ", ip, flush=True)
-                polvec_i = np.zeros(npol, dtype=complex)
-                polvec_f = np.zeros(npol, dtype=complex)
-                ei, ef = dipole_polvec_rixs(thin, thout, phi, alpha, beta,
-                                            scatter_axis, (it, jt))
-                # dipolar transition
-                if npol == 3:
-                    polvec_i[:] = ei
-                    polvec_f[:] = ef
-                # quadrupolar transition
-                elif npol == 5:
-                    ki = unit_wavevector(thin, phi, scatter_axis, direction='in')
-                    kf = unit_wavevector(thout, phi, scatter_axis, direction='out')
-                    polvec_i[:] = quadrupole_polvec(ei, ki)
-                    polvec_f[:] = quadrupole_polvec(ef, kf)
-                else:
-                    raise Exception("Have NOT implemented this type of transition operators")
-                trans_i = np.zeros((ntot, ntot), dtype=complex)
-                trans_f = np.zeros((ntot, ntot), dtype=complex)
-                for i in range(npol):
-                    trans_i[:, :] += trans_mat[i] * polvec_i[i]
-                write_emat(trans_i, 'transop_rixs_i.in')
-                for i in range(npol):
-                    trans_f[:, :] += trans_mat[i] * polvec_f[i]
-                write_emat(np.conj(np.transpose(trans_f)), 'transop_rixs_f.in')
-
-            # call RIXS solver in fedrixs
-            comm.Barrier()
-            rixs_fsolver(fcomm, rank, size)
-            comm.Barrier()
-
-            file_list = ['rixs_poles.' + str(i+1) for i in range(num_gs)]
-            pole_dict = read_poles_from_file(file_list)
-            poles_per_om.append(pole_dict)
-            rixs[iom, :, ip] = get_spectra_from_poles(pole_dict, eloss,
-                                                      gamma_final, temperature)
-
-        poles.append(poles_per_om)
-
-    return rixs, poles

-
-
- -
-
-
- -
- -
-

© Copyright 2019, Brookhaven National Lab.

-
- - Built with Sphinx using a - theme - provided by Read the Docs. - - -
-
-
-
-
- - - - \ No newline at end of file diff --git a/edrixs/_modules/edrixs/utils.html b/edrixs/_modules/edrixs/utils.html deleted file mode 100644 index 2e17cf1370..0000000000 --- a/edrixs/_modules/edrixs/utils.html +++ /dev/null @@ -1,1093 +0,0 @@ - - - - - - - - edrixs.utils — edrixs documentation - - - - - - - - - - - - - - - - - - - - - -
- - -
- -
-
-
- -
-
-
-
- -

Source code for edrixs.utils

-__all__ = ['beta_to_kelvin', 'kelvin_to_beta', 'boltz_dist', 'UJ_to_UdJH',
-           'UdJH_to_UJ', 'UdJH_to_F0F2F4', 'UdJH_to_F0F2F4F6', 'F0F2F4_to_UdJH',
-           'F0F2F4_to_UJ', 'F0F2F4F6_to_UdJH', 'CT_imp_bath',
-           'CT_imp_bath_core_hole', 'info_atomic_shell',
-           'case_to_shell_name', 'edge_to_shell_name', 'slater_integrals_name',
-           'get_atom_data', 'rescale']
-
-import numpy as np
-import json
-import pkg_resources
-
-
-

-[docs]
-def beta_to_kelvin(beta):
-    """
-    Convert :math:`\\beta` to Kelvin.
-
-    Parameters
-    ----------
-    beta: float
-        Inversion temperature.
-
-    Returns
-    -------
-    T: float
-        Temperature (K).
-    """
-
-    kb = 8.6173303E-5
-    ev = 1.0 / float(beta)
-    T = ev / kb
-    return T

-
-
-
-

-[docs]
-def kelvin_to_beta(k):
-    """
-    Convert temperature from Kelvin to :math:`\\beta`.
-
-    Parameters
-    ----------
-    k: float
-        Temperature in Kelvin.
-
-    Returns
-    -------
-    beta: float
-        Inversion temperature.
-    """
-
-    kb = 8.6173303E-5
-    beta = 1.0 / (kb * k)
-    return beta

-
-
-
-

-[docs]
-def boltz_dist(gs, T):
-    """
-    Return Boltzmann distributition.
-
-    Parameters
-    ----------
-    gs: 1d float array
-        Energy levels.
-    T: float
-        Temperature in Kelvin.
-
-    Returns
-    -------
-    res: 1d float array
-        The Boltzmann distributition.
-    """
-    tmp_gs = np.array(gs)
-    beta = kelvin_to_beta(T)
-    res = np.exp(-beta * (tmp_gs - min(tmp_gs))) / np.sum(np.exp(-beta * (tmp_gs - min(tmp_gs))))
-    return res

-
-
-
-

-[docs]
-def UJ_to_UdJH(U, J):
-    """
-    Given Kanamori :math:`U` and :math:`J`, return :math:`U_d` and :math:`J_H`,
-    for :math:`t2g`-orbitals.
-
-    Parameters
-    ----------
-    U: float
-        Coulomb interaction :math:`U`.
-    J: float
-        Hund's coupling :math:`J`.
-
-    Returns
-    -------
-    Ud: float
-        Coulomb interaction :math:`U_d`.
-    JH: float
-        Hund's coupling :math:`J_{H}`.
-    """
-
-    F2 = J / (3.0 / 49.0 + 20 * 0.625 / 441.0)
-    F4 = 0.625 * F2
-    JH = (F2 + F4) / 14.0
-    Ud = U - 4.0 / 49.0 * (F2 + F4)
-
-    return Ud, JH

-
-
-
-

-[docs]
-def UdJH_to_UJ(Ud, JH):
-    """
-    Given :math:`U_d` and :math:`J_H`, return Kanamori :math:`U` and :math:`J`,
-    for :math:`t2g`-orbitals.
-
-    Parameters
-    ----------
-    Ud: float
-        Coulomb interaction :math:`U_d`.
-    JH: float
-        Hund's coupling :math:`J_H`.
-
-    Returns
-    -------
-    U: float
-        Coulomb interaction :math:`U` in Kanamori form.
-    J: float
-        Hund's coupling :math:`J` in Kanamori form.
-    """
-
-    F2 = 14.0 / 1.625 * JH
-    F4 = 0.625 * F2
-    J = 3.0 / 49.0 * F2 + 20 / 441.0 * F4
-    U = Ud + 4.0 / 49.0 * (F2 + F4)
-
-    return U, J

-
-
-
-

-[docs]
-def UdJH_to_F0F2F4(Ud, JH):
-    """
-    Given :math:`U_d` and :math:`J_H`, return :math:`F_0`, :math:`F_2` and :math:`F_4`,
-    for :math:`d`-orbitals.
-
-    Parameters
-    ----------
-    Ud: float
-        Coulomb interaction :math:`U_d`.
-    JH: float
-        Hund's coupling :math:`J_H`.
-
-    Returns
-    -------
-    F0: float
-        Slater integral :math:`F_0`.
-    F2: float
-        Slater integral :math:`F_2`.
-    F4: float
-        Slater integral :math:`F_4`.
-    """
-
-    F0 = Ud
-    F2 = 14 / 1.625 * JH
-    F4 = 0.625 * F2
-
-    return F0, F2, F4

-
-
-
-

-[docs]
-def UdJH_to_F0F2F4F6(Ud, JH):
-    """
-    Given :math:`U_d` and :math:`J_H`, return :math:`F_0`, :math:`F_2`, :math:`F_4` and :math:`F_6`,
-    for :math:`f`-orbitals.
-
-    Parameters
-    ----------
-    Ud: float
-        Coulomb interaction :math:`U_d`.
-    JH: float
-        Hund's coupling :math:`J_H`.
-
-    Returns
-    -------
-    F0: float
-        Slater integral :math:`F_0`.
-    F2: float
-        Slater integral :math:`F_2`.
-    F4: float
-        Slater integral :math:`F_4`.
-    F6: float
-        Slater integral :math:`F_6`.
-
-    """
-
-    F0 = Ud
-    F2 = 6435 / (286.0 + (195 * 451) / 675.0 + (250 * 1001) / 2025.0) * JH
-    F4 = 451 / 675.0 * F2
-    F6 = 1001 / 2025.0 * F2
-
-    return F0, F2, F4, F6

-
-
-
-

-[docs]
-def F0F2F4_to_UdJH(F0, F2, F4):
-    """
-    Given :math:`F_0`, :math:`F_2` and :math:`F_4`, return :math:`U_d` and :math:`J_H`,
-    for :math:`d`-orbitals.
-
-    Parameters
-    ----------
-    F0: float
-        Slater integral :math:`F_0`.
-    F2: float
-        Slater integral :math:`F_2`.
-    F4: float
-        Slater integral :math:`F_4`.
-
-    Returns
-    -------
-    Ud: float
-        Coulomb interaction :math:`U_d`.
-    JH: float
-        Hund's coupling :math:`J_H`.
-    """
-
-    Ud = F0
-    JH = (F2 + F4) / 14.0
-
-    return Ud, JH

-
-
-
-

-[docs]
-def F0F2F4_to_UJ(F0, F2, F4):
-    """
-    Given :math:`F_0`, :math:`F_2` and :math:`F_4`, return :math:`U` and :math:`J`,
-    for :math:`t2g`-orbitals.
-
-    Parameters
-    ----------
-    F0: float
-        Slater integral :math:`F_0`.
-    F2: float
-        Slater integral :math:`F_2`.
-    F4: float
-        Slater integral :math:`F_4`.
-
-    Returns
-    -------
-    U: float
-        Coulomb interaction :math:`U`.
-    J: float
-        Hund's coupling :math:`J`.
-    """
-
-    U = F0 + 4.0 / 49.0 * (F2 + F4)
-    J = 3.0 / 49.0 * F2 + 20 / 441.0 * F4
-
-    return U, J

-
-
-
-

-[docs]
-def F0F2F4F6_to_UdJH(F0, F2, F4, F6):
-    """
-    Given :math:`F_0`, :math:`F_2`, :math:`F_4` and :math:`F_6`,
-    return :math:`U_d` and :math:`J_H`, for :math:`f`-orbitals.
-
-    Parameters
-    ----------
-    F0: float
-        Slater integral :math:`F_0`.
-    F2: float
-        Slater integral :math:`F_2`.
-    F4: float
-        Slater integral :math:`F_4`.
-    F6: float
-        Slater integral :math:`F_6`.
-
-    Returns
-    -------
-    Ud: float
-        Coulomb interaction :math:`U_d`.
-    JH: float
-        Hund's coupling :math:`J_H`.
-    """
-
-    Ud = F0
-    JH = (286 * F2 + 195 * F4 + 250 * F6) / 6435.0
-    return Ud, JH

-
-
-
-

-[docs]
-def CT_imp_bath(U_dd, Delta, n):
-    """
-    Compute energies of the impurity and bath for an
-    Anderson impurity or charge-transfer model
-    appropriate for a :math:`d`-shell transition metal compound.
-
-    Parameters
-    ----------
-    U_dd: float
-        Coulomb interaction :math:`U_{dd}`
-    Delta: float
-        Charge-transfer energy :math:`\\Delta`
-    n : integer
-        Number of electrons in the :math:`d`-shell
-
-    Returns
-    -------
-    E_d : float
-        Energy of the impurity states :math:`E_d`
-    E_L : float
-        Energy of the bath states :math:`E_L`
-
-    Notes
-    -----
-    We credit our approach to Maurits Hakverkort,
-    Heidelberg University.
-    We define the state with a full set of bath orbitals to be zero
-    energy and write the energy levels using the same definitions
-    as [1]_ [2]_ [3]_.
-
-    * :math:`d^{n}L^{10}` has energy :math:`0`
-
-    * :math:`d^{n+1}L^9` has energy :math:`\\Delta`
-
-    * :math:`d^{n+2}L^8` has energy :math:`2\\Delta + U_{dd}`
-
-    Using this we can write and solve three linear equations to get
-    :math:`E_d` and :math:`E_L` the energies of the impurity and bath.
-
-       .. math::
-           \\begin{aligned}
-           10 E_L + n     E_d + n(n-1) \\frac{U_{dd}}{2} &= 0 \\\\
-            9 E_L + (n+1) E_d + (n+1)n \\frac{U_{dd}}{2} &= \\Delta \\\\
-            8 E_L + (n+2) E_d + (n+1)(n+2) \\frac{U_{dd}}{2}
-            &= 2\\Delta + U_{dd}
-           \\end{aligned}
-
-    The solutions are:
-
-       .. math::
-           \\begin{aligned}
-           E_d &=
-           \\frac{10 \\Delta - n (19 +  n) U_{dd}/2}{10 + n}  \\\\
-           E_L &=
-           \\frac{n ((1+n) U_{dd}/2-\\Delta)}{10 + n}
-           \\end{aligned}.
-
-    References
-    ----------
-    .. [1] J. Zaanen, G. A. Sawatzky, and J. W. Allen
-           `Phys. Rev. Lett. 55, 418 (1985) <https://doi.org/10.1103/PhysRevLett.55.418>`_
-    .. [2] Maurits Haverkort et al.,
-           `Phys. Rev. B 85, 165113 (2012) <https://doi.org/10.1103/PhysRevB.85.165113>`_
-    .. [3] A. E. Bocquet et al.,
-           `Phys. Rev. B 53, 1161 (1996) <https://doi.org/10.1103/PhysRevB.53.1161>`_
-    """
-    E_d = (10*Delta - n*(19 + n)*U_dd/2)/(10 + n)
-    E_L = n*((1 + n)*U_dd/2-Delta)/(10 + n)
-    return E_d, E_L

-
-
-
-

-[docs]
-def CT_imp_bath_core_hole(U_dd, U_pd, Delta, n):
-    """
-    Compute energies of the impurity and bath for an
-    Anderson impurity or charge-transfer model
-    appropriate for a :math:`d`-shell transition metal compound
-    with a core hole.
-
-    Parameters
-    ----------
-    U_dd: float
-        Coulomb interaction :math:`U_{dd}`
-    U_pd: float
-        Coulomb interaction :math:`U_{pd}`
-    Delta: float
-        Charge-transfer energy :math:`\\Delta`
-    n : integer
-        Number of electrons in the :math:`d`-shell :math:`n`
-
-    Returns
-    -------
-    E_dc : float
-        Energy of the impurity states :math:`E_{dc}` with
-        a core hole
-    E_Lc : float
-        Energy of the bath states :math:`E_{Lc}` with a core
-        hole
-    E_p : float
-        Energy of the core hole state :math:`E_\\textrm{E_p}`
-
-    Notes
-    -----
-    We credit our approach to Maurits Hakverkort,
-    Heidelberg University.
-    We define the state with a full set of bath orbitals to be zero
-    energy and write the energy levels using the same definitions as
-    [1]_ [2]_ [3]_.
-
-    * :math:`2p^5 d^{n}L^{10}` has energy :math:`0`
-
-    * :math:`2p^5 d^{n+1}L^9` has energy :math:`\\Delta + U_{dd} - U_{pd}`
-
-    * :math:`2p^5 d^{n+2}L^8` has energy :math:`2\\Delta + 3 U_{dd} - 2 U_{pd}`
-
-    Using this we can write and solve linear equations to get
-    :math:`E_{dc}`, :math:`E_{Lc}` and :math:`E_p` the energies of the
-    impurity and bath with a core hole and the energy of the core hole.
-
-       .. math::
-           \\begin{aligned}
-               6 E_p + 10 E_{Lc} +  n    E_{dc} + n(n-1) \\frac{U_{dd}}{2}
-               + 6 n     U_{pd} &= 0    \\\\
-               6 E_p +  9 E_{Lc} + (n+1) E_{dc} + (n+1)n \\frac{U_{dd}}{2}
-               + 6 (n+1) U_{pd} &= \\Delta  \\\\
-               6 E_p +  8 E_{Lc} + (n+2) E_{d} + (n+1)(n+2) \\frac{U_{dd}}{2}
-               + 6 (n+2) U_{pd} &= 2 \\Delta+U_{dd} \\\\
-               5 E_p + 10 E_{Lc} + (n+1) E_{d} + (n+1) n \\frac{U_{dd}}{2}
-               + 5 (n+1) U_{pd} &= 0 \\\\
-               5 E_p +  9 E_{Lc} + (n+2) E_{dc} + (n+2)(n+1) \\frac{U_{dd}}{2}
-               + 5 (n+2) U_{pd} &= \\Delta+U_{dd}-U_{pd} \\\\
-               5 E_p +  8 E_{Lc} + (n+3) E_{dc} + (n+3)(n+2) \\frac{U_{dd}}{2}
-               + 5 (n+3) U_{pd} &= 2 \\Delta+3 U_{dd}-2 U_{pd}
-           \\end{aligned}
-
-    The solutions are:
-
-       .. math::
-           \\begin{aligned}
-              E_{dc} &= \\frac{10 \\Delta - n (31+n) \\frac{U_{dd}}{2}-90 U_{pd}}{16+n} \\\\
-              E_{Lc} &= \\frac{(1+n) (n \\frac{U_{dd}}{2}+6*U_{pd})-(6+n) \\Delta}{16+n} \\\\
-              E_p    &= \\frac{10 \\Delta + (1+n)(n\\frac{U_{dd}}{2}-(10+n)*U_{pd}}{16+n}
-           \\end{aligned}
-
-    References
-    ----------
-    .. [1] J. Zaanen, G. A. Sawatzky, and J. W. Allen
-           `Phys. Rev. Lett. 55, 418 (1985) <https://doi.org/10.1103/PhysRevLett.55.418>`_
-    .. [2] Maurits Haverkort et al.,
-           `Phys. Rev. B 85, 165113 (2012) <https://doi.org/10.1103/PhysRevB.85.165113>`_
-    .. [3] A. E. Bocquet et al.,
-           `Phys. Rev. B 53, 1161 (1996) <https://doi.org/10.1103/PhysRevB.53.1161>`_
-    """
-    E_dc = (10*Delta - n*(31 + n)*U_dd/2 - 90*U_pd) / (16 + n)
-    E_Lc = ((1 + n)*(n*U_dd/2 + 6*U_pd) - (6 + n)*Delta) / (16 + n)
-    E_p = (10*Delta + (1 + n)*(n*U_dd/2 - (10 + n)*U_pd)) / (16 + n)
-    return E_dc, E_Lc, E_p

-
-
-
-

-[docs]
-def info_atomic_shell():
-    """
-    Return a dict to describe the information of atomic shell.
-    The key is a string of the shell's name, the value is a 2-elments tuple,
-    the first value is the orbital angular moment number and
-    the second value is the dimension of the Hilbert space.
-
-    Returns
-    -------
-    info: dict
-        The dict describing info of atomic shell.
-    """
-
-    info = {'s':   (0, 2),
-            'p':   (1, 6),
-            'p12': (1, 2),
-            'p32': (1, 4),
-            't2g': (2, 6),
-            'd':   (2, 10),
-            'd32': (2, 4),
-            'd52': (2, 6),
-            'f':   (3, 14),
-            'f52': (3, 6),
-            'f72': (3, 8)
-            }
-
-    return info

-
-
-
-

-[docs]
-def case_to_shell_name(case):
-    """
-    Return the shell names for different cases.
-
-    Parameters
-    ----------
-    case: string
-        A string describing the shells included.
-
-    Returns
-    -------
-    shell_name: tuple of one or two strings
-        The name of shells.
-
-    Examples
-    --------
-    >>> import edrixs
-    >>> edrixs.case_to_shell_name('d')
-    ('d',)
-    >>> edrixs.case_to_shell_name('t2gp32')
-    ('t2g', 'p32')
-    """
-
-    shell = ['s', 'p', 'p12', 'p32', 't2g', 'd', 'd32', 'd52', 'f', 'f52', 'f72']
-
-    shell_name = {}
-    for str1 in shell:
-        shell_name[str1] = (str1,)
-
-    for str1 in shell:
-        for str2 in shell:
-            shell_name[str1+str2] = (str1, str2)
-
-    return shell_name[case.strip()]

-
-
-
-

-[docs]
-def edge_to_shell_name(edge_name, with_main_qn=False):
-    """
-    Given edge name, return shell name.
-    If one needs to include both spin-orbit split edges of one shell,
-    one can use string, for example, 'L23' means both L2 and L3 edges
-    are considered in the calculations, and the shell name will be 'p'.
-
-    Parameters
-    ----------
-    edge_name: string
-        Standard edge name.
-    with_main_qn: logical
-        If true, the shell name will include the main quantum number.
-    Returns
-    -------
-    shell_name: string
-        Shell name.
-
-        - with_main_qn=True, shell_name will include the main quantum number,
-          for example, *2p*
-
-        - with_main_qn=False, shell_name will not include the main quantum number,
-          for example, *p*
-    """
-    shell_name = {
-        'K': ('s', '1s'),
-        'L1': ('s', '2s'),
-        'L2': ('p12', '2p12'),
-        'L3': ('p32', '2p32'),
-        'L23': ('p', '2p'),
-        'M1': ('s', '3s'),
-        'M2': ('p12', '3p12'),
-        'M3': ('p32', '3p32'),
-        'M23': ('p', '3p'),
-        'M4': ('d32', '3d32'),
-        'M5': ('d52', '3d52'),
-        'M45': ('d', '3d'),
-        'N1': ('s', '4s'),
-        'N2': ('p12', '4p12'),
-        'N3': ('p32', '4p32'),
-        'N23': ('p', '4p'),
-        'N4': ('d32', '4d32'),
-        'N5': ('d52', '4d52'),
-        'N45': ('d', '4d'),
-        'N6': ('f52', '4f52'),
-        'N7': ('f72', '4f72'),
-        'N67': ('f', '4f'),
-        'O1': ('s', '5s'),
-        'O2': ('p12', '5p12'),
-        'O3': ('p32', '5p32'),
-        'O23': ('p', '5p'),
-        'O4': ('d32', '5d32'),
-        'O5': ('d52', '5d52'),
-        'O45': ('d', '5d'),
-        'P1': ('s', '6s'),
-        'P2': ('p12', '6p12'),
-        'P3': ('p32', '6p32'),
-        'P23': ('p', '6p')
-    }
-
-    if with_main_qn:
-        return shell_name[edge_name.strip()][1]
-    else:
-        return shell_name[edge_name.strip()][0]

-
-
-
-

-[docs]
-def slater_integrals_name(shell_name, label=None):
-    """
-    Given shell names, return the names of required Slater integrals.
-    The order of these names in the return list is according to the convention:
-
-    - For 1 shell: [FX_11]
-
-    - For 2 shells: [FX_11, FX_12, GX_12, FX_22]
-
-    - For 3 shells: [FX_11, FX_12, GX_12, FX_22, FX_13, GX_13, FX_23, GX_23, FX_33]
-
-    where, X=0, 2, 4, 6, or X=1, 3, 5, X shoule be in ascending order.
-
-    Parameters
-    ----------
-    shell_name: tuple of strings
-        Name of shells. Its length should be less and equal than 3.
-    label: tuple of strings
-        Label of shells, same shape as shell_name.
-
-        If not provided, label will be set to
-
-        - label=(1,) for one shell, or
-
-        - label=(1,2) for two shells, or
-
-        - label=(1,2,3) for three shells
-
-    Returns
-    -------
-    res: list of strings
-        Names of Slater integrals.
-    """
-    info = info_atomic_shell()
-    # one shell
-    if len(shell_name) == 1:
-        res = []
-        l1 = info[shell_name[0]][0]
-        if label is not None:
-            x = label[0]
-        else:
-            x = '1'
-        res.extend(['F' + str(i) + '_' + x + x for i in range(0, 2*l1+1, 2)])
-    elif len(shell_name) == 2:
-        res = []
-        l1 = info[shell_name[0]][0]
-        l2 = info[shell_name[1]][0]
-        if label is not None:
-            x, y = label[0], label[1]
-        else:
-            x, y = '1', '2'
-        res.extend(['F' + str(i) + '_' + x + x for i in range(0, 2*l1+1, 2)])
-        res.extend(['F' + str(i) + '_' + x + y for i in range(0, min(2*l1, 2*l2)+1, 2)])
-        res.extend(['G' + str(i) + '_' + x + y for i in range(abs(l1-l2), l1+l2+1, 2)])
-        res.extend(['F' + str(i) + '_' + y + y for i in range(0, 2*l2+1, 2)])
-    elif len(shell_name) == 3:
-        res = []
-        l1 = info[shell_name[0]][0]
-        l2 = info[shell_name[1]][0]
-        l3 = info[shell_name[2]][0]
-        if label is not None:
-            x, y, z = label[0], label[1], label[2]
-        else:
-            x, y, z = '1', '2', '3'
-        res.extend(['F' + str(i) + '_' + x + x for i in range(0, 2*l1+1, 2)])
-        res.extend(['F' + str(i) + '_' + x + y for i in range(0, min(2*l1, 2*l2)+1, 2)])
-        res.extend(['G' + str(i) + '_' + x + y for i in range(abs(l1-l2), l1+l2+1, 2)])
-        res.extend(['F' + str(i) + '_' + y + y for i in range(0, 2*l2+1, 2)])
-        res.extend(['F' + str(i) + '_' + x + z for i in range(0, min(2*l1, 2*l3)+1, 2)])
-        res.extend(['G' + str(i) + '_' + x + z for i in range(abs(l1-l3), l1+l3+1, 2)])
-        res.extend(['F' + str(i) + '_' + y + z for i in range(0, min(2*l2, 2*l3)+1, 2)])
-        res.extend(['G' + str(i) + '_' + y + z for i in range(abs(l2-l3), l2+l3+1, 2)])
-        res.extend(['F' + str(i) + '_' + z + z for i in range(0, 2*l3+1, 2)])
-    else:
-        raise Exception("Not implemented for this case: ", shell_name)
-
-    return res

-
-
-
-

-[docs]
-def get_atom_data(atom, v_name, v_noccu, edge=None, trans_to_which=1, label=None):
-    """
-    Return Slater integrals, spin-orbit coupling strength, edge energy and core hole
-    life-time broadening for an atom with given type of valence shells, occupancy of
-    valence shells and the x-ray resonant edge.
-
-    The Slater integrals and spin-orbit coupling are calculated by Cowan's code
-    (https://github.com/mretegan/atomic-parameters) based on Hartree-Fock
-    approximation. These numbers are just initial guess and serve as a start point,
-    you need to rescale them to reproduce your XAS or RIXS spectra.
-
-    NOTE: F0 is not calculated by Cowan's code, they are all set to be zero, you need to
-    set F0 by yourself.
-
-    Parameters
-    ----------
-    atom: string
-        Name of atom, for example, 'Ni', 'Cu', 'Ir', 'U'.
-    v_name: a string or a tuple of one or two strings
-        Names of one or two valence shells. Set it to a single string if there is only
-        one valence shell, or set it to a tuple of two strings if there are two valence
-        shells. For example, *v_name='3d'*, *v_name=('3d', '4p')*
-    v_noccu: a int or a tuple of one or two ints
-        Occupancy of valence shells before x-ray absorption (without a core hole).
-        Set it to a single int if there is only one valence shell, or set it to a
-        tuple of two ints if there are two valence shells. The order should be
-        consistent with that in v_name.
-        For example, *v_name='3d', v_noccu=8*, *v_name=('3d', '4p'), v_noccu=(8, 0)*.
-    edge: a string
-        X-ray resonant edge. If edge is not None, both the information of
-        the initial (without a core hole) and intermediate (with a core hole) Hamiltonians
-        will be returned, otherwise, only the information of the initial Hamiltonian
-        will be returned. It can be,
-
-        - K, L1, L2, L3, M1, M2, M3, M4, M5, N1, N2, N3, N4, N5,
-          N6, N7, O1, O2, O3, O4, O5, P1, P2, P3
-
-        - L23 (both L2 and L3), M23 (both M2 and M3), M45 (both M4 and M5)
-          N23 (both N2 and N3), N45 (both N4 and N5), N67 (both N6 and N7)
-          O23 (both O2 and O3), O45 (both O4 and O5), P23 (both P2 and P3)
-    trans_to_which: int
-        If there are two valence shells, this variable is used to indicate which
-        valence shell the photon transition happens.
-
-        - 1: to the first valence shell
-
-        - 2: to the second valence shell
-    label: a string or tuple of strings
-        User-defined symbols to label the names of Slater integrals.
-
-        - one element, for the valence shell
-
-        - two elements, 1st (2nd)-element for the 1st (2nd) valence shells or
-          1st-element for the 1st valence shell and the 2nd one for the core shell.
-
-        - three elements, the 1st (2nd)-element is for the 1st (2nd) valence shell,
-          the 3rd one is for the core shell
-
-
-    Returns
-    -------
-    res:  dict
-        Atomic data. A dict likes this if edge is None:
-
-        .. code-block:: python
-
-           res={
-               'slater_i': [(name of slater integrals, vaule of slater integrals), (), ...],
-               'v_soc': [list of SOC for each atomic shell in v_name],
-           }
-
-        otherwise,
-
-        .. code-block:: python
-
-           res={
-               'slater_i': [(name of slater integrals, value of slater integrals), (), ...],
-               'slater_n': [(name of slater integrals, value of slater integrals), (), ...],
-               'v_soc': [list of SOC for each atomic shell in v_name],
-               'c_soc': SOC of core shell,
-               'edge_ene': [list of edge energy, 2 elements if edge is any of
-               "L23, M23, M45, N23, N45, N67, O23, O45, P23", othewise only 1 element],
-               'gamma_c': [list of core hole life-time broadening, 2 elements if edge is any of
-               "L23, M23, M45, N23, N45, N67, O23, O45, P23", othewise only 1 element],
-           }
-
-
-    Examples
-    --------
-    >>> import edrixs
-    >>> res = edrixs.get_atom_data('Ni', v_name='3d', v_noccu=8)
-    >>> res
-    {'slater_i': [('F0_11', 0.0), ('F2_11', 13.886), ('F4_11', 8.67)],
-     'v_soc_i': [0.112]}
-    >>> name, slater = [list(i) for i in zip(*res['slater_i'])]
-    >>> name
-    ['F0_11', 'F2_11', 'F4_11']
-    >>> slater
-    [0.0, 13.886, 8.67]
-    >>> slater = [i * 0.8 for i in slater]
-    >>> slater[0] = edrixs.get_F0('d', slater[1], slater[2])
-    >>> slater
-    [0.5728507936507936, 11.1088, 6.936]
-
-    >>> import edrixs
-    >>> res=edrixs.get_atom_data('Ni', v_name='3d', v_noccu=8, edge='L3', label=('d', 'p'))
-    >>> res
-    {'slater_i': [('F0_dd', 0.0), ('F2_dd', 12.234), ('F4_dd', 7.598)],
-     'v_soc_i': [0.083],
-     'slater_n': [('F0_dd', 0.0), ('F2_dd', 12.234), ('F4_dd', 7.598),
-      ('F0_dp', 0.0), ('F2_dp', 7.721), ('G1_dp', 5.787), ('G3_dp', 3.291),
-      ('F0_pp', 0.0), ('F2_pp', 0.0)],
-     'v_soc_n': [0.102],
-     'c_soc': 11.507,
-     'edge_ene': [852.7],
-     'gamma_c': [0.275]}
-    >>> name_i, slater_i = [list(i) for i in zip(*res['slater_i'])]
-    >>> name_n, slater_n = [list(i) for i in zip(*res['slater_n'])]
-    >>> name_i
-    ['F0_dd', 'F2_dd', 'F4_dd']
-    >>> slater_i
-    [0.0, 12.234, 7.598]
-    >>> name_n
-    ['F0_dd', 'F2_dd', 'F4_dd', 'F0_dp', 'F2_dp', 'G1_dp', 'G3_dp', 'F0_pp', 'F2_pp']
-    >>> slater_n
-    [0.0, 12.234, 7.598, 0.0, 7.721, 5.787, 3.291, 0.0, 0.0]
-    >>> slater_n[0] = edrixs.get_F0('d', slater_n[1], slater_n[2])
-    >>> slater_n[3] = edrixs.get_F0('dp', slater_n[5], slater_n[6])
-    >>> slater_n
-    [0.6295873015873016, 12.234, 7.598, 0.5268428571428572, 7.721, 5.787, 3.291, 0.0, 0.0]
-
-    >>> import edrixs
-    >>> import collections
-    >>> res=edrixs.get_atom_data('Ni', v_name=('3d', '4p'), v_noccu=(8, 0),
-    ...     edge='K', trans_to_which=2, label=('d', 'p', 's'))
-    >>> res
-    {'slater_i': [('F0_dd', 0.0), ('F2_dd', 12.234), ('F4_dd', 7.598),
-      ('F0_dp', 0.0), ('F2_dp', 0.0), ('G1_dp', 0.0), ('G3_dp', 0.0),
-      ('F0_pp', 0.0), ('F2_pp', 0.0)],
-     'v_soc_i': [0.083, 0.0],
-     'slater_n': [('F0_dd', 0.0), ('F2_dd', 13.851), ('F4_dd', 8.643),
-      ('F0_dp', 0.0), ('F2_dp', 2.299), ('G1_dp', 0.828), ('G3_dp', 0.713),
-      ('F0_pp', 0.0), ('F2_pp', 0.0),
-      ('F0_ds', 0.0), ('G2_ds', 0.079),
-      ('F0_ps', 0.0), ('G1_ps', 0.194),
-      ('F0_ss', 0.0)],
-     'v_soc_n': [0.113, 0.093],
-     'c_soc': 0.0,
-     'edge_ene': [8333.0],
-     'gamma_c': [0.81]}
-    >>> slat_i = collections.OrderedDict(res['slater_i'])
-    >>> slat_n = collections.OrderedDict(res['slater_n'])
-    >>> list(slat_i.keys())
-    ['F0_dd', 'F2_dd', 'F4_dd', 'F0_dp', 'F2_dp', 'G1_dp', 'G3_dp', 'F0_pp', 'F2_pp']
-    >>> list(slat_i.values())
-    [0.0, 12.234, 7.598, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0]
-    >>> list(slat_n.keys())
-    ['F0_dd', 'F2_dd', 'F4_dd', 'F0_dp', 'F2_dp', 'G1_dp', 'G3_dp', 'F0_pp', 'F2_pp',
-     'F0_ds', 'G2_ds', 'F0_ps', 'G1_ps', 'F0_ss']
-    >>> list(slat_n.values())
-    [0.0, 13.851, 8.643, 0.0, 2.299, 0.828, 0.713, 0.0, 0.0,
-     0.0, 0.079, 0.0, 0.194, 0.0]
-    """
-    c_norb = {'s': 2, 'p': 6, 'd': 10, 'f': 14}
-    atom = atom.strip()
-    avail_atoms = ['Ti', 'V', 'Cr', 'Mn', 'Fe', 'Co', 'Ni', 'Cu',
-                   'Re', 'Os', 'Ir',
-                   'Sm',
-                   'U', 'Pu']
-    avail_shells = ['1s', '2s', '2p', '3s', '3p', '3d', '4s', '4p', '4d', '4f',
-                    '5s', '5p', '5d', '5f', '6s', '6p', '6d']
-
-    if atom not in avail_atoms:
-        raise Exception("Atom data is Not available for this atom: ", atom)
-
-    if not isinstance(v_name, (list, tuple)):
-        v_name = (v_name,)
-    if not isinstance(v_noccu, (list, tuple)):
-        v_noccu = (v_noccu,)
-
-    if label is not None:
-        if not isinstance(label, (list, tuple)):
-            label = (label,)
-
-    if len(v_name) != len(v_noccu):
-        raise Exception("The shape of v_name is not same as noccu")
-
-    for ishell in v_name:
-        if ishell not in avail_shells:
-            raise Exception("Not available for this shell: ", ishell)
-
-    fname = pkg_resources.resource_filename('edrixs', 'atom_data/'+atom+'.json')
-    with open(fname, 'r') as f:
-        atom_dict = json.load(f)
-
-    res = {}
-    shell_name = []
-    for name in v_name:
-        shell_name.append(name[-1])
-    if label is not None:
-        my_label = label[0:len(shell_name)]
-    else:
-        my_label = None
-    slater_name = slater_integrals_name(shell_name, label=my_label)
-
-    if len(v_name) == 1:
-        case = v_name[0] + str(v_noccu[0])
-    else:
-        case = v_name[0] + str(v_noccu[0]) + '_' + v_name[1] + str(v_noccu[1])
-    if case not in atom_dict:
-        raise Exception("This configuration is not available in atom_data", case)
-
-    nslat = len(slater_name)
-    slater_i = [0.0] * nslat
-    tmp = atom_dict[case]['slater']
-    slater_i[0:len(tmp)] = tmp
-    res['slater_i'] = list(zip(slater_name, slater_i))
-    res['v_soc_i'] = atom_dict[case]['soc']
-
-    if edge is not None:
-        edge = edge.strip()
-        edge_name = edge_to_shell_name(edge, with_main_qn=True)
-        shell_name = []
-        for name in v_name:
-            shell_name.append(name[-1])
-        shell_name.append(edge_name[1:2])
-        if label is not None:
-            my_label = label[0:len(shell_name)]
-        else:
-            my_label = None
-        slater_name = slater_integrals_name(shell_name, label=my_label)
-
-        if len(v_name) == 1:
-            case = (v_name[0] + str(v_noccu[0]+1) + '_' +
-                    edge_name[0:2] + str(c_norb[edge_name[1]]-1))
-        else:
-            if trans_to_which == 1:
-                case = (v_name[0] + str(v_noccu[0]+1) + '_' +
-                        v_name[1] + str(v_noccu[1]) + '_' +
-                        edge_name[0:2] + str(c_norb[edge_name[1]]-1))
-            else:
-                case = (v_name[0] + str(v_noccu[0]) + '_' +
-                        v_name[1] + str(v_noccu[1]+1) + '_' +
-                        edge_name[0:2] + str(c_norb[edge_name[1]]-1))
-        if case not in atom_dict:
-            raise Exception("This configuration is currently not available in atom_data", case)
-
-        nslat = len(slater_name)
-        slater_n = [0.0] * nslat
-        tmp = atom_dict[case]['slater']
-        slater_n[0:len(tmp)] = tmp
-        res['slater_n'] = list(zip(slater_name, slater_n))
-
-        res['v_soc_n'] = atom_dict[case]['soc'][0:-1]
-        res['c_soc'] = atom_dict[case]['soc'][-1]
-
-        res['edge_ene'] = atom_dict[edge]['ene']
-        res['gamma_c'] = [i / 2 for i in atom_dict[edge]['gamma']]
-
-    return res

-
-
-
-

-[docs]
-def rescale(old_list, scale=None):
-    """
-    Rescale a 1d list.
-
-    Parameters
-    ----------
-    old_list: 1d list of numerical values
-        Input list.
-    scale: a list of tuples
-        The scaling factors.
-
-    Returns
-    -------
-    new_list: 1d list
-        The rescaled list.
-    """
-
-    new_list = [i for i in old_list]
-    if scale is not None:
-        for pos, val in zip(scale[0], scale[1]):
-            new_list[pos] = new_list[pos] * val
-    return new_list

-
-
- -
-
-
- -
- -
-

© Copyright 2019, Brookhaven National Lab.

-
- - Built with Sphinx using a - theme - provided by Read the Docs. - - -
-
-
-
-
- - - - \ No newline at end of file diff --git a/edrixs/_modules/edrixs/wannier_ham.html b/edrixs/_modules/edrixs/wannier_ham.html deleted file mode 100644 index 6c5bae4c4e..0000000000 --- a/edrixs/_modules/edrixs/wannier_ham.html +++ /dev/null @@ -1,489 +0,0 @@ - - - - - - - - edrixs.wannier_ham — edrixs documentation - - - - - - - - - - - - - - - - - - - - - -
- - -
- -
-
-
- -
-
-
-
- -

Source code for edrixs.wannier_ham

-__all__ = ['HR', 'KVec', 'SymKVec', 'UniKVec']
-
-import numpy as np
-
-
-

-[docs]
-class HR():
-    """
-    Class for post-process of the Wannier90 tight-binding (TB) Hamiltonian in real space.
-
-    Parameters
-    ----------
-    nwann: int
-        Number of Wannier orbitals.
-    nrpt: int
-        Number of :math:`r` points.
-    irpt0: int
-        Index of the point (0,0,0).
-    rpts: float array
-        The coordinates of :math:`r` points.
-    deg_rpt: int array
-        Degenerancy of :math:`r` points.
-    hr: complex array
-        Hamiltonian :math:`H(r)` from Wannier90.
-    """
-
-    def __init__(self, nwann, nrpt, irpt0, rpts, deg_rpt, hr):
-        self.nwann = nwann
-        self.nrpt = nrpt
-        self.irpt0 = irpt0
-        self.deg_rpt = deg_rpt
-        self.rpts = rpts
-        self.hr = hr
-
-

-[docs]
-    @staticmethod
-    def from_file(fname='wannier90_hr.dat'):
-        """
-        Generate a TB Hamiltonian from Wannier90 output file "case_hr.dat".
-
-        Parameters
-        ----------
-        fname: str
-            The file that contains the Wannier90 output file: "case_hr.dat".
-
-        Returns
-        -------
-        HR: HR object
-            A HR object.
-        """
-
-        with open(fname, 'r') as f:
-            # skip the header (1 line)
-            f.readline()
-            nwann = int(f.readline().strip())
-            nrpt = int(f.readline().strip())
-            # the degeneracy of R points
-            nline = nrpt // 15 + 1
-            tmp = []
-            for i in range(nline):
-                tmp.extend(f.readline().strip().split())
-            tmp = [np.int(item) for item in tmp]
-            deg_rpt = np.array(tmp, dtype=int)
-            # read hr for each r-point
-            rpts = np.zeros((nrpt, 3), dtype=int)
-            hr = np.zeros((nrpt, nwann, nwann), dtype=np.complex128)
-            for i in range(nrpt):
-                for j in range(nwann):
-                    for k in range(nwann):
-                        rx, ry, rz, hr_i, hr_j, hr_real, hr_imag = f.readline().strip().split()
-                        rpts[i, :] = int(rx), int(ry), int(rz)
-                        if int(rx) == 0 and int(ry) == 0 and int(rz) == 0:
-                            irpt0 = i
-                        hr[i, k, j] = np.float64(
-                            hr_real) + np.float64(hr_imag) * 1j
-            # construct the HR instance
-            return HR(nwann, nrpt, irpt0, rpts, deg_rpt, hr)

-
-
-

-[docs]
-    @staticmethod
-    def copy_hr(other):
-        """
-        Copy instance of HR.
-
-        Parameters
-        ----------
-        other: HR object
-            A HR object to be copied.
-
-        Returns
-        -------
-        HR: HR object
-            Return a new HR object.
-        """
-
-        return HR(other.nwann,
-                  other.nrpt,
-                  other.irpt0,
-                  np.copy(other.rpts),
-                  np.copy(other.deg_rpt),
-                  np.copy(other.hr))

-
-
-

-[docs]
-    def get_hr0(self, ispin=False):
-        """
-        Return the on-site term :math:`H(r=0)`.
-
-        Parameters
-        ----------
-        ispin: logical
-            Whether to include spin degree of freedom or not (default: False).
-
-        Returns
-        -------
-        hr: 2d complex array
-            The on-site Hamiltonian.
-        """
-
-        if ispin:
-            norbs = 2 * self.nwann
-            hr0_spin = np.zeros((norbs, norbs), dtype=np.complex128)
-            hr0_spin[0:norbs:2, 0:norbs:2] = self.hr[self.irpt0, :, :]
-            hr0_spin[1:norbs:2, 1:norbs:2] = self.hr[self.irpt0, :, :]
-            return hr0_spin
-        else:
-            return self.hr[self.irpt0, :, :]

-
-
-

-[docs]
-    def get_hr(self, ispin):
-        """
-        Return the Hamiltonian of :math:`H(r)`.
-
-        Parameters
-        ----------
-        ispin: logical
-            Whether to include spin degree of freedom or not (default: False)
-
-        Returns
-        -------
-        hr: 3d complex array
-            The Hamiltonian :math:`H(r)`.
-        """
-
-        # with spin, spin order: up dn up dn up dn
-        if ispin == 1:
-            norbs = 2 * self.nwann
-            hr_spin = np.zeros((self.nrpt, norbs, norbs), dtype=np.complex128)
-            hr_spin[:, 0:norbs:2, 0:norbs:2] = self.hr
-            hr_spin[:, 1:norbs:2, 1:norbs:2] = self.hr
-            return hr_spin
-        # with spin, spin order: up up up dn dn dn
-        elif ispin == 2:
-            norbs = 2 * self.nwann
-            hr_spin = np.zeros((self.nrpt, norbs, norbs), dtype=np.complex128)
-            hr_spin[:, 0:self.nwann, 0:self.nwann] = self.hr
-            hr_spin[:, self.nwann:norbs, self.nwann:norbs] = self.hr
-            return hr_spin
-        # without spin
-        else:
-            return self.hr[:, :, :]

-

-
-
-
-

-[docs]
-class KVec():
-    """
-    Define :math:`k` points in BZ, high symmetry line or uniform grid.
-
-    Parameters
-    ----------
-    kpt_type: str
-        The type of :math:`k` points, 'uni' or 'sym'.
-    kbase: :math:`3 \\times 3` float array
-        The basis vectors of the primitive reciprocal space.
-    nkpt: int
-        Number of :math:`k` points.
-    kvec: float array
-        The :math:`k` points.
-    """
-
-    def __init__(self, kpt_type='uni', kbase=None, nkpt=None, kvec=None):
-        self.nkpt = nkpt
-        self.kbase = np.array(kbase, dtype=np.float64)
-        self.kvec = np.array(kvec, dtype=np.float64)
-        self.kpt_type = kpt_type
-
-

-[docs]
-    def set_base(self, kbase):
-        """
-        Set the basis of the primitive reciprocal.
-
-        Parameters
-        ----------
-        kbase: :math:`3 \\times 3` float array
-            The basis with respect to the global axis.
-        """
-
-        self.kbase = np.array(kbase, dtype=np.float64)

-
-
-

-[docs]
-    def kvec_from_file(self, fname):
-        """
-        Read :math:`k` points from file.
-
-        Parameters
-        ----------
-        fname: str
-            File name.
-        """
-
-        tmp = []
-        with open(fname, 'r') as f:
-            for line in f:
-                line = line.strip().split()
-                if line != []:
-                    tmp.append(line)
-            self.kvec = np.array(tmp, dtype=np.float64)
-            self.nkpt = len(tmp)

-

-
-
-
-

-[docs]
-class SymKVec(KVec):
-    """
-    Class for defining :math:`k` points in high symmetry line, derived from :class:`KVec`.
-
-    Parameters
-    ----------
-    kbase: :math:`3 \\times 3` float array
-        Basis of the primitive reciprocal lattice.
-    hsymkpt: float array
-        Starting and end :math:`k` points along high symmetry lines.
-    klen: float array
-        Length of segments of :math:`k` points line.
-    """
-
-    def __init__(self, kbase=None, hsymkpt=None, klen=None):
-        self.klen = np.array(klen, dtype=np.float64)
-        self.hsymkpt = np.array(hsymkpt, dtype=np.float64)
-        KVec.__init__(self, 'sym', kbase)
-
-

-[docs]
-    def get_klen(self):
-        """
-        Return length of :math:`k` points segments.
-        """
-
-        self.klen = np.zeros(self.nkpt, dtype=np.float64)
-        self.klen[0] = 0.0
-        prev_kpt = self.kvec[0]
-
-        for i in range(1, self.nkpt):
-            curr_kpt = self.kvec[i, :]
-            tmp_kpt = curr_kpt - prev_kpt
-            kx = np.dot(tmp_kpt, self.kbase[:, 0])
-            ky = np.dot(tmp_kpt, self.kbase[:, 1])
-            kz = np.dot(tmp_kpt, self.kbase[:, 2])
-            self.klen[i] = self.klen[i - 1] + \
-                np.sqrt(np.dot((kx, ky, kz), (kx, ky, kz)))
-            prev_kpt = curr_kpt

-
-
-

-[docs]
-    def from_hsymkpt(self, nkpt_per_path=20):
-        """
-        Given starting and end :math:`k` points of each segment,
-        and the number of points per each segment,
-        return the high symmetry :math:`k` points.
-
-        Parameters
-        ----------
-        nkpt_per_path: int
-            Number of :math:`k` points per each segment.
-        """
-
-        self.nkpt = nkpt_per_path * (len(self.hsymkpt) - 1)
-        self.kvec = np.zeros((self.nkpt, 3), dtype=np.float64)
-        for i in range(1, len(self.hsymkpt)):
-            kpt_prev = self.hsymkpt[i - 1, :]
-            kpt_curr = self.hsymkpt[i, :]
-            for j in range(nkpt_per_path):
-                ikpt = (i - 1) * nkpt_per_path + j
-                self.kvec[ikpt, :] = (float(j) / float(nkpt_per_path - 1) *
-                                      (kpt_curr - kpt_prev) + kpt_prev)

-
-
-

-[docs]
-    def from_hsymkpt_uni(self, step):
-        """
-        Given a step, return high symmetry :math:`k` points.
-
-        Parameters
-        ----------
-        step: float
-            Step size.
-        """
-
-        kvec = []
-        self.hsym_dis = np.zeros(len(self.hsymkpt), dtype=np.float64)
-        self.hsym_dis[0] = 0.0
-        for i in range(0, len(self.hsymkpt) - 1):
-            kpt_prev = self.hsymkpt[i, :]
-            kpt_curr = self.hsymkpt[i + 1, :]
-            tmp = np.dot(self.kbase.transpose(), kpt_curr - kpt_prev)
-            dis = np.sqrt(np.dot(tmp, tmp))
-            self.hsym_dis[i + 1] = self.hsym_dis[i] + dis
-            pts = np.arange(0, dis, step) / dis
-            for ipt in pts:
-                kvec.append(ipt * (kpt_curr - kpt_prev) + kpt_prev)
-        self.kvec = np.array(kvec, dtype=np.float64)
-        self.nkpt = len(self.kvec)

-

-
-
-
-

-[docs]
-class UniKVec(KVec):
-    """
-    Class for defining uniform :math:`k` points grid, derived from :class:`KVec`.
-
-    Parameters
-    ----------
-    grid: 3-elements tuple
-        Three numbers defining a uniform grid, for example: :math:`11 \\times 11 \\times 11`.
-    """
-
-    def __init__(self, grid=None):
-        self.grid = grid
-        KVec.__init__(self, 'uni')
-
-

-[docs]
-    def from_grid(self):
-        """
-        Return uniform :math:`k` points.
-        """
-
-        delta = 0.001
-        nx, ny, nz = self.grid
-        self.nkpt = nx * ny * nz
-        self.kvec = np.zeros((self.nkpt, 3), dtype=np.float64)
-        ikpt = 0
-        for i in range(nx):
-            if nx == 1:
-                kx = 0.0
-            else:
-                kx = float(i) / float(nx)
-            for j in range(ny):
-                if ny == 1:
-                    ky = 0.0
-                else:
-                    ky = float(j) / float(ny)
-                for k in range(nz):
-                    if nz == 1:
-                        kz = 0.0
-                    else:
-                        kz = float(k) / float(nz)
-                    ikpt = ikpt + 1
-                    self.kvec[ikpt - 1, :] = kx + delta, ky + delta, kz + delta

-

-
-
- -
-
-
- -
- -
-

© Copyright 2019, Brookhaven National Lab.

-
- - Built with Sphinx using a - theme - provided by Read the Docs. - - -
-
-
-
-
- - - - \ No newline at end of file diff --git a/edrixs/_modules/index.html b/edrixs/_modules/index.html deleted file mode 100644 index 520a66631e..0000000000 --- a/edrixs/_modules/index.html +++ /dev/null @@ -1,122 +0,0 @@ - - - - - - - - Overview: module code — edrixs documentation - - - - - - - - - - - - - - - - - - - - - -
- - -
- -
-
-
-
    -
    • -
  • Overview: module code
    • -
  • -
    • -
-
-
-
- -
-
- -
- -
-

© Copyright 2019, Brookhaven National Lab.

-
- - Built with Sphinx using a - theme - provided by Read the Docs. - - -
-
-
-
-
- - - - \ No newline at end of file diff --git a/edrixs/_sources/auto_examples/example_0_ed_calculator.rst.txt b/edrixs/_sources/auto_examples/example_0_ed_calculator.rst.txt deleted file mode 100644 index c4a833541a..0000000000 --- a/edrixs/_sources/auto_examples/example_0_ed_calculator.rst.txt +++ /dev/null @@ -1,584 +0,0 @@ - -.. DO NOT EDIT. -.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. -.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: -.. "auto_examples/example_0_ed_calculator.py" -.. LINE NUMBERS ARE GIVEN BELOW. - -.. only:: html - - .. note:: - :class: sphx-glr-download-link-note - - :ref:`Go to the end ` - to download the full example code. - -.. rst-class:: sphx-glr-example-title - -.. _sphx_glr_auto_examples_example_0_ed_calculator.py: - - -Exact diagonalization -===================================== -Here we show how to find the eigenvalues and eigenvectors of a many-body -Hamiltonian of fermions with Coulomb interactions. We then determine their spin -and orbital angular momentum and how this changes when we switch on spin-orbit -coupling. - -.. GENERATED FROM PYTHON SOURCE LINES 12-13 - -Import the necessary modules. - -.. GENERATED FROM PYTHON SOURCE LINES 13-18 - -.. code-block:: Python - - import numpy as np - import matplotlib.pyplot as plt - import scipy - import edrixs - - - - - - - - -.. GENERATED FROM PYTHON SOURCE LINES 19-24 - -Parameters ------------------------------------------------------------------------------- -Define the orbital angular momentum number :math:`l=1` (i.e. a `p` shell), -the number of spin-orbitals, the occupancy and the Slater integrals. -:math:`F^{k}` with :math:`k=0,2`: - -.. GENERATED FROM PYTHON SOURCE LINES 24-29 - -.. code-block:: Python - - l = 1 - norb = 6 - noccu = 2 - F0, F2 = 4.0, 1.0 - - - - - - - - -.. GENERATED FROM PYTHON SOURCE LINES 30-36 - -Coulomb interactions ------------------------------------------------------------------------------- -The Coulomb interactions in EDRIXS are described by a tensor. Understanding this -in full is complicated and requires careful consideration of the symmetry of the -interactions. See example 6 for more discussion if desired. -EDRIXS can construct the matrix via - -.. GENERATED FROM PYTHON SOURCE LINES 36-38 - -.. code-block:: Python - - umat = edrixs.get_umat_slater('p', F0, F2) - - - - - - - - -.. GENERATED FROM PYTHON SOURCE LINES 39-47 - -Create basis ------------------------------------------------------------------------------- -Now we build the binary form of the Fock basis :math:`|F>` (we consider it -preferable to use the standard :math:`F` and trust the reader to avoid -confusing it with the interaction parameters.) -The Fock basis is the simplest legitimate form for the basis and it consists -of a series of 1s and 0s where 1 means occupied and -0 means empty. These are in order up, down, up, down, up, down. - -.. GENERATED FROM PYTHON SOURCE LINES 47-49 - -.. code-block:: Python - - basis = edrixs.get_fock_bin_by_N(norb, noccu) - print(np.array(basis)) - - - - -.. rst-class:: sphx-glr-script-out - - .. code-block:: none - - [[1 1 0 0 0 0] - [1 0 1 0 0 0] - [1 0 0 1 0 0] - [1 0 0 0 1 0] - [1 0 0 0 0 1] - [0 1 1 0 0 0] - [0 1 0 1 0 0] - [0 1 0 0 1 0] - [0 1 0 0 0 1] - [0 0 1 1 0 0] - [0 0 1 0 1 0] - [0 0 1 0 0 1] - [0 0 0 1 1 0] - [0 0 0 1 0 1] - [0 0 0 0 1 1]] - - - - -.. GENERATED FROM PYTHON SOURCE LINES 50-53 - -We expect the number of these states to be given by the mathematical -combination of two electrons distributed among six states (three spin-orbitals -with two spins per orbital). - -.. GENERATED FROM PYTHON SOURCE LINES 53-56 - -.. code-block:: Python - - message = ("We predict C(norb={}, noccu={})={:.0f} states and we got {:d}, " - "which is reassuring!") - print(message.format(norb, noccu, edrixs.combination(norb, noccu), len(basis))) - - - - -.. rst-class:: sphx-glr-script-out - - .. code-block:: none - - We predict C(norb=6, noccu=2)=15 states and we got 15, which is reassuring! - - - - -.. GENERATED FROM PYTHON SOURCE LINES 57-59 - -Note that in more complicated problems with both valence and core -electrons, the edrixs convention is to list the valence electrons first. - -.. GENERATED FROM PYTHON SOURCE LINES 61-74 - -Transform interactions into Fock basis ------------------------------------------------------------------------------- -edrixs works by initiailly creating a Hamiltonian matrix -:math:`\hat{H}` in the single particle basis and then transforming into -our chosen Fock basis. In the single particle basis, we have four fermion -interactions with this form - - .. math:: - \hat{H} = - -generated as - -.. GENERATED FROM PYTHON SOURCE LINES 74-77 - -.. code-block:: Python - - n_fermion = 4 - H = edrixs.build_opers(n_fermion, umat, basis) - - - - - - - - -.. GENERATED FROM PYTHON SOURCE LINES 78-80 - -We needed to specify :code:`n_fermion = 4` because the -:code:`edrixs.build_opers` function can also make two fermion terms. - -.. GENERATED FROM PYTHON SOURCE LINES 82-88 - -Diagonalize the matrix ------------------------------------------------------------------------------- -For a small problem such as this it is convenient to use the native -`scipy `_ diagonalization routine. This returns eigenvalues -:code:`e` and eignvectors :code:`v` where eigenvalue :code:`e[i]` corresponds -to eigenvector :code:`v[:,i]`. - -.. GENERATED FROM PYTHON SOURCE LINES 88-93 - -.. code-block:: Python - - e, v = scipy.linalg.eigh(H) - print("{} eignvalues and {} eigvenvectors {} elements long.".format(len(e), - v.shape[1], - v.shape[0])) - - - - - -.. rst-class:: sphx-glr-script-out - - .. code-block:: none - - 15 eignvalues and 15 eigvenvectors 15 elements long. - - - - -.. GENERATED FROM PYTHON SOURCE LINES 94-101 - -Computing expectation values ------------------------------------------------------------------------------- -To interpret the results, it is informative to compute the expectations values -related to the spin :math:`\mathbf{S}`, orbital :math:`\mathbf{L}`, -and total :math:`\mathbf{J}`, angular momentum. We first load the relevant -matrices for these quantities for a `p` atomic shell. We need to specify -that we would like to include spin when loading the orbital operator. - -.. GENERATED FROM PYTHON SOURCE LINES 101-105 - -.. code-block:: Python - - orb_mom = edrixs.get_orb_momentum(l, ispin=True) - spin_mom = edrixs.get_spin_momentum(l) - tot_mom = orb_mom + spin_mom - - - - - - - - -.. GENERATED FROM PYTHON SOURCE LINES 106-107 - -We again transform these matrices to our Fock basis to build the operators - -.. GENERATED FROM PYTHON SOURCE LINES 107-111 - -.. code-block:: Python - - n_fermion = 2 - opL, opS, opJ = edrixs.build_opers(n_fermion, [orb_mom, spin_mom, tot_mom], - basis) - - - - - - - - -.. GENERATED FROM PYTHON SOURCE LINES 112-121 - -Recall that quantum mechanics forbids us from knowing all three Cartesian -components of angular momentum at once, so we want to compute the squares of -these operators i.e. - - .. math:: - \mathbf{S}^2 = S^2_x + S^2_y + S^2_z\\ - \mathbf{L}^2 = L^2_x + L^2_y + L^2_z\\ - \mathbf{J}^2 = J^2_x + J^2_y + J^2_z - - -.. GENERATED FROM PYTHON SOURCE LINES 121-125 - -.. code-block:: Python - - L2 = np.dot(opL[0], opL[0]) + np.dot(opL[1], opL[1]) + np.dot(opL[2], opL[2]) - S2 = np.dot(opS[0], opS[0]) + np.dot(opS[1], opS[1]) + np.dot(opS[2], opS[2]) - J2 = np.dot(opJ[0], opJ[0]) + np.dot(opJ[1], opJ[1]) + np.dot(opJ[2], opJ[2]) - - - - - - - - -.. GENERATED FROM PYTHON SOURCE LINES 126-129 - -Remember that the eigenvalues of :math:`\mathbf{S}^2` are in the form -:math:`S(S+1)` etc. and that they can be obtained by calculating the -projection of the operators onto our eigenvectors. - -.. GENERATED FROM PYTHON SOURCE LINES 129-132 - -.. code-block:: Python - - L2_val = edrixs.cb_op(L2, v).diagonal().real - S2_val = edrixs.cb_op(S2, v).diagonal().real - J2_val = edrixs.cb_op(J2, v).diagonal().real - - - - - - - -.. GENERATED FROM PYTHON SOURCE LINES 133-135 - -We can determine the degeneracy of the eigenvalues numerically and print out -the values as follows - -.. GENERATED FROM PYTHON SOURCE LINES 135-143 - -.. code-block:: Python - - e = np.round(e, decimals=6) - degeneracy = [sum(eval == e) for eval in e] - header = "{:<3s}\t{:>8s}\t{:>8s}\t{:>8s}\t{:>8s}" - print(header.format("# ", "E ", "S(S+1)", "L(L+1)", "Degen.")) - for i, eigenvalue in enumerate(e): - values_list = [i, eigenvalue, S2_val[i], L2_val[i], degeneracy[i]] - print("{:<3d}\t{:8.3f}\t{:8.3f}\t{:8.3f}\t{:>3d}".format(*values_list)) - - - - - -.. rst-class:: sphx-glr-script-out - - .. code-block:: none - - # E S(S+1) L(L+1) Degen. - 0 3.800 2.000 2.000 9 - 1 3.800 2.000 2.000 9 - 2 3.800 2.000 2.000 9 - 3 3.800 2.000 2.000 9 - 4 3.800 2.000 2.000 9 - 5 3.800 2.000 2.000 9 - 6 3.800 2.000 2.000 9 - 7 3.800 2.000 2.000 9 - 8 3.800 2.000 2.000 9 - 9 4.040 0.000 6.000 5 - 10 4.040 0.000 6.000 5 - 11 4.040 0.000 6.000 5 - 12 4.040 0.000 6.000 5 - 13 4.040 0.000 6.000 5 - 14 4.400 0.000 -0.000 1 - - - - -.. GENERATED FROM PYTHON SOURCE LINES 144-153 - -We see :math:`S=0` and :math:`S=1` states coming from the -two combinations of the spin 1/2 particles. :math:`L` can take values of -0, 1, 2. Remember that spin states have degeneracy of :math:`2S+1` and the -same is true for orbital states. -We must multiply these :math:`S` and -:math:`L` degeneracies to get the total degeneracy. -Since these particles are fermions, the -overall state must be antisymmetric, which dictates the allowed combinations -of :math:`S` and :math:`L`. - -.. GENERATED FROM PYTHON SOURCE LINES 155-158 - -Energy level diagram ------------------------------------------------------------------------------- -Let us show our findings graphically - -.. GENERATED FROM PYTHON SOURCE LINES 158-178 - -.. code-block:: Python - - fig, ax = plt.subplots() - for i, eigenvalue in enumerate(np.unique(e)): - art = ax.plot([0, 1], [eigenvalue, eigenvalue], '-', color='C{}'.format(i)) - ind = np.where(eigenvalue == e)[0][0] - L = (-1 + np.sqrt(1 + 4*L2_val[ind]))/2 - S = (-1 + np.sqrt(1 + 4*S2_val[ind]))/2 - message = "L={:.0f}, S={:.0f} ({:.0f})" - ax.text(1, eigenvalue, message.format(L, S, degeneracy[ind]), - horizontalalignment='right', - verticalalignment='bottom', - color='C{}'.format(i)) - - ax.set_ylabel('Energy') - for loc in ['right', 'top', 'bottom']: - ax.spines[loc].set_visible(False) - - ax.yaxis.set_ticks_position('left') - ax.set_xticks([]) - plt.show() - - - - -.. image-sg:: /auto_examples/images/sphx_glr_example_0_ed_calculator_001.png - :alt: example 0 ed calculator - :srcset: /auto_examples/images/sphx_glr_example_0_ed_calculator_001.png - :class: sphx-glr-single-img - - - - - -.. GENERATED FROM PYTHON SOURCE LINES 179-182 - -We see Hund's rules in action! Rule 1 says that the highest spin :math:`S=1` -state has the lowest energy. Of the two :math:`S=0` states, the state with -larger :math:`L=1` is lower energy following rule 2. - -.. GENERATED FROM PYTHON SOURCE LINES 184-191 - -Spin orbit coupling ------------------------------------------------------------------------------- -For fun, we can see how this changes when we add spin orbit coupling (SOC). -This is a two-fermion operator that we create, transform into the Fock basis -and add to the prior Hamiltonian. To make things easy, let us make the SOC -small so that the LS coupling approximation is valid and we can -still track the states. - -.. GENERATED FROM PYTHON SOURCE LINES 191-195 - -.. code-block:: Python - - soc = edrixs.atom_hsoc('p', 0.1) - n_fermion = 2 - H2 = H + edrixs.build_opers(n_fermion, soc, basis) - - - - - - - - -.. GENERATED FROM PYTHON SOURCE LINES 196-197 - -Then, we redo the diagonalization and print the results. - -.. GENERATED FROM PYTHON SOURCE LINES 197-211 - -.. code-block:: Python - - e2, v2 = scipy.linalg.eigh(H2) - e2 = np.round(e2, decimals=6) - degeneracy2 = [sum(eval == e2) for eval in e2] - print() - message = "With SOC\n {:<3s}\t{:>8s}\t{:>8s}\t{:>8s}\t{:>8s}\t{:>8s}" - print(message.format("#", "E", "S(S+1)", "L(L+1)", "J(J+1)", "degen.")) - J2_val_soc = edrixs.cb_op(J2, v2).diagonal().real - L2_val_soc = edrixs.cb_op(L2, v2).diagonal().real - S2_val_soc = edrixs.cb_op(S2, v2).diagonal().real - for i, eigenvalue in enumerate(e2): - values_list = [i, eigenvalue, S2_val_soc[i], L2_val_soc[i], J2_val_soc[i], - degeneracy2[i]] - print("{:<3d}\t{:8.3f}\t{:8.3f}\t{:8.3f}\t{:8.3f}\t{:8.3f}".format(*values_list)) - - - - - -.. rst-class:: sphx-glr-script-out - - .. code-block:: none - - - With SOC - # E S(S+1) L(L+1) J(J+1) degen. - 0 3.673 1.927 1.927 -0.000 1.000 - 1 3.750 2.000 2.000 2.000 3.000 - 2 3.750 2.000 2.000 2.000 3.000 - 3 3.750 2.000 2.000 2.000 3.000 - 4 3.827 1.802 2.396 6.000 5.000 - 5 3.827 1.802 2.396 6.000 5.000 - 6 3.827 1.802 2.396 6.000 5.000 - 7 3.827 1.802 2.396 6.000 5.000 - 8 3.827 1.802 2.396 6.000 5.000 - 9 4.063 0.198 5.604 6.000 5.000 - 10 4.063 0.198 5.604 6.000 5.000 - 11 4.063 0.198 5.604 6.000 5.000 - 12 4.063 0.198 5.604 6.000 5.000 - 13 4.063 0.198 5.604 6.000 5.000 - 14 4.427 0.073 0.073 -0.000 1.000 - - - - -.. GENERATED FROM PYTHON SOURCE LINES 212-213 - -and we make an equivalent energy level diagram. - -.. GENERATED FROM PYTHON SOURCE LINES 213-233 - -.. code-block:: Python - - - fig, ax = plt.subplots() - for i, eigenvalue in enumerate(np.unique(e2)): - art = ax.plot([0, 1], [eigenvalue, eigenvalue], '-', color='C{}'.format(i)) - ind = np.where(eigenvalue == e2)[0][0] - J = (-1 + np.sqrt(1+4*J2_val_soc[ind]))/2 - message = "J={:.0f} ({:.0f})" - ax.text(1, eigenvalue, message.format(J, degeneracy2[ind]), - horizontalalignment='right', - verticalalignment='bottom', - color='C{}'.format(i)) - - ax.set_ylabel('Energy') - for loc in ['right', 'top', 'bottom']: - ax.spines[loc].set_visible(False) - - ax.yaxis.set_ticks_position('left') - ax.set_xticks([]) - plt.show() - - - - -.. image-sg:: /auto_examples/images/sphx_glr_example_0_ed_calculator_002.png - :alt: example 0 ed calculator - :srcset: /auto_examples/images/sphx_glr_example_0_ed_calculator_002.png - :class: sphx-glr-single-img - - - - - -.. GENERATED FROM PYTHON SOURCE LINES 234-238 - -It is clear that we have split the :math:`S=1` state, which branches into -three states from :math:`J=|L-S|, |L-S|+1, ..., |L+S|`. Since the shell is -less than half full, Hund's third rule dictates that the smaller :math:`J` -states have the lower energies. - - -.. rst-class:: sphx-glr-timing - - **Total running time of the script:** (0 minutes 0.138 seconds) - - -.. _sphx_glr_download_auto_examples_example_0_ed_calculator.py: - -.. only:: html - - .. container:: sphx-glr-footer sphx-glr-footer-example - - .. container:: sphx-glr-download sphx-glr-download-jupyter - - :download:`Download Jupyter notebook: example_0_ed_calculator.ipynb ` - - .. container:: sphx-glr-download sphx-glr-download-python - - :download:`Download Python source code: example_0_ed_calculator.py ` - - .. container:: sphx-glr-download sphx-glr-download-zip - - :download:`Download zipped: example_0_ed_calculator.zip ` - - -.. only:: html - - .. rst-class:: sphx-glr-signature - - `Gallery generated by Sphinx-Gallery `_ diff --git a/edrixs/_sources/auto_examples/example_1_crystal_field.rst.txt b/edrixs/_sources/auto_examples/example_1_crystal_field.rst.txt deleted file mode 100644 index a907cfff9a..0000000000 --- a/edrixs/_sources/auto_examples/example_1_crystal_field.rst.txt +++ /dev/null @@ -1,608 +0,0 @@ - -.. DO NOT EDIT. -.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. -.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: -.. "auto_examples/example_1_crystal_field.py" -.. LINE NUMBERS ARE GIVEN BELOW. - -.. only:: html - - .. note:: - :class: sphx-glr-download-link-note - - :ref:`Go to the end ` - to download the full example code. - -.. rst-class:: sphx-glr-example-title - -.. _sphx_glr_auto_examples_example_1_crystal_field.py: - - -Crystal fields -===================================== -This example explains how to implement crystal fields in edrixs. - -.. GENERATED FROM PYTHON SOURCE LINES 8-9 - -We need to import these modules. - -.. GENERATED FROM PYTHON SOURCE LINES 9-15 - -.. code-block:: Python - - import edrixs - import numpy as np - import scipy - - np.set_printoptions(precision=2, suppress=True, linewidth=90) - - - - - - - - -.. GENERATED FROM PYTHON SOURCE LINES 16-22 - -Crystal field matrices ------------------------------------------------------------------------------- -Let us start by considering the common case of a :math:`d` atomic shell in a -cubic crystal field. This is controlled by parameter :math:`10D_q` and is -described in terms of a matrix which we will assign to :code:`cfmat`. edrixs -can make this matrix via. - -.. GENERATED FROM PYTHON SOURCE LINES 22-25 - -.. code-block:: Python - - ten_dq = 10 - cfmat = edrixs.angular_momentum.cf_cubic_d(ten_dq) - - - - - - - - -.. GENERATED FROM PYTHON SOURCE LINES 26-33 - -Note this matrix is in a complex harmonic basis :math:`Y^m_l` where :math:`m` -goes from :math:`-l,-l+1,...,l-1, l`. There is an up spin and a down -spin for each :math:`Y^m_l`. This matrix is not diagonal in the complex -harmonic basis, but it would be diagonal in the real harmonic basis -:math:`d_{3z^2-r^2}, d_{xz}, d_{yz}, d_{x^2-y^2}, d_{xy}`. -Let us diagonalize this matrix as a check and print out the energies -and their degeneracies. - -.. GENERATED FROM PYTHON SOURCE LINES 33-44 - -.. code-block:: Python - - - e, v = scipy.linalg.eigh(cfmat) - e = e.round(decimals=6) - unique_e = np.unique(e) - degeneracies = [sum(evalue == e) for evalue in unique_e] - - print("E \tDegeneracy") - for evalue, degenvalue in zip(unique_e, degeneracies): - print("{:.1f}\t{:.0f}".format(evalue, degenvalue)) - print("{} distinct energies".format(len(unique_e))) - - - - - -.. rst-class:: sphx-glr-script-out - - .. code-block:: none - - E Degeneracy - -4.0 6 - 6.0 4 - 2 distinct energies - - - - -.. GENERATED FROM PYTHON SOURCE LINES 45-47 - -This makes sense! We see two different energies split by :math:`10D_q=10`. Let -us look at the six columns corresponding to the lower energy eigenvalues. - -.. GENERATED FROM PYTHON SOURCE LINES 47-50 - -.. code-block:: Python - - - print(v[:, :6].real) - - - - - -.. rst-class:: sphx-glr-script-out - - .. code-block:: none - - [[ 0. 0. 0. 0. 0.71 0. ] - [ 0. 0. 0. 0. 0. 0.71] - [ 0. 0. -1. 0. 0. 0. ] - [ 0. 0. 0. -1. 0. 0. ] - [ 0. 0. 0. 0. 0. 0. ] - [ 0. 0. 0. 0. 0. 0. ] - [ 1. 0. 0. 0. 0. 0. ] - [ 0. 1. 0. 0. 0. 0. ] - [ 0. 0. 0. 0. -0.71 0. ] - [ 0. 0. 0. 0. 0. -0.71]] - - - - -.. GENERATED FROM PYTHON SOURCE LINES 51-54 - -These are the set of so-called :math:`t_{2g}` orbitals, composed of -:math:`Y^2_2, Y^{-2}_2, Y^{1}_2, Y^{-1}_2`. The rest of the eigenvectors -(the last four) are - -.. GENERATED FROM PYTHON SOURCE LINES 54-56 - -.. code-block:: Python - - print(v[:, 6:].real) - - - - - -.. rst-class:: sphx-glr-script-out - - .. code-block:: none - - [[-0.71 0. 0. 0. ] - [ 0. -0.71 0. 0. ] - [ 0. 0. 0. 0. ] - [ 0. 0. 0. 0. ] - [ 0. 0. 0. 1. ] - [ 0. 0. 1. 0. ] - [ 0. 0. 0. 0. ] - [ 0. 0. 0. 0. ] - [-0.71 0. 0. 0. ] - [ 0. -0.71 0. 0. ]] - - - - -.. GENERATED FROM PYTHON SOURCE LINES 57-69 - -These are the set of so-called :math:`e_{g}` orbitals, composed of -:math:`Y^2_2, Y^{-2}_2, Y^{0}_2`. We can use edrixs to prove that -:code:`cfmat` would be diagonal in the real -harmonic basis. An operator :math:`\hat{O}` can be transformed into an -operator in another basis :math:`\hat{O}^{\prime}` using a unitary -transformation matrix :math:`T` as - - .. math:: - - \hat{O}^{\prime} = (T)^{\dagger} \hat{O} (T). - -This is computed as follows - -.. GENERATED FROM PYTHON SOURCE LINES 69-72 - -.. code-block:: Python - - cfmat_rhb = edrixs.cb_op(cfmat, edrixs.tmat_c2r('d', ispin=True)) - print(cfmat_rhb.real) - - - - - -.. rst-class:: sphx-glr-script-out - - .. code-block:: none - - [[ 6. 0. 0. 0. 0. 0. 0. 0. 0. 0.] - [ 0. 6. 0. 0. 0. 0. 0. 0. 0. 0.] - [ 0. 0. -4. 0. 0. 0. 0. 0. 0. 0.] - [ 0. 0. 0. -4. 0. 0. 0. 0. 0. 0.] - [ 0. 0. 0. 0. -4. 0. 0. 0. 0. 0.] - [ 0. 0. 0. 0. 0. -4. 0. 0. 0. 0.] - [ 0. 0. 0. 0. 0. 0. 6. 0. 0. 0.] - [ 0. 0. 0. 0. 0. 0. 0. 6. 0. 0.] - [ 0. 0. 0. 0. 0. 0. 0. 0. -4. 0.] - [ 0. 0. 0. 0. 0. 0. 0. 0. 0. -4.]] - - - - -.. GENERATED FROM PYTHON SOURCE LINES 73-78 - -where :code:`edrixs.tmat_c2r('d', ispin=True)` is the transformation matrix. -We needed to tell edrixs that we are working with a :math:`d`-shell and that it -should include spin. We could also have transformed :code:`v` to see how these -eignevectors are composed of the real harmonic basis. We will see an example -of this later. - -.. GENERATED FROM PYTHON SOURCE LINES 80-84 - -Crystal field on an atom ------------------------------------------------------------------------------- -To simulate the solid state, we need to combine the crystal field with Coulomb -interactions. Let us choose an atomic model for Ni. - -.. GENERATED FROM PYTHON SOURCE LINES 84-90 - -.. code-block:: Python - - l = 2 - norb = 10 - noccu = 8 - basis = edrixs.get_fock_bin_by_N(norb, noccu) - slater = edrixs.get_atom_data('Ni', '3d', noccu, edge='L3')['slater_i'] - - - - - - - - -.. GENERATED FROM PYTHON SOURCE LINES 91-94 - -Let us implement a tetragonal crystal field, for which we need to pass -:code:`d1` the splitting of :math:`d_{yz}/d_{zx}` and :math:`d_{xy}` and -:code:`d3` the splitting of :math:`d_{3z^2-r^2}` and :math:`d_{x^2-y^2}`. - -.. GENERATED FROM PYTHON SOURCE LINES 94-96 - -.. code-block:: Python - - ten_dq, d1, d3 = 2.5, 0.9, .2 - - - - - - - - -.. GENERATED FROM PYTHON SOURCE LINES 97-101 - -To determine the eigenvalues and eigenvectors we need to transform both our -Coulomb matrix and our crystal field matrix into the same basis. See the -example on exact diagonalization if needed. In this case, we put this -procedure into a function, with the option to scale the Coulomb interactions. - -.. GENERATED FROM PYTHON SOURCE LINES 101-115 - -.. code-block:: Python - - - - def diagonlize(scaleU=1): - umat = edrixs.get_umat_slater('d', - slater[0][1]*scaleU, - slater[1][1]*scaleU, - slater[2][1]*scaleU) - cfmat = edrixs.angular_momentum.cf_tetragonal_d(ten_dq=ten_dq, d1=d1, d3=d3) - H = edrixs.build_opers(2, cfmat, basis) + edrixs.build_opers(4, umat, basis) - e, v = scipy.linalg.eigh(H) - e = e - np.min(e) # define ground state as zero energy - return e, v - - - - - - - - - -.. GENERATED FROM PYTHON SOURCE LINES 116-121 - -Let us look what happens when we run the function with the Coulomb -interactions switched off and check the degeneracy of the output. Look at this -python -`string formatting tutorial `_ -if the code is confusing. - -.. GENERATED FROM PYTHON SOURCE LINES 121-131 - -.. code-block:: Python - - e, v = diagonlize(scaleU=0) - e = e.round(decimals=6) - unique_e = np.unique(e) - degeneracies = [sum(evalue == e) for evalue in unique_e] - - print("E \tDegeneracy") - for evalue, degenvalue in zip(unique_e, degeneracies): - print("{:.1f}\t{:.0f}".format(evalue, degenvalue)) - print("{} distinct energies".format(len(unique_e))) - - - - - -.. rst-class:: sphx-glr-script-out - - .. code-block:: none - - E Degeneracy - 0.0 1 - 0.2 4 - 0.4 1 - 2.5 4 - 2.7 4 - 3.4 8 - 3.6 8 - 5.0 1 - 5.9 8 - 6.8 6 - 10 distinct energies - - - - -.. GENERATED FROM PYTHON SOURCE LINES 132-146 - -We see 10 distinct energies, which is the number of ways one can arrange -two holes among 4 energy levels -- which makes sense as the tetragonal field -involves four levels :math:`zx/zy, xy, 3z^2-r^2, x^2-y^2`. To see what is going -on in more detail, we can also calculate the expectation -values of the occupancy number of the orbitals -:math:`3z^2-r^2, zx, zy, x^2-y^2, xy`. -To create the operator, first write the matrix in the real harmonics basis -:math:`|3z^2-r^2,\uparrow>`, :math:`|3z^2-r^2,\downarrow>`, -:math:`|zx,\uparrow>`, :math:`|zx,\downarrow>`, etc. -In this basis, they take a simple form: only the diagonal terms have element -1. We therefore make a 3D empty array and assign the diagonal as 1. Check -out the -`numpy indexing notes `_ -if needed. - -.. GENERATED FROM PYTHON SOURCE LINES 146-150 - -.. code-block:: Python - - nd_real_harmoic_basis = np.zeros((norb, norb, norb), dtype=complex) - indx = np.arange(norb) - nd_real_harmoic_basis[indx, indx, indx] = 1 - - - - - - - - -.. GENERATED FROM PYTHON SOURCE LINES 151-153 - -Recalling the necessity to put everything in the same basis, we transform -into the complex harmonic basis and then transform into our Fock basis - -.. GENERATED FROM PYTHON SOURCE LINES 153-158 - -.. code-block:: Python - - nd_complex_harmoic_basis = edrixs.cb_op(nd_real_harmoic_basis, - edrixs.tmat_r2c('d', True)) - nd_op = edrixs.build_opers(2, nd_complex_harmoic_basis, basis) - - - - - - - - - -.. GENERATED FROM PYTHON SOURCE LINES 159-163 - -We apply the operator and print out as follows. Check the -`numpy docs `_ -if the details of how the spin pairs have been added up is not immediately -transparent. - -.. GENERATED FROM PYTHON SOURCE LINES 163-173 - -.. code-block:: Python - - nd_expt = np.array([edrixs.cb_op(nd_vec, v).diagonal().real for nd_vec in nd_op]) - - message = "{:>3s}" + "\t{:>6s}"*5 - print(message.format(*"E 3z2-r2 zx zy x2-y2 xy".split(" "))) - - message = "{:>3.1f}" + "\t{:>6.1f}"*5 - for evalue, row in zip(e, nd_expt.T): - spin_pairs = row.reshape(-1, 2).sum(1) - print(message.format(evalue, *spin_pairs)) - - - - - -.. rst-class:: sphx-glr-script-out - - .. code-block:: none - - E 3z2-r2 zx zy x2-y2 xy - 0.0 2.0 2.0 2.0 0.0 2.0 - 0.2 1.0 2.0 2.0 1.0 2.0 - 0.2 1.0 2.0 2.0 1.0 2.0 - 0.2 1.0 2.0 2.0 1.0 2.0 - 0.2 1.0 2.0 2.0 1.0 2.0 - 0.4 0.0 2.0 2.0 2.0 2.0 - 2.5 2.0 2.0 2.0 1.0 1.0 - 2.5 2.0 2.0 2.0 1.0 1.0 - 2.5 2.0 2.0 2.0 1.0 1.0 - 2.5 2.0 2.0 2.0 1.0 1.0 - 2.7 1.0 2.0 2.0 2.0 1.0 - 2.7 1.0 2.0 2.0 2.0 1.0 - 2.7 1.0 2.0 2.0 2.0 1.0 - 2.7 1.0 2.0 2.0 2.0 1.0 - 3.4 2.0 1.5 1.5 1.0 2.0 - 3.4 2.0 1.5 1.5 1.0 2.0 - 3.4 2.0 1.5 1.5 1.0 2.0 - 3.4 2.0 1.5 1.5 1.0 2.0 - 3.4 2.0 1.5 1.5 1.0 2.0 - 3.4 2.0 1.5 1.5 1.0 2.0 - 3.4 2.0 1.5 1.5 1.0 2.0 - 3.4 2.0 1.5 1.5 1.0 2.0 - 3.6 1.0 1.6 1.4 2.0 2.0 - 3.6 1.0 1.5 1.5 2.0 2.0 - 3.6 1.0 1.5 1.5 2.0 2.0 - 3.6 1.0 1.6 1.4 2.0 2.0 - 3.6 1.0 1.3 1.7 2.0 2.0 - 3.6 1.0 1.5 1.5 2.0 2.0 - 3.6 1.0 1.4 1.6 2.0 2.0 - 3.6 1.0 1.6 1.4 2.0 2.0 - 5.0 2.0 2.0 2.0 2.0 0.0 - 5.9 2.0 1.5 1.5 2.0 1.0 - 5.9 2.0 1.5 1.5 2.0 1.0 - 5.9 2.0 1.5 1.5 2.0 1.0 - 5.9 2.0 1.5 1.5 2.0 1.0 - 5.9 2.0 1.5 1.5 2.0 1.0 - 5.9 2.0 1.5 1.5 2.0 1.0 - 5.9 2.0 1.5 1.5 2.0 1.0 - 5.9 2.0 1.5 1.5 2.0 1.0 - 6.8 2.0 0.4 1.6 2.0 2.0 - 6.8 2.0 1.3 0.7 2.0 2.0 - 6.8 2.0 1.0 1.0 2.0 2.0 - 6.8 2.0 1.0 1.0 2.0 2.0 - 6.8 2.0 1.0 1.0 2.0 2.0 - 6.8 2.0 1.2 0.8 2.0 2.0 - - - - -.. GENERATED FROM PYTHON SOURCE LINES 174-178 - -The lowest energy state involves putting both holes in the :math:`x^2-y^2` -orbital, which makes sense. Now, let us redo the proceedure including Coulomb -repulsion, which imposes an energy cost to putting multiple electrons in the -same orbital. - -.. GENERATED FROM PYTHON SOURCE LINES 178-191 - -.. code-block:: Python - - - e, v = diagonlize(scaleU=1) - - nd_expt = np.array([edrixs.cb_op(nd_vec, v).diagonal().real for nd_vec in nd_op]) - - message = "{:>3s}" + "\t{:>6s}"*5 - print(message.format(*"E 3z2-r2 zx zy x2-y2 xy".split(" "))) - - message = "{:>3.1f}" + "\t{:>6.1f}"*5 - for evalue, row in zip(e, nd_expt.T): - spin_pairs = row.reshape(-1, 2).sum(1) - print(message.format(evalue, *spin_pairs)) - - - - - -.. rst-class:: sphx-glr-script-out - - .. code-block:: none - - E 3z2-r2 zx zy x2-y2 xy - 0.0 1.0 2.0 2.0 1.0 2.0 - 0.0 1.0 2.0 2.0 1.0 2.0 - 0.0 1.0 2.0 2.0 1.0 2.0 - 2.4 1.2 2.0 2.0 0.8 2.0 - 2.5 1.0 2.0 2.0 1.0 2.0 - 2.5 1.0 2.0 2.0 2.0 1.0 - 2.5 1.0 2.0 2.0 2.0 1.0 - 2.5 1.0 2.0 2.0 2.0 1.0 - 3.2 1.8 1.2 1.8 1.2 2.0 - 3.2 1.8 1.7 1.3 1.2 2.0 - 3.2 1.8 1.5 1.5 1.2 2.0 - 3.2 1.8 1.7 1.3 1.2 2.0 - 3.2 1.8 1.4 1.6 1.2 2.0 - 3.2 1.8 1.5 1.5 1.2 2.0 - 4.0 2.0 1.9 1.9 1.1 1.1 - 4.0 2.0 1.9 1.9 1.1 1.1 - 4.0 2.0 1.9 1.9 1.1 1.1 - 4.3 0.9 2.0 2.0 1.2 1.9 - 4.7 1.4 1.4 1.6 1.9 1.7 - 4.7 1.4 1.5 1.5 1.9 1.7 - 4.7 1.4 1.6 1.4 1.9 1.7 - 4.7 1.4 1.4 1.6 1.9 1.7 - 4.7 1.4 1.4 1.6 1.9 1.7 - 4.7 1.4 1.6 1.4 1.9 1.7 - 4.9 1.0 2.0 2.0 2.0 1.0 - 5.5 2.0 2.0 2.0 1.0 1.0 - 5.6 1.8 1.6 1.4 1.2 2.0 - 5.6 1.8 1.4 1.6 1.2 2.0 - 6.5 1.2 1.6 1.4 1.8 2.0 - 6.5 1.2 1.4 1.6 1.8 2.0 - 6.8 1.8 1.4 1.6 1.9 1.3 - 6.8 1.8 1.5 1.5 1.9 1.3 - 6.8 1.8 1.5 1.5 1.9 1.3 - 6.8 1.8 1.6 1.4 1.9 1.3 - 6.8 1.8 1.4 1.6 1.9 1.3 - 6.8 1.8 1.6 1.4 1.9 1.3 - 7.4 2.0 1.1 1.1 1.9 1.9 - 7.4 2.0 1.1 1.1 1.9 1.9 - 7.4 2.0 1.1 1.1 1.9 1.9 - 8.0 2.0 1.8 1.8 2.0 0.4 - 8.5 2.0 1.5 1.5 2.0 1.0 - 8.5 2.0 1.5 1.5 2.0 1.0 - 9.3 2.0 1.0 1.0 2.0 2.0 - 9.4 2.0 1.0 1.0 2.0 2.0 - 12.8 1.9 1.2 1.2 1.9 1.7 - - - - -.. GENERATED FROM PYTHON SOURCE LINES 192-197 - -Now the lowest energy state involves splitting the holes between the two -highest energy :math:`x^2-y^2` and :math:`3z^2-r^2` orbitals. i.e. we have -gone from low-spin to high spin. Working out the balance between these two -states is often one of the first things one wants to determine upon the -discovery of an interesting new material [1]_. - -.. GENERATED FROM PYTHON SOURCE LINES 199-202 - -.. rubric:: Footnotes - -.. [1] M. Rossi et al., `arXiv:2011.00595 (2021) `_. - - -.. rst-class:: sphx-glr-timing - - **Total running time of the script:** (0 minutes 0.149 seconds) - - -.. _sphx_glr_download_auto_examples_example_1_crystal_field.py: - -.. only:: html - - .. container:: sphx-glr-footer sphx-glr-footer-example - - .. container:: sphx-glr-download sphx-glr-download-jupyter - - :download:`Download Jupyter notebook: example_1_crystal_field.ipynb ` - - .. container:: sphx-glr-download sphx-glr-download-python - - :download:`Download Python source code: example_1_crystal_field.py ` - - .. container:: sphx-glr-download sphx-glr-download-zip - - :download:`Download zipped: example_1_crystal_field.zip ` - - -.. only:: html - - .. rst-class:: sphx-glr-signature - - `Gallery generated by Sphinx-Gallery `_ diff --git a/edrixs/_sources/auto_examples/example_2_single_atom_RIXS.rst.txt b/edrixs/_sources/auto_examples/example_2_single_atom_RIXS.rst.txt deleted file mode 100644 index 7bb847adff..0000000000 --- a/edrixs/_sources/auto_examples/example_2_single_atom_RIXS.rst.txt +++ /dev/null @@ -1,483 +0,0 @@ - -.. DO NOT EDIT. -.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. -.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: -.. "auto_examples/example_2_single_atom_RIXS.py" -.. LINE NUMBERS ARE GIVEN BELOW. - -.. only:: html - - .. note:: - :class: sphx-glr-download-link-note - - :ref:`Go to the end ` - to download the full example code. - -.. rst-class:: sphx-glr-example-title - -.. _sphx_glr_auto_examples_example_2_single_atom_RIXS.py: - - -RIXS calculations for an atomic model -===================================== -Here we show how to compute RIXS for a single site atomic model with crystal -field and electron-electron interactions. We take the case of -Sr\ :sub:`2`\ YIrO\ :sub:`6` -from Ref. [1]_ as the material in question. The aim of this example is to -illustrate the proceedure and to provide what we hope is useful advice. What is -written is not meant to be a replacement for reading the docstrings of the -functions, which can always be accessed on the -`edrixs website `_ or -by executing functions with ?? in IPython. - -.. GENERATED FROM PYTHON SOURCE LINES 15-19 - -.. code-block:: Python - - import edrixs - import numpy as np - import matplotlib.pyplot as plt - - - - - - - - -.. GENERATED FROM PYTHON SOURCE LINES 20-26 - -Specify active core and valence orbitals ------------------------------------------------------------------------------- -Sr\ :sub:`2`\ YIrO\ :sub:`6`\ has a :math:`5d^4` electronic configuration and -we want to calculate the :math:`L_3` edge spectrum i.e. resonating with a -:math:`2p_{3/2}` core hole. We will start by including only the -:math:`t_{2g}` valance orbitals. - -.. GENERATED FROM PYTHON SOURCE LINES 26-29 - -.. code-block:: Python - - shell_name = ('t2g', 'p32') - v_noccu = 4 - - - - - - - - -.. GENERATED FROM PYTHON SOURCE LINES 30-38 - -Slater parameters ------------------------------------------------------------------------------- -Here we want to use Hund's interaction -:math:`J_H` and spin orbit coupling :math:`\lambda` as adjustable parameters -to match experiment. We will take -the core hole interaction parameter from the Hartree Fock numbers EDRIXS has -in its database. These need to be converted and arranged into the order -required by EDRIXS. - -.. GENERATED FROM PYTHON SOURCE LINES 38-58 - -.. code-block:: Python - - Ud = 2 - JH = 0.25 - lam = 0.42 - F0_d, F2_d, F4_d = edrixs.UdJH_to_F0F2F4(Ud, JH) - info = edrixs.utils.get_atom_data('Ir', '5d', v_noccu, edge='L3') - G1_dp = info['slater_n'][5][1] - G3_dp = info['slater_n'][6][1] - F0_dp = edrixs.get_F0('dp', G1_dp, G3_dp) - F2_dp = info['slater_n'][4][1] - - slater_i = [F0_d, F2_d, F4_d] # Fk for d - slater_n = [ - F0_d, F2_d, F4_d, # Fk for d - F0_dp, F2_dp, # Fk for dp - G1_dp, G3_dp, # Gk for dp - 0.0, 0.0 # Fk for p - ] - slater = [slater_i, slater_n] - v_soc = (lam, lam) - - - - - - - - -.. GENERATED FROM PYTHON SOURCE LINES 59-70 - -Diagonalization ------------------------------------------------------------------------------- -We obtain the ground and intermediate state eigenenergies and the transition -operators via matrix diagonalization. Note that the calculation does not know -the core hole energy, so we need to adjust the energy that the resonance will -appear at by hand. We know empirically that the resonance is at 11215 eV -and that putting four electrons into the valance band costs about -:math:`4 F^0_d\approx6` eV. In this case -we are assuming a perfectly cubic crystal field, which we have already -implemented when we specified the use of the :math:`t_{2g}` subshell only -so we do not need to pass an additional :code:`v_cfmat` matrix. - -.. GENERATED FROM PYTHON SOURCE LINES 70-76 - -.. code-block:: Python - - - off = 11215 - 6 - out = edrixs.ed_1v1c_py(shell_name, shell_level=(0, -off), v_soc=v_soc, - c_soc=info['c_soc'], v_noccu=v_noccu, slater=slater) - eval_i, eval_n, trans_op = out - - - - - -.. rst-class:: sphx-glr-script-out - - .. code-block:: none - - edrixs >>> Running ED ... - - Summary of Slater integrals: - ------------------------------ - Terms, Initial Hamiltonian, Intermediate Hamiltonian - F0_vv : 2.0000000000 2.0000000000 - F2_vv : 2.1538461538 2.1538461538 - F4_vv : 1.3461538462 1.3461538462 - F0_vc : 0.0000000000 0.0881857143 - F2_vc : 0.0000000000 1.0700000000 - G1_vc : 0.0000000000 0.9570000000 - G3_vc : 0.0000000000 0.5690000000 - F0_cc : 0.0000000000 0.0000000000 - F2_cc : 0.0000000000 0.0000000000 - - edrixs >>> Dimension of the initial Hamiltonian: 15 - edrixs >>> Dimension of the intermediate Hamiltonian: 24 - edrixs >>> Building Many-body Hamiltonians ... - edrixs >>> Done ! - edrixs >>> Exact Diagonalization of Hamiltonians ... - edrixs >>> Done ! - edrixs >>> ED Done ! - - - - -.. GENERATED FROM PYTHON SOURCE LINES 77-105 - -Compute XAS ------------------------------------------------------------------------------- -To calculate XAS we need to correctly specify the orientation of the x-rays -with respect to the sample. By default, the :math:`x, y, z` coordinates -of the sample's crystal field, will be aligned with our lab frame, passing -:code:`loc_axis` to :code:`ed_1v1c_py` can be used to specify a different -convention. The experimental geometry is specified following the angles -shown in Figure 1 of Y. Wang et al., -`Computer Physics Communications 243, 151-165 (2019) -`_. The default -setting has x-rays along :math:`z` for :math:`\theta=\pi/2` rad -and the x-ray beam along :math:`-x` for -:math:`\theta=\phi=0`. Parameter :code:`scatter_axis` can be passed to -:code:`xas_1v1c_py` to specify a different geometry if desired. - -Variable :code:`pol_type` specifies a list of different x-ray -polarizations to calculate. Here we will use so-called :math:`\pi`-polarization -where the x-rays are parallel to the plane spanned by the incident -beam and the sample :math:`z`-axis. - -EDRIXS represents the system's ground state using a set of -low energy eigenstates weighted by Boltzmann thermal factors. -These eigenstates are specified by :code:`gs_list`, -which is of the form :math:`[0, 1, 2, 3, \dots]`. In this example, we -calculate these states as those that have non-negligible thermal -population. The function :code:`xas_1v1c_py` assumes that the spectral -broadening is dominated by the inverse core hole lifetime :code:`gamma_c`, -which is the Lorentzian half width at half maximum. - -.. GENERATED FROM PYTHON SOURCE LINES 105-121 - -.. code-block:: Python - - - ominc = np.linspace(11200, 11230, 50) - temperature = 300 # in K - prob = edrixs.boltz_dist(eval_i, temperature) - gs_list = [n for n, prob in enumerate(prob) if prob > 1e-6] - - thin = 30*np.pi/180 - phi = 0 - pol_type = [('linear', 0)] - - xas = edrixs.xas_1v1c_py( - eval_i, eval_n, trans_op, ominc, gamma_c=info['gamma_c'], - thin=thin, phi=phi, pol_type=pol_type, - gs_list=gs_list) - - - - - - -.. rst-class:: sphx-glr-script-out - - .. code-block:: none - - edrixs >>> Running XAS ... - edrixs >>> XAS Done ! - - - - -.. GENERATED FROM PYTHON SOURCE LINES 122-136 - -Compute RIXS ------------------------------------------------------------------------------- -Calculating RIXS is overall similar to XAS, but with a few additional -considerations. The spectral width in the energy loss axis of RIXS it -not set by the core hole lifetime, but by either the final state lifetime -or the experimental resolution and is parameterized by :code:`gamma_f` --- the Lorentzian half width at half maximum. - -The angle and polarization of the emitted beam must also be specified, so -we pass :code:`pol_type_rixs` to the function, which specifies the -includes the incoming and outgoing x-ray states. If, as is common in -experiments, the emitted polarization is not resolved -one needs to add both emitted polarization channels, which is what we will -do later on in this example. - -.. GENERATED FROM PYTHON SOURCE LINES 136-151 - -.. code-block:: Python - - - eloss = np.linspace(-.5, 6, 400) - pol_type_rixs = [('linear', 0, 'linear', 0), ('linear', 0, 'linear', np.pi/2)] - - thout = 60*np.pi/180 - gamma_f = 0.02 - - rixs = edrixs.rixs_1v1c_py( - eval_i, eval_n, trans_op, ominc, eloss, - gamma_c=info['gamma_c'], gamma_f=gamma_f, - thin=thin, thout=thout, phi=phi, - pol_type=pol_type_rixs, gs_list=gs_list, - temperature=temperature - ) - - - - - -.. rst-class:: sphx-glr-script-out - - .. code-block:: none - - edrixs >>> Running RIXS ... - edrixs >>> RIXS Done ! - - - - -.. GENERATED FROM PYTHON SOURCE LINES 152-154 - -The array :code:`xas` will have shape -:code:`(len(ominc_xas), len(pol_type))` - -.. GENERATED FROM PYTHON SOURCE LINES 156-162 - -Plot XAS and RIXS ------------------------------------------------------------------------------- -Let's plot everything. We will use a function so we can reuse the code later. -Note that the rixs array :code:`rixs` has shape -:code:`(len(ominc_xas), len(ominc_xas), len(pol_type))`. We will use some numpy -tricks to sum over the two different emitted polarizations. - -.. GENERATED FROM PYTHON SOURCE LINES 162-196 - -.. code-block:: Python - - - fig, axs = plt.subplots(2, 2, figsize=(10, 10)) - - - def plot_it(axs, ominc, xas, eloss, rixscut, rixsmap=None, label=None): - axs[0].plot(ominc, xas[:, 0], label=label) - axs[0].set_xlabel('Energy (eV)') - axs[0].set_ylabel('Intensity') - axs[0].set_title('XAS') - - axs[1].plot(eloss, rixscut, label=f"{label}") - axs[1].set_xlabel('Energy loss (eV)') - axs[1].set_ylabel('Intensity') - axs[1].set_title(f'RIXS at resonance') - - if rixsmap is not None: - art = axs[2].pcolormesh(ominc, eloss, rixsmap.T, shading='auto') - plt.colorbar(art, ax=axs[2], label='Intensity') - axs[2].set_xlabel('Incident energy (eV)') - axs[2].set_ylabel('Energy loss') - axs[2].set_title('RIXS map') - - - rixs_pol_sum = rixs.sum(-1) - cut_index = np.argmax(rixs_pol_sum[:, eloss < 2].sum(1)) - rixscut = rixs_pol_sum[cut_index] - - plot_it(axs.ravel(), ominc, xas, eloss, rixscut, rixsmap=rixs_pol_sum) - axs[0, 1].set_xlim(right=3) - axs[1, 0].set_ylim(top=3) - axs[1, 1].remove() - - plt.show() - - - - -.. image-sg:: /auto_examples/images/sphx_glr_example_2_single_atom_RIXS_001.png - :alt: XAS, RIXS at resonance, RIXS map - :srcset: /auto_examples/images/sphx_glr_example_2_single_atom_RIXS_001.png - :class: sphx-glr-single-img - - - - - -.. GENERATED FROM PYTHON SOURCE LINES 197-204 - -Full d shell calculation ------------------------------------------------------------------------------- -Some researchers have questioned the appropriateness of only including the -:math:`t_{2g}` subshell for iridates [2]_. Let's test this. We specify that -the full :math:`d` shell should be used and apply cubic crystal field matrix -:code:`v_cfmat`. We shift the energy offset by :math:`\frac{2}{5}10D_q`, which -is the amount the crystal field moves the :math:`t_{2g}` subshell. - -.. GENERATED FROM PYTHON SOURCE LINES 204-234 - -.. code-block:: Python - - - ten_dq = 3.5 - v_cfmat = edrixs.cf_cubic_d(ten_dq) - off = 11215 - 6 + ten_dq*2/5 - out = edrixs.ed_1v1c_py(('d', 'p32'), shell_level=(0, -off), v_soc=v_soc, - v_cfmat=v_cfmat, - c_soc=info['c_soc'], v_noccu=v_noccu, slater=slater) - eval_i, eval_n, trans_op = out - - xas_full_d_shell = edrixs.xas_1v1c_py( - eval_i, eval_n, trans_op, ominc, gamma_c=info['gamma_c'], - thin=thin, phi=phi, pol_type=pol_type, - gs_list=gs_list) - - rixs_full_d_shell = edrixs.rixs_1v1c_py( - eval_i, eval_n, trans_op, np.array([11215]), eloss, - gamma_c=info['gamma_c'], gamma_f=gamma_f, - thin=thin, thout=thout, phi=phi, - pol_type=pol_type_rixs, gs_list=gs_list, - temperature=temperature) - - fig, axs = plt.subplots(1, 2, figsize=(10, 4)) - plot_it(axs, ominc, xas, eloss, rixscut, label='$t_{2g}$ subshell') - rixscut = rixs_full_d_shell.sum((0, -1)) - plot_it(axs, ominc, xas_full_d_shell, eloss, rixscut, label='$d$ shell') - - axs[0].legend() - axs[1].legend() - plt.show() - - - - -.. image-sg:: /auto_examples/images/sphx_glr_example_2_single_atom_RIXS_002.png - :alt: XAS, RIXS at resonance - :srcset: /auto_examples/images/sphx_glr_example_2_single_atom_RIXS_002.png - :class: sphx-glr-single-img - - -.. rst-class:: sphx-glr-script-out - - .. code-block:: none - - edrixs >>> Running ED ... - - Summary of Slater integrals: - ------------------------------ - Terms, Initial Hamiltonian, Intermediate Hamiltonian - F0_vv : 2.0000000000 2.0000000000 - F2_vv : 2.1538461538 2.1538461538 - F4_vv : 1.3461538462 1.3461538462 - F0_vc : 0.0000000000 0.0881857143 - F2_vc : 0.0000000000 1.0700000000 - G1_vc : 0.0000000000 0.9570000000 - G3_vc : 0.0000000000 0.5690000000 - F0_cc : 0.0000000000 0.0000000000 - F2_cc : 0.0000000000 0.0000000000 - - edrixs >>> Dimension of the initial Hamiltonian: 210 - edrixs >>> Dimension of the intermediate Hamiltonian: 1008 - edrixs >>> Building Many-body Hamiltonians ... - edrixs >>> Done ! - edrixs >>> Exact Diagonalization of Hamiltonians ... - edrixs >>> Done ! - edrixs >>> ED Done ! - edrixs >>> Running XAS ... - edrixs >>> XAS Done ! - edrixs >>> Running RIXS ... - edrixs >>> RIXS Done ! - - - - -.. GENERATED FROM PYTHON SOURCE LINES 235-239 - -As expected, we see the appearance of excitations on the energy scale of -:math:`10D_q` in the XAS and RIXS. The low energy manifold is qualitatively, -but not quantiatively similar. This makes it clear that the parameterization -of Sr\ :sub:`2`\ YIrO\ :sub:`6`\ is dependent on the model. - -.. GENERATED FROM PYTHON SOURCE LINES 241-248 - -.. rubric:: Footnotes - -.. [1] Bo Yuan et al., - `Phys. Rev. B 95, 235114 (2017) `_. - -.. [2] Georgios L. Stamokostas and Gregory A. Fiete - `Phys. Rev. B 97, 085150 (2018) `_. - - -.. rst-class:: sphx-glr-timing - - **Total running time of the script:** (0 minutes 3.985 seconds) - - -.. _sphx_glr_download_auto_examples_example_2_single_atom_RIXS.py: - -.. only:: html - - .. container:: sphx-glr-footer sphx-glr-footer-example - - .. container:: sphx-glr-download sphx-glr-download-jupyter - - :download:`Download Jupyter notebook: example_2_single_atom_RIXS.ipynb ` - - .. container:: sphx-glr-download sphx-glr-download-python - - :download:`Download Python source code: example_2_single_atom_RIXS.py ` - - .. container:: sphx-glr-download sphx-glr-download-zip - - :download:`Download zipped: example_2_single_atom_RIXS.zip ` - - -.. only:: html - - .. rst-class:: sphx-glr-signature - - `Gallery generated by Sphinx-Gallery `_ diff --git a/edrixs/_sources/auto_examples/example_3_AIM_XAS.rst.txt b/edrixs/_sources/auto_examples/example_3_AIM_XAS.rst.txt deleted file mode 100644 index 5f3111c264..0000000000 --- a/edrixs/_sources/auto_examples/example_3_AIM_XAS.rst.txt +++ /dev/null @@ -1,748 +0,0 @@ - -.. DO NOT EDIT. -.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. -.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: -.. "auto_examples/example_3_AIM_XAS.py" -.. LINE NUMBERS ARE GIVEN BELOW. - -.. only:: html - - .. note:: - :class: sphx-glr-download-link-note - - :ref:`Go to the end ` - to download the full example code. - -.. rst-class:: sphx-glr-example-title - -.. _sphx_glr_auto_examples_example_3_AIM_XAS.py: - - -Anderson impurity model for NiO XAS -================================================================================ -Here we calculate the :math:`L`-edge XAS spectrum of an Anderson impurity model, -which is sometimes also called a charge-transfer multiplet model. This model -considers a set of correlated orbitals, often called the impurity or metal -states, that hybridize with a set of uncorrelated orbitals, often called the -ligands or bath states. Everyone's favorite test case for x-ray spectroscopic -calculations of the Anderson impurity model is NiO and we won't risk being -original! This means that our correlated states will -be the Ni :math:`3d` orbitals and the uncorrelated states come from O -:math:`2p` orbitals. The fact that we include these interactions means that our -spectrum can include processes where electrons transition from the bath to the -impurity, as such the Anderson Impurity Model is often more accurate than atomic -models, especially if the material has strong covalency. - -When defining the bath states, it is useful to use the so-called -symmetry adapted linear combinations of orbitals as the basis. These states -take into account the symmetry relationships between the different bath atom -orbitals and the fact that there are bath orbital combinations that do not -interact with the impurity by symmetry. By doing this the problem can be -represented with fewer orbitals, which makes the calculation far more efficient. -The standard EDRIXS solver that we will use assumes that the bath states are -represented by an integer number of bath sites set by :code:`nbath`, each of -which hosts the same number of spin-orbits as the impurity e.g. 10 for a -:math:`d`-electron material. - -NiO has a rocksalt structure in which all Ni atoms are surrounded by six O -atoms. This NiO cluster used to simulate the -crystal would then contain 10 Ni :math:`3d` spin-orbitals and :math:`6` -spin-orbitals per O :math:`\times 6` O atoms :math:`=36` oxygen -spin-orbitals. As explained by, for example, Maurits Haverkort -et al. in [1]_ symmetry allows us to represent the bath with 10 symmetry -adapted linear combinations of the different O :math:`p_x, p_y, p_z` states. -The crystal field and hopping parameters for -such a calculation can be obtained by post-processing DFT calculations. We will -use values for NiO from [1]_. If you use values from a paper the relevant -references should, of course, be cited. - -.. GENERATED FROM PYTHON SOURCE LINES 42-46 - -.. code-block:: Python - - import edrixs - import numpy as np - import matplotlib.pyplot as plt - - - - - - - - -.. GENERATED FROM PYTHON SOURCE LINES 47-56 - -Number of electrons ------------------------------------------------------------------------------- -When formulating problems of this type, one usually thinks of a nominal -valence for the impurity atom in this case :code:`nd = 8` and assume that the -bath is full. The solver that we will -use can simulate multiple bath sites. In our case we specify -:code:`nbath = 1` sites. Electrons will be able to transition from O to Ni -during our calculation, but the total number of valance electrons -:code:`v_noccu` will be conserved. - -.. GENERATED FROM PYTHON SOURCE LINES 56-63 - -.. code-block:: Python - - nd = 8 - norb_d = 10 - norb_bath = 10 - nbath = 1 - v_noccu = nd + nbath*norb_d - shell_name = ('d', 'p') # valence and core shells for XAS calculation - - - - - - - - -.. GENERATED FROM PYTHON SOURCE LINES 64-70 - -Coulomb interactions ------------------------------------------------------------------------------- -The atomic Coulomb interactions are usually initialized based on Hartree-Fock -calculations from, for example, -`Cowan's code `_. -edrixs has a database of these. - -.. GENERATED FROM PYTHON SOURCE LINES 70-72 - -.. code-block:: Python - - info = edrixs.utils.get_atom_data('Ni', '3d', nd, edge='L3') - - - - - - - - -.. GENERATED FROM PYTHON SOURCE LINES 73-77 - -The atomic values are typically scaled to account for screening in the solid. -Here we use 80% scaling. Let's write these out in full, so that nothing is -hidden. Values for :math:`U_{dd}` and :math:`U_{dp}` are those of Ref. [1]_ -obtained by comparing theory and experiment [2]_ [3]_. - -.. GENERATED FROM PYTHON SOURCE LINES 77-93 - -.. code-block:: Python - - scale_dd = 0.8 - F2_dd = info['slater_i'][1][1] * scale_dd - F4_dd = info['slater_i'][2][1] * scale_dd - U_dd = 7.3 - F0_dd = U_dd + edrixs.get_F0('d', F2_dd, F4_dd) - - scale_dp = 0.8 - F2_dp = info['slater_n'][4][1] * scale_dp - G1_dp = info['slater_n'][5][1] * scale_dp - G3_dp = info['slater_n'][6][1] * scale_dp - U_dp = 8.5 - F0_dp = U_dp + edrixs.get_F0('dp', G1_dp, G3_dp) - - slater = ([F0_dd, F2_dd, F4_dd], # initial - [F0_dd, F2_dd, F4_dd, F0_dp, F2_dp, G1_dp, G3_dp]) # with core hole - - - - - - - - -.. GENERATED FROM PYTHON SOURCE LINES 94-112 - -Energy of the bath states ------------------------------------------------------------------------------- -In the notation used in EDRIXS, :math:`\Delta` sets the energy difference -between the bath and impurity states. :math:`\Delta` is defined in the atomic -limit without crystal field (i.e. in terms of the centers of the impurity and -bath states before hybridization is considered) as the energy for a -:math:`d^{n_d} \rightarrow d^{n_d + 1} \underline{L}` transition. -Note that as electrons are moved one has to pay energy -costs associated with the Coulomb interactions. The -energy splitting between the bath and impurity is consequently not simply -:math:`\Delta`. One must therefore determine the energies by solving -a set of linear equations. See the :ref:`edrixs.utils functions ` -for details. We can call these functions to get the impurity energy -:math:`E_d`, bath energy :math:`E_L`, impurity energy with a core hole -:math:`E_{dc}`, bath energy with a core hole :math:`E_{Lc}` and the -core hole energy :math:`E_p`. The -:code:`if __name__ == '__main__'` code specifies that this command -should only be executed if the file is explicitly run. - -.. GENERATED FROM PYTHON SOURCE LINES 112-124 - -.. code-block:: Python - - Delta = 4.7 - E_d, E_L = edrixs.CT_imp_bath(U_dd, Delta, nd) - E_dc, E_Lc, E_p = edrixs.CT_imp_bath_core_hole(U_dd, U_dp, Delta, nd) - message = ("E_d = {:.3f} eV\n" - "E_L = {:.3f} eV\n" - "E_dc = {:.3f} eV\n" - "E_Lc = {:.3f} eV\n" - "E_p = {:.3f} eV\n") - if __name__ == '__main__': - print(message.format(E_d, E_L, E_dc, E_Lc, E_p)) - - - - - - -.. rst-class:: sphx-glr-script-out - - .. code-block:: none - - E_d = -41.189 eV - E_L = 12.511 eV - E_dc = -77.367 eV - E_Lc = 27.333 eV - E_p = -44.467 eV - - - - - -.. GENERATED FROM PYTHON SOURCE LINES 125-128 - -The spin-orbit coupling for the valence electrons in the ground state, the -valence electrons with the core hole present, and for the core hole itself -are initialized using the atomic values. - -.. GENERATED FROM PYTHON SOURCE LINES 128-132 - -.. code-block:: Python - - zeta_d_i = info['v_soc_i'][0] - zeta_d_n = info['v_soc_n'][0] - c_soc = info['c_soc'] - - - - - - - - -.. GENERATED FROM PYTHON SOURCE LINES 133-143 - -Build matrices describing interactions ------------------------------------------------------------------------------- -edrixs uses complex spherical harmonics as its default basis set. If we want to -use another basis set, we need to pass a matrix to the solver, which transforms -from complex spherical harmonics into the basis we use. -The solver will use this matrix when implementing the Coulomb interactions -using the :code:`slater` list of Coulomb parameters. -Here it is easiest to -use real harmonics. We make the complex harmonics to real harmonics transformation -matrix via - -.. GENERATED FROM PYTHON SOURCE LINES 143-145 - -.. code-block:: Python - - trans_c2n = edrixs.tmat_c2r('d',True) - - - - - - - - -.. GENERATED FROM PYTHON SOURCE LINES 146-156 - -The crystal field and SOC needs to be passed to the solver by constructing -the impurity matrix in the real harmonic basis. For cubic symmetry, we need -to set the energies of the orbitals along the -diagonal of the matrix. These need to be in pairs as there are two -spin-orbitals for each orbital energy. Python -`list comprehension `_ -and -`numpy indexing `_ -are used here. See :ref:`sphx_glr_auto_examples_example_1_crystal_field.py` -for more details if needed. - -.. GENERATED FROM PYTHON SOURCE LINES 156-171 - -.. code-block:: Python - - ten_dq = 0.56 - CF = np.zeros((norb_d, norb_d), dtype=complex) - diagonal_indices = np.arange(norb_d) - - orbital_energies = np.array([e for orbital_energy in - [+0.6 * ten_dq, # dz2 - -0.4 * ten_dq, # dzx - -0.4 * ten_dq, # dzy - +0.6 * ten_dq, # dx2-y2 - -0.4 * ten_dq] # dxy) - for e in [orbital_energy]*2]) - - - CF[diagonal_indices, diagonal_indices] = orbital_energies - - - - - - - - -.. GENERATED FROM PYTHON SOURCE LINES 172-174 - -The valence band SOC is constructed in the normal way and transformed into the -real harmonic basis. - -.. GENERATED FROM PYTHON SOURCE LINES 174-176 - -.. code-block:: Python - - soc = edrixs.cb_op(edrixs.atom_hsoc('d', zeta_d_i), edrixs.tmat_c2r('d', True)) - - - - - - - - -.. GENERATED FROM PYTHON SOURCE LINES 177-181 - -The total impurity matrices for the ground and core-hole states are then -the sum of crystal field and spin-orbit coupling. We further needed to apply -an energy shift along the matrix diagonal, which we do using the -:code:`np.eye` function which creates a diagonal matrix of ones. - -.. GENERATED FROM PYTHON SOURCE LINES 181-186 - -.. code-block:: Python - - E_d_mat = E_d*np.eye(norb_d) - E_dc_mat = E_dc*np.eye(norb_d) - imp_mat = CF + soc + E_d_mat - imp_mat_n = CF + soc + E_dc_mat - - - - - - - - -.. GENERATED FROM PYTHON SOURCE LINES 187-191 - -The energy level of the bath(s) is described by a matrix where the row index -denotes which bath and the column index denotes which orbital. Here we have -only one bath, with 10 spin-orbitals. We initialize the matrix to -:code:`norb_d` and then split the energies according to :code:`ten_dq_bath`. - -.. GENERATED FROM PYTHON SOURCE LINES 191-203 - -.. code-block:: Python - - ten_dq_bath = 1.44 - bath_level = np.full((nbath, norb_d), E_L, dtype=complex) - bath_level[0, :2] += ten_dq_bath*.6 # 3z2-r2 - bath_level[0, 2:6] -= ten_dq_bath*.4 # zx/yz - bath_level[0, 6:8] += ten_dq_bath*.6 # x2-y2 - bath_level[0, 8:] -= ten_dq_bath*.4 # xy - bath_level_n = np.full((nbath, norb_d), E_Lc, dtype=complex) - bath_level_n[0, :2] += ten_dq_bath*.6 # 3z2-r2 - bath_level_n[0, 2:6] -= ten_dq_bath*.4 # zx/yz - bath_level_n[0, 6:8] += ten_dq_bath*.6 # x2-y2 - bath_level_n[0, 8:] -= ten_dq_bath*.4 # xy - - - - - - - - -.. GENERATED FROM PYTHON SOURCE LINES 204-209 - -The hybridization matrix describes the hopping between the bath -and the impurity. This is called either :math:`V` or :math:`T` in the -literature and matrix sign can either be positive or negative based. -This is the same shape as the bath matrix. We take our -values from Maurits Haverkort et al.'s DFT calculations [1]_. - -.. GENERATED FROM PYTHON SOURCE LINES 209-218 - -.. code-block:: Python - - Veg = 2.06 - Vt2g = 1.21 - - hyb = np.zeros((nbath, norb_d), dtype=complex) - hyb[0, :2] = Veg # 3z2-r2 - hyb[0, 2:6] = Vt2g # zx/yz - hyb[0, 6:8] = Veg # x2-y2 - hyb[0, 8:] = Vt2g # xy - - - - - - - - -.. GENERATED FROM PYTHON SOURCE LINES 219-221 - -We now need to define the parameters describing the XAS. X-ray polarization -can be linear, circular or isotropic (appropriate for a powder). - -.. GENERATED FROM PYTHON SOURCE LINES 221-222 - -.. code-block:: Python - - poltype_xas = [('isotropic', 0)] - - - - - - - -.. GENERATED FROM PYTHON SOURCE LINES 223-225 - -edrixs uses the temperature in Kelvin to work out the population of the low-lying -states via a Boltzmann distribution. - -.. GENERATED FROM PYTHON SOURCE LINES 225-226 - -.. code-block:: Python - - temperature = 300 - - - - - - - -.. GENERATED FROM PYTHON SOURCE LINES 227-228 - -The x-ray beam is specified by the incident angle and azimuthal angle in radians - -.. GENERATED FROM PYTHON SOURCE LINES 228-230 - -.. code-block:: Python - - thin = 0 / 180.0 * np.pi - phi = 0.0 - - - - - - - -.. GENERATED FROM PYTHON SOURCE LINES 231-234 - -these are with respect to the crystal field :math:`z` and :math:`x` axes -written above. (That is, unless you specify the :code:`loc_axis` parameter -described in the :code:`edrixs.xas_siam_fort` function documentation.) - -.. GENERATED FROM PYTHON SOURCE LINES 236-242 - -The spectrum in the raw calculation is offset by the energy involved with the -core hole state, which is roughly :math:`5 E_p`, so we offset the spectrum by -this and use :code:`om_shift` as an adjustable parameters for comparing -theory to experiment. We also use this to specify :code:`ominc_xas` -the range we want to compute the spectrum over. The core hole lifetime -broadening also needs to be set via :code:`gamma_c_stat`. - -.. GENERATED FROM PYTHON SOURCE LINES 242-246 - -.. code-block:: Python - - om_shift = 857.6 - c_level = -om_shift - 5*E_p - ominc_xas = om_shift + np.linspace(-15, 25, 1000) - - - - - - - - -.. GENERATED FROM PYTHON SOURCE LINES 247-251 - -The final state broadening is specified in terms of half-width at half-maximum -You can either pass a constant value or an array the same size as -:code:`om_shift` with varying values to simulate, for example, different state -lifetimes for higher energy states. - -.. GENERATED FROM PYTHON SOURCE LINES 251-253 - -.. code-block:: Python - - gamma_c = np.full(ominc_xas.shape, 0.48/2) - - - - - - - - -.. GENERATED FROM PYTHON SOURCE LINES 254-263 - -Magnetic field is a three-component vector in eV specified with respect to the -same local axis as the x-ray beam. Since we are considering a powder here -we create an isotropic normalized vector. :code:`on_which = 'both'` specifies to -apply the operator to the total spin plus orbital angular momentum as is -appropriate for a physical external magnetic field. You can use -:code:`on_which = 'spin'` to apply the operator to spin in order to simulate -magnetic order in the sample. The value of the Bohr Magneton can -be useful for converting here :math:`\mu_B = 5.7883818012\times 10^{−5}`. -For this example, we will account for magnetic order in the sample by - -.. GENERATED FROM PYTHON SOURCE LINES 263-266 - -.. code-block:: Python - - ext_B = np.array([0.00, 0.00, 0.12]) - on_which = 'spin' - - - - - - - - -.. GENERATED FROM PYTHON SOURCE LINES 267-278 - -The number crunching uses -`mpi4py `_. You can safely ignore -this for most purposes, but see -`Y. L. Wang et al., Computer Physics Communications 243, 151-165 (2019) `_ -if you would like more details. -The main thing to remember is that you should call this script via:: - - mpirun -n python example_AIM_XAS.py - -where :code:`` is the number of processors -you'd like to us. Running it as normal will work, it will just be slower. - -.. GENERATED FROM PYTHON SOURCE LINES 278-284 - -.. code-block:: Python - - if __name__ == '__main__': - from mpi4py import MPI - comm = MPI.COMM_WORLD - rank = comm.Get_rank() - size = comm.Get_size() - - - - - - - - -.. GENERATED FROM PYTHON SOURCE LINES 285-293 - -Calling the :code:`edrixs.ed_siam_fort` solver will find the ground state and -write input files, *hopping_i.in*, *hopping_n.in*, *coulomb_i.in*, *coulomb_n.in* -for following XAS (or RIXS) calculation. We need to specify :code:`siam_type=0` -which says that we will pass *imp_mat*, *bath_level* and *hyb*. -We need to specify :code:`do_ed = 1`. For this example, we cannot use -:code:`do_ed = 0` for a ground state search as we have set the impurity and -bath energy levels artificially, which means edrixs will have trouble to know -which subspace to search to find the ground state. - -.. GENERATED FROM PYTHON SOURCE LINES 293-301 - -.. code-block:: Python - - if __name__ == '__main__': - do_ed = 1 - eval_i, denmat, noccu_gs = edrixs.ed_siam_fort( - comm, shell_name, nbath, siam_type=0, imp_mat=imp_mat, imp_mat_n=imp_mat_n, - bath_level=bath_level, bath_level_n=bath_level_n, hyb=hyb, c_level=c_level, - c_soc=c_soc, slater=slater, ext_B=ext_B, - on_which=on_which, trans_c2n=trans_c2n, v_noccu=v_noccu, do_ed=do_ed, - ed_solver=2, neval=50, nvector=3, ncv=100, idump=True) - - - - -.. rst-class:: sphx-glr-script-out - - .. code-block:: none - - edrixs >>> Running ED ... - - Summary of Slater integrals: - ------------------------------ - Terms, Initial Hamiltonian, Intermediate Hamiltonian - F0_vv : 7.8036698413 7.8036698413 - F2_vv : 9.7872000000 9.7872000000 - F4_vv : 6.0784000000 6.0784000000 - F0_vc : 0.0000000000 8.9214742857 - F2_vc : 0.0000000000 6.1768000000 - G1_vc : 0.0000000000 4.6296000000 - G3_vc : 0.0000000000 2.6328000000 - F0_cc : 0.0000000000 0.0000000000 - F2_cc : 0.0000000000 0.0000000000 - - edrixs >>> do_ed=1, perform ED at noccu: 18 - - - - -.. GENERATED FROM PYTHON SOURCE LINES 302-304 - -Let's check that we have all the electrons we think we have and print how -the electron are distributed between the Ni (impurity) and O (bath). - -.. GENERATED FROM PYTHON SOURCE LINES 304-310 - -.. code-block:: Python - - if __name__ == '__main__': - assert np.abs(noccu_gs - v_noccu) < 1e-6 - impurity_occupation = np.sum(denmat[0].diagonal()[0:norb_d]).real - bath_occupation = np.sum(denmat[0].diagonal()[norb_d:]).real - print('Impurity occupation = {:.6f}\n'.format(impurity_occupation)) - print('Bath occupation = {:.6f}\n'.format(bath_occupation)) - - - - -.. rst-class:: sphx-glr-script-out - - .. code-block:: none - - Impurity occupation = 8.179506 - - Bath occupation = 9.820494 - - - - - -.. GENERATED FROM PYTHON SOURCE LINES 311-317 - -We see that 0.18 electrons move from the O to the Ni in the ground state. - -We can now construct the XAS spectrum edrixs by applying a transition -operator to create the excited state. We need to be careful to specify how -many of the low energy states are thermally populated. In this case -:code:`num_gs=3`. This can be determined by inspecting the function output. - -.. GENERATED FROM PYTHON SOURCE LINES 317-322 - -.. code-block:: Python - - if __name__ == '__main__': - xas, xas_poles = edrixs.xas_siam_fort( - comm, shell_name, nbath, ominc_xas, gamma_c=gamma_c, v_noccu=v_noccu, thin=thin, - phi=phi, num_gs=3, nkryl=200, pol_type=poltype_xas, temperature=temperature - ) - - - - -.. rst-class:: sphx-glr-script-out - - .. code-block:: none - - edrixs >>> Running XAS ... - edrixs >>> Loop over for polarization: 0 isotropic - edrixs >>> Isotropic, component: 0 - edrixs >>> Loop over for polarization: 0 isotropic - edrixs >>> Isotropic, component: 1 - edrixs >>> Loop over for polarization: 0 isotropic - edrixs >>> Isotropic, component: 2 - - - - -.. GENERATED FROM PYTHON SOURCE LINES 323-324 - -Let's plot the data and save it just in case - -.. GENERATED FROM PYTHON SOURCE LINES 324-335 - -.. code-block:: Python - - if __name__ == '__main__': - fig, ax = plt.subplots() - - ax.plot(ominc_xas, xas) - ax.set_xlabel('Energy (eV)') - ax.set_ylabel('XAS intensity') - ax.set_title('Anderson impurity model for NiO') - plt.show() - - np.savetxt('xas.dat', np.concatenate((np.array([ominc_xas]).T, xas), axis=1)) - - - - -.. image-sg:: /auto_examples/images/sphx_glr_example_3_AIM_XAS_001.png - :alt: Anderson impurity model for NiO - :srcset: /auto_examples/images/sphx_glr_example_3_AIM_XAS_001.png - :class: sphx-glr-single-img - - - - - -.. GENERATED FROM PYTHON SOURCE LINES 336-344 - -.. rubric:: Footnotes - -.. [1] Maurits Haverkort et al - `Phys. Rev. B 85, 165113 (2012) `_. -.. [2] A. E. Bocquet et al., - `Phys. Rev. B 53, 1161 (1996) `_. -.. [3] Arata Tanaka, and Takeo Jo, - `J. Phys. Soc. Jpn. 63, 2788-2807(1994) `_. - - -.. rst-class:: sphx-glr-timing - - **Total running time of the script:** (0 minutes 0.548 seconds) - - -.. _sphx_glr_download_auto_examples_example_3_AIM_XAS.py: - -.. only:: html - - .. container:: sphx-glr-footer sphx-glr-footer-example - - .. container:: sphx-glr-download sphx-glr-download-jupyter - - :download:`Download Jupyter notebook: example_3_AIM_XAS.ipynb ` - - .. container:: sphx-glr-download sphx-glr-download-python - - :download:`Download Python source code: example_3_AIM_XAS.py ` - - .. container:: sphx-glr-download sphx-glr-download-zip - - :download:`Download zipped: example_3_AIM_XAS.zip ` - - -.. only:: html - - .. rst-class:: sphx-glr-signature - - `Gallery generated by Sphinx-Gallery `_ diff --git a/edrixs/_sources/auto_examples/example_4_GS_analysis.rst.txt b/edrixs/_sources/auto_examples/example_4_GS_analysis.rst.txt deleted file mode 100644 index c04d815ec1..0000000000 --- a/edrixs/_sources/auto_examples/example_4_GS_analysis.rst.txt +++ /dev/null @@ -1,387 +0,0 @@ - -.. DO NOT EDIT. -.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. -.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: -.. "auto_examples/example_4_GS_analysis.py" -.. LINE NUMBERS ARE GIVEN BELOW. - -.. only:: html - - .. note:: - :class: sphx-glr-download-link-note - - :ref:`Go to the end ` - to download the full example code. - -.. rst-class:: sphx-glr-example-title - -.. _sphx_glr_auto_examples_example_4_GS_analysis.py: - - -Ground state analysis for NiO -================================================================================ -This example follows the :ref:`sphx_glr_auto_examples_example_3_AIM_XAS.py` -example and considers the same model. This time we show how to analyze -the wavevectors in terms of a -:math:`\alpha |d^8L^{10}> + \beta |d^9L^9> \gamma |d^{10}L^8>` -representation. - -In doing this we will go through the exercise of building and diagonalizing -the Hamiltonian in a way that hopefully clarifies how to analyze other -properties of the model. - -.. GENERATED FROM PYTHON SOURCE LINES 15-23 - -.. code-block:: Python - - import edrixs - import numpy as np - import matplotlib.pyplot as plt - import scipy - import example_3_AIM_XAS - import importlib - _ = importlib.reload(example_3_AIM_XAS) - - - - - - - - -.. GENERATED FROM PYTHON SOURCE LINES 24-40 - -Hamiltonian ------------------------------------------------------------------------------- -edrixs builds model Hamiltonians based on two fermion and four fermion terms. -The four fermion terms come from Coulomb interactions and will be -assigned to :code:`umat`. All other interactions contribute to two fermion -terms in :code:`emat`. - - .. math:: - \begin{equation} - \hat{H}_{i} = \sum_{\alpha,\beta} t_{\alpha,\beta} - \hat{f}^{\dagger}_{\alpha} \hat{f}_{\beta} - + \sum_{\alpha,\beta,\gamma,\delta} U_{\alpha,\beta,\gamma,\delta} - \hat{f}^{\dagger}_{\alpha}\hat{f}^{\dagger}_{\beta} - \hat{f}_{\gamma}\hat{f}_{\delta}, - \end{equation} - - -.. GENERATED FROM PYTHON SOURCE LINES 42-48 - -Import parameters ------------------------------------------------------------------------------- -Let's get the parammeters we need from the -:ref:`sphx_glr_auto_examples_example_3_AIM_XAS.py` example. We need to -consider :code:`ntot=20` spin-orbitals -for this problem. - -.. GENERATED FROM PYTHON SOURCE LINES 48-54 - -.. code-block:: Python - - - from example_3_AIM_XAS import (F0_dd, F2_dd, F4_dd, - nd, norb_d, norb_bath, v_noccu, - imp_mat, bath_level, - hyb, ext_B, trans_c2n) - ntot = 20 - - - - - - - -.. GENERATED FROM PYTHON SOURCE LINES 55-64 - -Four fermion matrix ------------------------------------------------------------------------------- -The Coulomb interactions in the :math:`d` shell of this problem are described -by a :math:`10\times10\times10\times10` matrix. We -need to specify a :math:`20\times20\times20\times 20` matrix since we need to -include the full problem with :code:`ntot=20` spin-orbitals. The edrixs -convention is to put the :math:`d` orbitals first, so we assign them to the -first :math:`10\times10\times10\times 10` indices of the matrix. edrixs -creates this matrix in the complex harmmonic basis by default. - -.. GENERATED FROM PYTHON SOURCE LINES 64-69 - -.. code-block:: Python - - umat_delectrons = edrixs.get_umat_slater('d', F0_dd, F2_dd, F4_dd) - umat = np.zeros((ntot, ntot, ntot, ntot), dtype=complex) - umat[:norb_d, :norb_d, :norb_d, :norb_d] += umat_delectrons - - - - - - - - - -.. GENERATED FROM PYTHON SOURCE LINES 70-76 - -Two fermion matrix ------------------------------------------------------------------------------- -Previously we made a :math:`10\times10` two-fermion matrix describing the -:math:`d`-shell interactions. Keep in mind we did this in the real harmonic -basis. We need to specify the two-fermion matrix for -the full problem :math:`20\times20` spin-orbitals in size. - -.. GENERATED FROM PYTHON SOURCE LINES 76-79 - -.. code-block:: Python - - emat_rhb = np.zeros((ntot, ntot), dtype='complex') - emat_rhb[0:norb_d, 0:norb_d] += imp_mat - - - - - - - - -.. GENERATED FROM PYTHON SOURCE LINES 80-82 - -The :code:`bath_level` energies need to be applied to the diagonal of the -last :math:`10\times10` region of the matrix. - -.. GENERATED FROM PYTHON SOURCE LINES 82-85 - -.. code-block:: Python - - indx = np.arange(norb_d, norb_d*2) - emat_rhb[indx, indx] += bath_level[0] - - - - - - - - -.. GENERATED FROM PYTHON SOURCE LINES 86-88 - -The :code:`hyb` terms mix the impurity and bath states and are therefore -applied to the off-diagonal terms of :code:`emat`. - -.. GENERATED FROM PYTHON SOURCE LINES 88-93 - -.. code-block:: Python - - indx1 = np.arange(norb_d) - indx2 = np.arange(norb_d, norb_d*2) - emat_rhb[indx1, indx2] += hyb[0] - emat_rhb[indx2, indx1] += np.conj(hyb[0]) - - - - - - - - -.. GENERATED FROM PYTHON SOURCE LINES 94-97 - -We now need to transform into the complex harmonic basis. We assign -the two diagonal blocks of a :math:`20\times20` matrix to the -conjugate transpose of the transition matrix. - -.. GENERATED FROM PYTHON SOURCE LINES 97-104 - -.. code-block:: Python - - tmat = np.eye(ntot, dtype=complex) - for i in range(2): - off = i * norb_d - tmat[off:off+norb_d, off:off+norb_d] = np.conj(np.transpose(trans_c2n)) - - emat_chb = edrixs.cb_op(emat_rhb, tmat) - - - - - - - - -.. GENERATED FROM PYTHON SOURCE LINES 105-107 - -The spin exchange is built from the spin operators and the effective field -is applied to the :math:`d`-shell region of the matrix. - -.. GENERATED FROM PYTHON SOURCE LINES 107-114 - -.. code-block:: Python - - v_orbl = 2 - sx = edrixs.get_sx(v_orbl) - sy = edrixs.get_sy(v_orbl) - sz = edrixs.get_sz(v_orbl) - zeeman = ext_B[0] * (2 * sx) + ext_B[1] * (2 * sy) + ext_B[2] * (2 * sz) - emat_chb[0:norb_d, 0:norb_d] += zeeman - - - - - - - - -.. GENERATED FROM PYTHON SOURCE LINES 115-122 - -Build the Fock basis and Hamiltonain and Diagonalize ------------------------------------------------------------------------------- -We create the fock basis and build the Hamiltonian using the full set of -:math:`20` spin orbitals, specifying that they are occuplied by :math:`18` -electrons. See the :ref:`sphx_glr_auto_examples_example_0_ed_calculator.py` -example for more details if needed. We also set the ground state to zero -energy. - -.. GENERATED FROM PYTHON SOURCE LINES 122-129 - -.. code-block:: Python - - basis = np.array(edrixs.get_fock_bin_by_N(ntot, v_noccu)) - H = (edrixs.build_opers(2, emat_chb, basis) - + edrixs.build_opers(4, umat, basis)) - - e, v = scipy.linalg.eigh(H) - e -= e[0] - - - - - - - - -.. GENERATED FROM PYTHON SOURCE LINES 130-136 - -State analysis ------------------------------------------------------------------------------- -Now we have all the eigenvectors in the Fock basis, we need to pick out the -states that have 8, 9 and 10 :math:`d`-electrons, respectively. -The modulus squared of these coeffcients need to be added up to get -:math:`\alpha`, :math:`\beta`, and :math:`\gamma`. - -.. GENERATED FROM PYTHON SOURCE LINES 136-143 - -.. code-block:: Python - - - num_d_electrons = basis[:, :norb_d].sum(1) - - alphas = np.sum(np.abs(v[num_d_electrons==8, :])**2, axis=0) - betas = np.sum(np.abs(v[num_d_electrons==9, :])**2, axis=0) - gammas = np.sum(np.abs(v[num_d_electrons==10, :])**2, axis=0) - - - - - - - - -.. GENERATED FROM PYTHON SOURCE LINES 144-145 - -The ground state is the first entry. - -.. GENERATED FROM PYTHON SOURCE LINES 145-148 - -.. code-block:: Python - - message = "Ground state\nalpha={:.3f}\tbeta={:.3f}\tgamma={:.3f}" - print(message.format(alphas[0], betas[0], gammas[0])) - - - - - -.. rst-class:: sphx-glr-script-out - - .. code-block:: none - - Ground state - alpha=0.825 beta=0.171 gamma=0.004 - - - - -.. GENERATED FROM PYTHON SOURCE LINES 149-153 - -Plot ------------------------------------------------------------------------------- -Let's look how :math:`\alpha`, :math:`\beta`, and :math:`\gamma` vary with -energy. - -.. GENERATED FROM PYTHON SOURCE LINES 153-166 - -.. code-block:: Python - - - fig, ax = plt.subplots() - - ax.plot(e, alphas, label=r'$\alpha$ $d^8L^{10}$') - ax.plot(e, betas, label=r'$\beta$ $d^9L^{9}$') - ax.plot(e, gammas, label=r'$\gamma$ $d^{10}L^{8}$') - - ax.set_xlabel('Energy (eV)') - ax.set_ylabel('Population') - ax.set_title('NiO') - ax.legend() - plt.show() - - - - -.. image-sg:: /auto_examples/images/sphx_glr_example_4_GS_analysis_001.png - :alt: NiO - :srcset: /auto_examples/images/sphx_glr_example_4_GS_analysis_001.png - :class: sphx-glr-single-img - - - - - -.. GENERATED FROM PYTHON SOURCE LINES 167-171 - -We see that the ligand states are mixed into the ground state, but the -majority of the weight for the :math:`L^9` and :math:`L^8` states -live at :math:`\Delta` and :math:`2\Delta`. With a lot of additional -structure from the other interactions. - - -.. rst-class:: sphx-glr-timing - - **Total running time of the script:** (0 minutes 0.421 seconds) - - -.. _sphx_glr_download_auto_examples_example_4_GS_analysis.py: - -.. only:: html - - .. container:: sphx-glr-footer sphx-glr-footer-example - - .. container:: sphx-glr-download sphx-glr-download-jupyter - - :download:`Download Jupyter notebook: example_4_GS_analysis.ipynb ` - - .. container:: sphx-glr-download sphx-glr-download-python - - :download:`Download Python source code: example_4_GS_analysis.py ` - - .. container:: sphx-glr-download sphx-glr-download-zip - - :download:`Download zipped: example_4_GS_analysis.zip ` - - -.. only:: html - - .. rst-class:: sphx-glr-signature - - `Gallery generated by Sphinx-Gallery `_ diff --git a/edrixs/_sources/auto_examples/example_5_Hubbard_dimer.rst.txt b/edrixs/_sources/auto_examples/example_5_Hubbard_dimer.rst.txt deleted file mode 100644 index f784fff8d1..0000000000 --- a/edrixs/_sources/auto_examples/example_5_Hubbard_dimer.rst.txt +++ /dev/null @@ -1,371 +0,0 @@ - -.. DO NOT EDIT. -.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. -.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: -.. "auto_examples/example_5_Hubbard_dimer.py" -.. LINE NUMBERS ARE GIVEN BELOW. - -.. only:: html - - .. note:: - :class: sphx-glr-download-link-note - - Click :ref:`here ` - to download the full example code - -.. rst-class:: sphx-glr-example-title - -.. _sphx_glr_auto_examples_example_5_Hubbard_dimer.py: - - -Hubbard Dimer -===================================== -This exercise will demonstrate how to handle hopping and multi-site problems within -edrixs using the example of a Hubbard dimer. We want to solve the equation - - .. math:: - \begin{equation} - \hat{H} = \sum_{i,j} \sum_{\sigma} t_{i,j} \hat{f}^{\dagger}_{i,\sigma} \hat{f}_{j, \sigma} - + U \sum_{i} \hat{n}_{i,\uparrow}\hat{n}_{i,\downarrow}, - \end{equation} - -which involves two sites labeled with indices :math:`i` or :math:`j` with two -electrons of spin :math:`\sigma\in{\uparrow,\downarrow}`. :math:`t_{i,j}` -is the hopping between sites, :math:`\hat{f}^{\dagger}_{i,\sigma}` is the -creation operators, and -:math:`\hat{n}^{\dagger}_{i,\sigma}=\hat{f}^{\dagger}_{i,\sigma}\hat{f}_{i,\sigma}` -is the number operator. The main task is to represent this Hamiltonian and -the related spin operator using the EDRIXS two-fermion and four-fermion form -where :math:`\alpha,\beta,\delta,\gamma` are the indices of the single -particle basis. - - .. math:: - \begin{equation} - \hat{H} = \sum_{\alpha,\beta} t_{\alpha,\beta} \hat{f}^{\dagger}_{\alpha} \hat{f}_{\beta} - + \sum_{\alpha,\beta,\gamma,\delta} U_{\alpha,\beta,\gamma,\delta} - \hat{f}^{\dagger}_{\alpha}\hat{f}^{\dagger}_{\beta}\hat{f}_{\gamma}\hat{f}_{\delta}. - \end{equation} - -.. GENERATED FROM PYTHON SOURCE LINES 34-39 - -Initialize matrices ------------------------------------------------------------------------------- -We start by noting that each of the two sites is like an :math:`l=0` -:math:`s`-orbital with two spin-orbitals each. We will include -two electron occupation and build the Fock basis. - -.. GENERATED FROM PYTHON SOURCE LINES 39-52 - -.. code-block:: default - - import numpy as np - import matplotlib.pyplot as plt - import scipy - import edrixs - np.set_printoptions(precision=4) - - - ll = 0 - case = 's' - norb = 4 - noccu = 2 - basis = edrixs.get_fock_bin_by_N(norb, noccu) - - - - - - - - -.. GENERATED FROM PYTHON SOURCE LINES 53-69 - -Create function to populate and diagonalize matrices ------------------------------------------------------------------------------- -The Coulomb and hopping matrices :code:`umat` and :code:`emat` will be -represented by :math:`4\times4\times4\times4` and :math:`4\times4` matrices, -respectively. Note that we needed to specify -that these are, in general, complex, although -they happen to contain only real numbers in this case. We follow the convention -that these are ordered first by site and then by spin: -:math:`|0\uparrow>, |0\downarrow>, |1\uparrow>, |1\downarrow>`. -Consequently the :math:`2\times2` and :math:`2\times2\times2\times2` block -diagonal structures of the matrices will contain the on-site interactions. -The converse is true for the hopping between the sites. -From here let us generate a function to build and diagonalize the Hamiltonian. -We need to generate the Coulomb matrix for the on-site interactions and -apply it to the block diagonal. The hopping connects off-site indices with -the same spin. - -.. GENERATED FROM PYTHON SOURCE LINES 69-86 - -.. code-block:: default - - def diagonalize(U, t, extra_emat=None): - """Diagonalize 2 site Hubbard Hamiltonian""" - umat = np.zeros((norb, norb, norb, norb), dtype=np.complex128) - emat = np.zeros((norb, norb), dtype=np.complex128) - U_mat_1site = edrixs.get_umat_slater('s', U) - umat[:2, :2, :2, :2,] = umat[2:, 2:, 2:, 2:] = U_mat_1site - emat[2, 0] = emat[3, 1] = emat[0, 2] = emat[1, 3] = t - - if extra_emat is not None: - emat = emat + extra_emat - - H = (edrixs.build_opers(2, emat, basis) - + edrixs.build_opers(4, umat, basis)) - - e, v = scipy.linalg.eigh(H) - return e, v - - - - - - - - -.. GENERATED FROM PYTHON SOURCE LINES 87-90 - -The large :math:`U` limit ------------------------------------------------------------------------------- -Let us see what happens with :math:`U \gg t`. - -.. GENERATED FROM PYTHON SOURCE LINES 90-94 - -.. code-block:: default - - e, v = diagonalize(1000, 1) - print("Energies are") - print(e) - - - - - -.. rst-class:: sphx-glr-script-out - - Out: - - .. code-block:: none - - Energies are - [ -0.004 0. 0. 0. 1000. 1000.004] - - - - -.. GENERATED FROM PYTHON SOURCE LINES 95-98 - -To analyze what is going on we can determine the spin expectation values -of the cluster. Building the operators follows the same form as the -Hamiltonian and the previous example. - -.. GENERATED FROM PYTHON SOURCE LINES 98-106 - -.. code-block:: default - - spin_mom_one_site = edrixs.get_spin_momentum(ll) - spin_mom = np.zeros((3, norb, norb), dtype=np.complex128) - spin_mom[:, :2, :2] = spin_mom[:, 2:, 2:] = spin_mom_one_site - - opS = edrixs.build_opers(2, spin_mom, basis) - opS_squared = (np.dot(opS[0], opS[0]) + np.dot(opS[1], opS[1]) - + np.dot(opS[2], opS[2])) - - - - - - - - -.. GENERATED FROM PYTHON SOURCE LINES 107-110 - -This time let us include a tiny magnetic field along the :math:`z`-axis, so -that we have a well-defined measurement axis and print out the expectation -values. - -.. GENERATED FROM PYTHON SOURCE LINES 110-122 - -.. code-block:: default - - zeeman = np.zeros((norb, norb), dtype=np.complex128) - zeeman[:2, :2] = zeeman[2:, 2:] = 1e-8*spin_mom_one_site[2] - e, v = diagonalize(1000, 1, extra_emat=zeeman) - - Ssq_exp = edrixs.cb_op(opS_squared, v).diagonal().real - Sz_exp = edrixs.cb_op(opS[2], v).diagonal().real - - header = "{:<10s}\t{:<6s}\t{:<6s}" - print(header.format("E", "S(S+1)", "")) - for i in range(len(e)): - print("{:<2f}\t{:.1f}\t{:.1f}".format(e[i], Ssq_exp[i], Sz_exp[i])) - - - - - -.. rst-class:: sphx-glr-script-out - - Out: - - .. code-block:: none - - E S(S+1) - -0.004000 0.0 0.0 - -0.000000 2.0 -1.0 - -0.000000 2.0 0.0 - 0.000000 2.0 1.0 - 1000.000000 0.0 0.0 - 1000.004000 0.0 0.0 - - - - -.. GENERATED FROM PYTHON SOURCE LINES 123-127 - -For :math:`U \gg t` the two states with double occupancy acquire an energy of -approximately :math:`U`. The low energy states are a :math:`S=0` singlet and -and :math:`S=1` triplet, which are split by :math:`4t^2/U`, which is the -magnetic exchange term. - -.. GENERATED FROM PYTHON SOURCE LINES 129-132 - -:math:`U` dependence ------------------------------------------------------------------------------- -Let us plot the changes in energy with :math:`U`. - -.. GENERATED FROM PYTHON SOURCE LINES 132-143 - -.. code-block:: default - - plt.figure() - - t = 1 - Us = np.linspace(0.01, 10, 50) - Es = np.array([diagonalize(U, t, extra_emat=zeeman)[0] for U in Us]) - - plt.plot(Us/t, Es/t) - plt.xlabel('U/t') - plt.ylabel('Eigenstate energies/t') - plt.show() - - - - -.. image-sg:: /auto_examples/images/sphx_glr_example_5_Hubbard_dimer_001.png - :alt: example 5 Hubbard dimer - :srcset: /auto_examples/images/sphx_glr_example_5_Hubbard_dimer_001.png - :class: sphx-glr-single-img - - - - - -.. GENERATED FROM PYTHON SOURCE LINES 144-146 - -To help interpret this, we can represent the eigenvectors in terms of a sum -of the single particle states. - -.. GENERATED FROM PYTHON SOURCE LINES 146-164 - -.. code-block:: default - - - def get_single_particle_repesentations(v): - reps = [] - for i in range(6): - rep = sum([vec*weight for weight, vec - in zip(v[:, i], np.array(basis))]) - reps.append(rep) - - return np.array(reps) - - t = 1 - for U in [10000, 0.0001]: - e, v = diagonalize(U, t, extra_emat=zeeman) - repesentations = get_single_particle_repesentations(v) - print("For U={} t={} states are".format(U, t)) - print(repesentations.round(3).real) - print("\n") - - - - - -.. rst-class:: sphx-glr-script-out - - Out: - - .. code-block:: none - - For U=10000 t=1 states are - [[-0.707 0.707 0.707 -0.707] - [ 0. 1. 0. 1. ] - [ 0.707 0.707 0.707 0.707] - [-1. 0. -1. 0. ] - [-0.707 -0.707 0.707 0.707] - [ 0.707 0.707 0.707 0.707]] - - - For U=0.0001 t=1 states are - [[-0. 1. 1. -0. ] - [ 0. 1. 0. 1. ] - [ 0.707 0.707 0.707 0.707] - [-1. 0. -1. 0. ] - [-0.707 -0.707 0.707 0.707] - [-1. -0. -0. -1. ]] - - - - - - -.. GENERATED FROM PYTHON SOURCE LINES 165-170 - -For :math:`U \gg t` the ground state maximizes its magnetic exchange -energy saving. In the :math:`U \ll t` condition the ground state maximizes -its kinetic energy saving. Since both states share the same parity, the -cross-over between them is smooth. This type of physics is at play in current -research on quantum materials [1]_ [2]_. - -.. GENERATED FROM PYTHON SOURCE LINES 172-176 - -.. rubric:: Footnotes - -.. [1] Y. Wang et al., `Phys. Rev. Lett. 122, 106401 (2019) `_. -.. [2] A. Revelli et al., `Science Advances 5, eaav4020 (2019) `_. - - -.. rst-class:: sphx-glr-timing - - **Total running time of the script:** ( 0 minutes 0.191 seconds) - - -.. _sphx_glr_download_auto_examples_example_5_Hubbard_dimer.py: - - -.. only :: html - - .. container:: sphx-glr-footer - :class: sphx-glr-footer-example - - - - .. container:: sphx-glr-download sphx-glr-download-python - - :download:`Download Python source code: example_5_Hubbard_dimer.py ` - - - - .. container:: sphx-glr-download sphx-glr-download-jupyter - - :download:`Download Jupyter notebook: example_5_Hubbard_dimer.ipynb ` - - -.. only:: html - - .. rst-class:: sphx-glr-signature - - `Gallery generated by Sphinx-Gallery `_ diff --git a/edrixs/_sources/auto_examples/example_5_charge_transfer.rst.txt b/edrixs/_sources/auto_examples/example_5_charge_transfer.rst.txt deleted file mode 100644 index 12e8b5eae6..0000000000 --- a/edrixs/_sources/auto_examples/example_5_charge_transfer.rst.txt +++ /dev/null @@ -1,290 +0,0 @@ - -.. DO NOT EDIT. -.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. -.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: -.. "auto_examples/example_5_charge_transfer.py" -.. LINE NUMBERS ARE GIVEN BELOW. - -.. only:: html - - .. note:: - :class: sphx-glr-download-link-note - - :ref:`Go to the end ` - to download the full example code. - -.. rst-class:: sphx-glr-example-title - -.. _sphx_glr_auto_examples_example_5_charge_transfer.py: - - -Charge-transfer energy for NiO -================================================================================ -This example follows the :ref:`sphx_glr_auto_examples_example_3_AIM_XAS.py` -example and considers the same model. This time we outline how to determine -the charge transfer energy in the sense defined by Zaanen, Sawatzky, and Allen -[1]_. That is, a :math:`d^{n_d} \rightarrow d^{n_d + 1} \underline{L}` transition -in the atomic limit, after considering Coulomb interactions and crystal field. Although -this can be determined analytically in some cases, the easiest way is often just to -calculate it, as we will do here. - -.. GENERATED FROM PYTHON SOURCE LINES 13-21 - -.. code-block:: Python - - import edrixs - import numpy as np - import matplotlib.pyplot as plt - import scipy - import example_3_AIM_XAS - import importlib - _ = importlib.reload(example_3_AIM_XAS) - - - - - - - - -.. GENERATED FROM PYTHON SOURCE LINES 22-27 - -Determine eigenvectors and occupations ------------------------------------------------------------------------------- -The first step repeats what was done in -:ref:`sphx_glr_auto_examples_example_4_GS_analysis.py` but it does not apply -the hybridization between the impurity and both states. - -.. GENERATED FROM PYTHON SOURCE LINES 27-63 - -.. code-block:: Python - - - from example_3_AIM_XAS import (F0_dd, F2_dd, F4_dd, - nd, norb_d, norb_bath, v_noccu, - imp_mat, bath_level, - hyb, ext_B, trans_c2n) - ntot = 20 - umat_delectrons = edrixs.get_umat_slater('d', F0_dd, F2_dd, F4_dd) - umat = np.zeros((ntot, ntot, ntot, ntot), dtype=complex) - umat[:norb_d, :norb_d, :norb_d, :norb_d] += umat_delectrons - emat_rhb = np.zeros((ntot, ntot), dtype='complex') - emat_rhb[0:norb_d, 0:norb_d] += imp_mat - indx = np.arange(norb_d, norb_d*2) - emat_rhb[indx, indx] += bath_level[0] - tmat = np.eye(ntot, dtype=complex) - for i in range(2): - off = i * norb_d - tmat[off:off+norb_d, off:off+norb_d] = np.conj(np.transpose(trans_c2n)) - - emat_chb = edrixs.cb_op(emat_rhb, tmat) - v_orbl = 2 - sx = edrixs.get_sx(v_orbl) - sy = edrixs.get_sy(v_orbl) - sz = edrixs.get_sz(v_orbl) - zeeman = ext_B[0] * (2 * sx) + ext_B[1] * (2 * sy) + ext_B[2] * (2 * sz) - emat_chb[0:norb_d, 0:norb_d] += zeeman - basis = np.array(edrixs.get_fock_bin_by_N(ntot, v_noccu)) - H = (edrixs.build_opers(2, emat_chb, basis) - + edrixs.build_opers(4, umat, basis)) - - e, v = scipy.linalg.eigh(H) - e -= e[0] - - num_d_electrons = basis[:, :norb_d].sum(1) - alphas = np.sum(np.abs(v[num_d_electrons == 8, :])**2, axis=0) - betas = np.sum(np.abs(v[num_d_electrons == 9, :])**2, axis=0) - - - - - - - - -.. GENERATED FROM PYTHON SOURCE LINES 64-67 - -Energy to lowest energy ligand orbital ------------------------------------------------------------------------------- -Let's vizualize :math:`\alpha` and :math:`\beta`. - -.. GENERATED FROM PYTHON SOURCE LINES 67-79 - -.. code-block:: Python - - - fig, ax = plt.subplots() - - ax.plot(e, alphas, '.-', label=r'$\alpha$ $d^8L^{10}$') - ax.plot(e, betas, '.-', label=r'$\beta$ $d^9L^{9}$') - - ax.set_xlabel('Energy (eV)') - ax.set_ylabel('Population') - ax.set_title('NiO') - ax.legend() - plt.show() - - - - -.. image-sg:: /auto_examples/images/sphx_glr_example_5_charge_transfer_001.png - :alt: NiO - :srcset: /auto_examples/images/sphx_glr_example_5_charge_transfer_001.png - :class: sphx-glr-single-img - - - - - -.. GENERATED FROM PYTHON SOURCE LINES 80-82 - -One can see that the mixing between impurity and bath states has disappered -because we have turned off the hybridization. The energy required to - -.. GENERATED FROM PYTHON SOURCE LINES 82-88 - -.. code-block:: Python - - - GS_energy = min(e[np.isclose(alphas, 1)]) - lowest_energy_to_transfer_electron = min(e[np.isclose(betas, 1)]) - E_to_ligand = lowest_energy_to_transfer_electron - GS_energy - print(f"Energy to lowest energy ligand state is {E_to_ligand:.3f} eV") - - - - - -.. rst-class:: sphx-glr-script-out - - .. code-block:: none - - Energy to lowest energy ligand state is 5.517 eV - - - - -.. GENERATED FROM PYTHON SOURCE LINES 89-91 - -where we have used :code:`np.isclose` to avoid errors from finite numerical -precision. - -.. GENERATED FROM PYTHON SOURCE LINES 93-98 - -Diagonalizing by blocks ------------------------------------------------------------------------------- -When working on a problem with a large basis, one can take advantage of the -lack of hybridization and separately diagonalize the impurity and bath -states - -.. GENERATED FROM PYTHON SOURCE LINES 98-116 - -.. code-block:: Python - - - energies = [] - - for n_ligand_holes in [0, 1]: - basis_d = edrixs.get_fock_bin_by_N(10, nd + n_ligand_holes) - Hd = (edrixs.build_opers(2, emat_chb[:10, :10], basis_d) - + edrixs.build_opers(4, umat[:10, :10, :10, :10], basis_d)) - ed = scipy.linalg.eigh(Hd, eigvals_only=True, subset_by_index=[0, 0])[0] - - basis_L = edrixs.get_fock_bin_by_N(10, 10 - n_ligand_holes) - HL = (edrixs.build_opers(2, emat_chb[10:, 10:], basis_L) - + edrixs.build_opers(4, umat[10:, 10:, 10:, 10:], basis_L)) - eL = scipy.linalg.eigh(HL, eigvals_only=True, subset_by_index=[0, 0])[0] - - energies.append(ed + eL) - - print(f"Energy to lowest energy ligand state is {energies[1] - energies[0]:.3f} eV") - - - - - -.. rst-class:: sphx-glr-script-out - - .. code-block:: none - - Energy to lowest energy ligand state is 5.517 eV - - - - -.. GENERATED FROM PYTHON SOURCE LINES 117-118 - -which yields the same result. - -.. GENERATED FROM PYTHON SOURCE LINES 120-130 - -Energy splitting in ligand states ------------------------------------------------------------------------------- -The last thing to consider is that our definition of the charge transfer -energy refers to the atomic limit with all hopping terms switched off, whereas -the ligand states in the model are already split by the oxygen-oxygen hopping -term :math:`T_{pp}` as illustrated below. So the final charge transer energy -needs to account for this. - - .. image:: /_static/energy_level.png - - -.. GENERATED FROM PYTHON SOURCE LINES 130-135 - -.. code-block:: Python - - - T_pp = 1 - print(f"Charge transfer is {energies[1] - energies[0] + T_pp:.3f} eV") - - - - - - -.. rst-class:: sphx-glr-script-out - - .. code-block:: none - - Charge transfer is 6.517 eV - - - - -.. GENERATED FROM PYTHON SOURCE LINES 136-140 - -.. rubric:: Footnotes - -.. [1] J. Zaanen, G. A. Sawatzky, and J. W. Allen, - `Phys. Rev. Lett. 55, 418 (1985) `_. - - -.. rst-class:: sphx-glr-timing - - **Total running time of the script:** (0 minutes 0.478 seconds) - - -.. _sphx_glr_download_auto_examples_example_5_charge_transfer.py: - -.. only:: html - - .. container:: sphx-glr-footer sphx-glr-footer-example - - .. container:: sphx-glr-download sphx-glr-download-jupyter - - :download:`Download Jupyter notebook: example_5_charge_transfer.ipynb ` - - .. container:: sphx-glr-download sphx-glr-download-python - - :download:`Download Python source code: example_5_charge_transfer.py ` - - .. container:: sphx-glr-download sphx-glr-download-zip - - :download:`Download zipped: example_5_charge_transfer.zip ` - - -.. only:: html - - .. rst-class:: sphx-glr-signature - - `Gallery generated by Sphinx-Gallery `_ diff --git a/edrixs/_sources/auto_examples/example_5_transitions.rst.txt b/edrixs/_sources/auto_examples/example_5_transitions.rst.txt deleted file mode 100644 index d50e104e09..0000000000 --- a/edrixs/_sources/auto_examples/example_5_transitions.rst.txt +++ /dev/null @@ -1,359 +0,0 @@ - -.. DO NOT EDIT. -.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. -.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: -.. "auto_examples/example_5_transitions.py" -.. LINE NUMBERS ARE GIVEN BELOW. - -.. only:: html - - .. note:: - :class: sphx-glr-download-link-note - - :ref:`Go to the end ` - to download the full example code. - -.. rst-class:: sphx-glr-example-title - -.. _sphx_glr_auto_examples_example_5_transitions.py: - - -X-ray transitions -================================================================================ -This example explains how to calculate x-ray transition amplitudes between -specific orbital and spin states. We take the case of a cuprate which has one -hole in the :math:`d_{x^2-y^2}` orbital and a spin ordering direction along the -in-plane diagaonal direction and compute the angular dependence of spin-flip -and non-spin-flip processes. - -This case was chosen because the eigenvectors in question are simple enough -for us to write them out more-or-less by hand, so this example helps the reader -to understand what happens under the hood in more complex cases. - -Some of the code here is credited to Yao Shen who used this approach for the -analysis of a low valence nickelate material [1]_. The task performed repeats -analysis done by many researchers e.g. Luuk Ament et al [2]_ as well as -several other groups. - -.. GENERATED FROM PYTHON SOURCE LINES 20-25 - -.. code-block:: Python - - import edrixs - import numpy as np - import matplotlib.pyplot as plt - import scipy - - - - - - - - -.. GENERATED FROM PYTHON SOURCE LINES 26-42 - -Eigenvectors ------------------------------------------------------------------------------- -Let us start by determining the eigenvectors involved in the transitions. -The spin direction can be set using a vector -:math:`\vec{B}` to represent a magnetic field in terms of generalized spin -operator :math:`\tilde{\sigma}=\vec{B}\cdot\sigma` based on the Pauli matrices -:math:`\sigma`. Let's put the spin along the :math:`[1, 1, 0]` direction -and formuate the problem in the hole basis. -For one particle, we know that the Hamiltonian will be diagonal in the real -harmonic basis. -We can generate the required eigenvectors by making a diagonal -matrix, transforming it to the required -complex harmonic basis (as is standard for EDRIXS) and diagonalizing it. -As long as the crystal field splitting is much larger than the magnetic -field, the eigenvectors will be independent of the exact values of both -these parameters. - -.. GENERATED FROM PYTHON SOURCE LINES 42-56 - -.. code-block:: Python - - - B = 1e-3*np.array([1, 1, 0]) - cf_splitting = 1e3 - zeeman = sum(s*b for s, b in zip(edrixs.get_spin_momentum(2), B)) - dd_levels = np.array([energy for dd_level in cf_splitting*np.arange(5) - for energy in [dd_level]*2], dtype=complex) - emat_rhb = np.diag(dd_levels) - emat = edrixs.cb_op(emat_rhb, edrixs.tmat_r2c('d', ispin=True)) + zeeman - _, eigenvectors = np.linalg.eigh(emat) - - def get_eigenvector(orbital_index, spin_index): - return eigenvectors[:, 2*orbital_index + spin_index] - - - - - - - - - -.. GENERATED FROM PYTHON SOURCE LINES 57-63 - -Let's examine the :math:`d_{x^2-y^2}` orbital first. Recall from the -:ref:`sphx_glr_auto_examples_example_1_crystal_field.py` -example that edrixs uses the standard orbital order of -:math:`d_{3z^2-r^2}, d_{xz}, d_{yz}, d_{x^2-y^2}, d_{xy}`. So we want -:code:`orbital_index = 3` element. Using this, we can build spin-up and -down -eigenvectors. - -.. GENERATED FROM PYTHON SOURCE LINES 63-68 - -.. code-block:: Python - - orbital_index = 3 - - groundstate_vector = get_eigenvector(orbital_index, 0) - excitedstate_vector = get_eigenvector(orbital_index, 1) - - - - - - - - -.. GENERATED FROM PYTHON SOURCE LINES 69-96 - -Transition operators and scattering matrix ------------------------------------------------------------------------------- -Here we are considering the :math:`L_3`-edge. This means -a :math:`2p_{3/2} \rightarrow 3d` -absoprtion transition and a :math:`2p_{3/2} \rightarrow 3d` -emission transition. We can read the relevant matrix from the edrixs database, -keeping in mind that there are in fact three operations for -:math:`x, y,` & :math:`z` directions. Note that edrixs provides operators -in electron notation. If we define :math:`D` as the transition operator in -electron language, :math:`D^\dagger` is the operator in the hole language -we are using for this example. -The angular dependence of a RIXS transition can be conveniently described -using the scattering matrix, which is a :math:`3\times3` element object that -specifies the transition amplitude for each incoming and outgoing x-ray -polarization. Correspondingly, we have - - .. math:: - \begin{equation} - \mathcal{F}=\sum_n\langle f|D|n\rangle\langle n|D^{\dagger}|g\rangle - \end{equation}. - -In matrix form this is - - .. math:: - \begin{equation} - \mathcal{F}(m,n)=\{f^{\dagger} \cdot D(m)\} \cdot \{D^{\dagger}(n) \cdot g\} - \end{equation}. - -.. GENERATED FROM PYTHON SOURCE LINES 96-107 - -.. code-block:: Python - - - D_Tmat = edrixs.get_trans_oper('dp32') - - def get_F(vector_i, vector_f): - F = np.zeros((3, 3), dtype=complex) - for i in range(3): - for j in range(3): - F[i, j] = np.dot(np.dot(np.conj(vector_f.T), D_Tmat[i]), - np.dot(np.conj(D_Tmat[j].T), vector_i)) - return F - - - - - - - - -.. GENERATED FROM PYTHON SOURCE LINES 108-111 - -Using this function, we can obtain non-spin-flip (NSF) and spin-flip (SF) -scattering matrices by choosing whether we return to the ground state or -whether we access the excited state with the spin flipped. - -.. GENERATED FROM PYTHON SOURCE LINES 111-114 - -.. code-block:: Python - - F_NSF = get_F(groundstate_vector, groundstate_vector) - F_SF = get_F(groundstate_vector, excitedstate_vector) - - - - - - - - -.. GENERATED FROM PYTHON SOURCE LINES 115-122 - -Angular dependence ------------------------------------------------------------------------------- -Let's consider the common case of fixing the total scattering angle at -:code:`two_theta = 90` and choosing a series of incident angles :code:`thins`. -Since the detector does not resolve polarization, we need to add both outgoing -polarizations. It is then convenient to use function :func:`.dipole_polvec_rixs` -to obtain the incoming and outgoing polarization vectors. - -.. GENERATED FROM PYTHON SOURCE LINES 122-137 - -.. code-block:: Python - - thins = np.linspace(0, 90) - two_theta = 90 - phi = 0 - - - def get_I(thin, alpha, F): - intensity = 0 - for beta in [0, np.pi/2]: - thout = two_theta - thin - ei, ef = edrixs.dipole_polvec_rixs(thin*np.pi/180, thout*np.pi/180, - phi*np.pi/180, alpha, beta) - intensity += np.abs(np.dot(ef, np.dot(F, ei)))**2 - return intensity - - - - - - - - - -.. GENERATED FROM PYTHON SOURCE LINES 138-143 - -Plot ------------------------------------------------------------------------------- -We now run through a few configurations specified in terms of incoming -polarization angle :math:`\alpha` (defined in radians w.r.t. the scattering -plane), :math:`F`, plotting label, and plotting color. - -.. GENERATED FROM PYTHON SOURCE LINES 143-159 - -.. code-block:: Python - - fig, ax = plt.subplots() - - config = [[0, F_NSF, r'$\pi$ NSF', 'C0'], - [np.pi/2, F_NSF, r'$\sigma$ NSF', 'C1'], - [0, F_SF, r'$\pi$ SF', 'C2'], - [np.pi/2, F_SF, r'$\sigma$ SF', 'C3']] - - for alpha, F, label, color in config: - Is = np.array([get_I(thin, alpha, F) for thin in thins]) - ax.plot(thins, Is, label=label, color=color) - - ax.legend() - ax.set_xlabel(r'Theta ($^\circ$)') - ax.set_ylabel('Relative intensity') - plt.show() - - - - -.. image-sg:: /auto_examples/images/sphx_glr_example_5_transitions_001.png - :alt: example 5 transitions - :srcset: /auto_examples/images/sphx_glr_example_5_transitions_001.png - :class: sphx-glr-single-img - - - - - -.. GENERATED FROM PYTHON SOURCE LINES 160-164 - -Run through orbitals ------------------------------------------------------------------------------- -For completeness, let's look at transitions from :math:`x^2-y^2` to all other -orbitals. - -.. GENERATED FROM PYTHON SOURCE LINES 164-191 - -.. code-block:: Python - - fig, axs = plt.subplots(5, 1, figsize=(7, 7), - sharex=True, sharey=True) - - orbitals = ['$d_{3z^2-r^2}$', '$d_{xz}$', '$d_{yz}$', - '$d_{x^2-y^2}$', '$d_{xy}$'] - orbital_order = [4, 1, 2, 0, 3] - - plot_index = 0 - for ax, orbital_index in zip(axs, orbital_order): - for spin_index, spin_label in zip([0, 1], ['NSF', 'SF']): - excitedstate_vector = get_eigenvector(orbital_index, spin_index) - F = get_F(groundstate_vector, excitedstate_vector) - for alpha, pol_label in zip([0, np.pi/2], [r'$\pi$', r'$\sigma$']): - Is = np.array([get_I(thin, alpha, F) for thin in thins]) - ax.plot(thins, Is*10, label=f'{pol_label} {spin_label}', - color=f'C{plot_index%4}') - plot_index += 1 - ax.legend(title=orbitals[orbital_index], bbox_to_anchor=(1.1, 1), - loc="upper left", fontsize=8) - - - axs[-1].set_xlabel(r'$\theta$ ($^\circ$)') - axs[2].set_ylabel('Scattering intensity') - - fig.subplots_adjust(hspace=0, left=.3, right=.6) - plt.show() - - - - -.. image-sg:: /auto_examples/images/sphx_glr_example_5_transitions_002.png - :alt: example 5 transitions - :srcset: /auto_examples/images/sphx_glr_example_5_transitions_002.png - :class: sphx-glr-single-img - - - - - -.. GENERATED FROM PYTHON SOURCE LINES 192-198 - -.. rubric:: Footnotes - -.. [1] Yao Shen et al., - `arXiv:2110.08937 (2022) `_. -.. [2] Luuk J. P. Ament et al., - `Phys. Rev. Lett. 103, 117003 (2009) `_ - - -.. rst-class:: sphx-glr-timing - - **Total running time of the script:** (0 minutes 0.559 seconds) - - -.. _sphx_glr_download_auto_examples_example_5_transitions.py: - -.. only:: html - - .. container:: sphx-glr-footer sphx-glr-footer-example - - .. container:: sphx-glr-download sphx-glr-download-jupyter - - :download:`Download Jupyter notebook: example_5_transitions.ipynb ` - - .. container:: sphx-glr-download sphx-glr-download-python - - :download:`Download Python source code: example_5_transitions.py ` - - .. container:: sphx-glr-download sphx-glr-download-zip - - :download:`Download zipped: example_5_transitions.zip ` - - -.. only:: html - - .. rst-class:: sphx-glr-signature - - `Gallery generated by Sphinx-Gallery `_ diff --git a/edrixs/_sources/auto_examples/example_6_Coulomb.rst.txt b/edrixs/_sources/auto_examples/example_6_Coulomb.rst.txt deleted file mode 100644 index 0242ad43fb..0000000000 --- a/edrixs/_sources/auto_examples/example_6_Coulomb.rst.txt +++ /dev/null @@ -1,649 +0,0 @@ - -.. DO NOT EDIT. -.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. -.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: -.. "auto_examples/example_6_Coulomb.py" -.. LINE NUMBERS ARE GIVEN BELOW. - -.. only:: html - - .. note:: - :class: sphx-glr-download-link-note - - Click :ref:`here ` - to download the full example code - -.. rst-class:: sphx-glr-example-title - -.. _sphx_glr_auto_examples_example_6_Coulomb.py: - - -Coulomb interactions -===================================== -In this example we provide more details on how Coulomb interactions are -implemented in multiplet calculations and EDRIXS in particular. We aim -to clarify the form of the matrices, how they are parametrized, -and how the breaking of spherical symmetry can switch on additional elements -that one might not anticipate. Our example is based on a :math:`d` atomic shell. - -.. GENERATED FROM PYTHON SOURCE LINES 13-50 - -Create matrix ------------------------------------------------------------------------------- -The Coulomb interaction between two particles can be written as - - .. math:: - \begin{equation} - \hat{H} = \frac{1}{2} - \int d\mathbf{r} \int d\mathbf{r}^\prime - \Sigma_{\sigma, \sigma^\prime} - |\hat{\psi}^\sigma(\mathbf{r})|^2 \frac{e^2}{R} - |\hat{\psi}^{\sigma^\prime}(\mathbf{r})|^2, - \end{equation} - -where :math:`\hat{\psi}^\sigma(\mathbf{r})` is the electron wavefunction, with -spin :math:`\sigma`, and :math:`R=|r-r^\prime|` is the electron separation. -Solving our problem in this form is difficult due to the need to symmeterize -the wavefunction to follow fermionic statistics. -Using second quantization, we can use operators to impose the required -particle exchange statistics and write the equation in terms of -a tensor :math:`U` - - .. math:: - \begin{equation} - \hat{H} = \sum_{\alpha,\beta,\gamma,\delta,\sigma,\sigma^\prime} - U_{\alpha\sigma,\beta\sigma^\prime,\gamma\sigma^\prime,\delta\sigma} - \hat{f}^{\dagger}_{\alpha\sigma} - \hat{f}^{\dagger}_{\beta\sigma^\prime} - \hat{f}_{t\sigma^\prime}\hat{f}_{\delta\sigma}, - \end{equation} - -where :math:`\alpha`, :math:`\beta`, :math:`\gamma`, :math:`\delta` are -orbital indices and :math:`\hat{f}^{\dagger}` -(:math:`\hat{f}`) are the creation (anihilation) operators. -For a :math:`d`-electron system, we have :math:`10` distinct spin-orbitals -(:math:`5` orbitals each with :math:`2` spins), which makes matrix the -:math:`10\times10\times10\times10` in total size. -In EDRIXS the matrix can be created as follows: - -.. GENERATED FROM PYTHON SOURCE LINES 50-58 - -.. code-block:: default - - import edrixs - import numpy as np - import scipy - import matplotlib.pyplot as plt - import itertools - - F0, F2, F4 = 6.94, 14.7, 4.41 - umat_chb = edrixs.get_umat_slater('d', F0, F2, F4) - - - - - - - -.. GENERATED FROM PYTHON SOURCE LINES 59-61 - -We stored this under variable :code:`umat_chb` where "cbh" stands for -complex harmonic basis, which is the default basis in EDRIXS. - -.. GENERATED FROM PYTHON SOURCE LINES 63-92 - -Parameterizing interactions ------------------------------------------------------------------------------- -EDRIXS parameterizes the interactions in :math:`U` via Slater integral -parameters :math:`F^{k}`. These relate to integrals of various spherical -Harmonics as well as Clebsch-Gordon coefficients, Gaunt coefficients, -and Wigner 3J symbols. Textbooks such as [1]_ can be used for further -reference. If you are interested in the details of how -EDRIXS does this (and you probably aren't) function :func:`.umat_slater`, -constructs the required matrix via Gaunt coeficents from -:func:`.get_gaunt`. Two alternative parameterizations are common. -The first are the Racah parameters, which are - - .. math:: - \begin{eqnarray} - A &=& F^0 - \frac{49}{441} F^4 \\ - B &=& \frac{1}{49}F^2 - \frac{5}{441}F^4 \\ - C &=& \frac{35}{441}F^4. - \end{eqnarray} - -or an alternative form for the Slater integrals - - .. math:: - \begin{eqnarray} - F_0 &=& F^0 \\ - F_2 &=& \frac{1}{49}F^2 \\ - F_4 &=& \frac{1}{441}F^4, - \end{eqnarray} - -which involves different normalization parameters. - -.. GENERATED FROM PYTHON SOURCE LINES 94-101 - -Basis transform ------------------------------------------------------------------------------- -If we want to use the real harmonic basis, we can use a tensor -transformation, which imposes the following orbital order -:math:`3z^2-r^2, xz, yz, x^2-y^2, xy`, each of which involves -:math:`\uparrow, \downarrow` spin pairs. Let's perform this transformation and -store a list of these orbitals. - -.. GENERATED FROM PYTHON SOURCE LINES 101-104 - -.. code-block:: default - - umat = edrixs.transform_utensor(umat_chb, edrixs.tmat_c2r('d', True)) - orbitals = ['3z^2-r^2', 'xz', 'yz', 'x^2-y^2', 'xy'] - - - - - - - - -.. GENERATED FROM PYTHON SOURCE LINES 105-158 - -Interactions ------------------------------------------------------------------------------- -Tensor :math:`U` is a series of matrix -elements - - .. math:: - \begin{equation} - \langle\psi_{\gamma,\delta}^{\bar{\sigma},\bar{\sigma}^\prime} - |\hat{H}| - \psi_{\alpha,\beta}^{\sigma,\sigma^\prime}\rangle - \end{equation} - -the combination of which defines the energetic cost of pairwise -electron-electron interactions between states :math:`\alpha,\sigma` -and :math:`\beta,\sigma^\prime`. In EDRIXS we follow the convention of -summing over all orbital pairs. Some other texts count each pair of -indices only once. The matrix elements here will consequently -be half the magnitude of those in other references. -We can express the interactions in terms of -the orbitals involved. It is common to distinguish "direct Coulomb" and -"exchange" interactions. The former come from electrons in the same orbital -and the later involve swapping orbital labels for electrons. We will use -:math:`U_0` and :math:`J` as a shorthand for distinguishing these. - -Before we describe the different types of interactions, we note that since -the Coulomb interaction is real, and due to the spin symmmetry properties -of the process :math:`U` always obeys - - .. math:: - \begin{equation} - U_{\alpha\sigma,\beta\sigma^\prime,\gamma\sigma^\prime,\delta\sigma} = - U_{\beta\sigma,\alpha\sigma^\prime,\delta\sigma^\prime,\gamma\sigma} = - U_{\delta\sigma,\gamma\sigma^\prime,\beta\sigma^\prime,\alpha\sigma} = - U_{\gamma\sigma,\delta\sigma^\prime,\alpha\sigma^\prime,\beta\sigma}. - \end{equation} - - -1. Intra orbital -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -The direct Coulomb energy cost to double-occupy an orbital comes from terms -like :math:`U_{\alpha\sigma,\alpha\bar\sigma,\alpha\bar\sigma,\alpha\sigma}`. -In this notation, we use :math:`\sigma^\prime` to denote that the matrix -element is summed over all pairs and :math:`\bar{\sigma}` to denote sums -over all opposite spin pairs. Due to rotational symmetry, all these -elements are the same and equal to - - .. math:: - \begin{eqnarray} - U_0 &=& \frac{A}{2} + 2B + \frac{3C}{2}\\ - &=& \frac{F_0}{2} + 2F_2 + 18F_4 - \end{eqnarray} - -Let's print these to demonstrate where these live in the array - -.. GENERATED FROM PYTHON SOURCE LINES 158-162 - -.. code-block:: default - - for i in range(0, 5): - val = umat[i*2, i*2 + 1, i*2 + 1, i*2].real - print(f"{orbitals[i]:<8} \t {val:.3f}") - - - - - -.. rst-class:: sphx-glr-script-out - - Out: - - .. code-block:: none - - 3z^2-r^2 4.250 - xz 4.250 - yz 4.250 - x^2-y^2 4.250 - xy 4.250 - - - - -.. GENERATED FROM PYTHON SOURCE LINES 163-170 - -2. Inter orbital Coulomb interactions -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -Direct Coulomb repulsion between different orbitals depends on terms like -:math:`U_{\alpha\sigma,\beta\sigma^\prime,\beta\sigma^\prime,\alpha\sigma}`. -Expresions for these parameters are provided in column :math:`U` in -:ref:`table_2_orbital`. We can print the values from :code:`umat` -like this: - -.. GENERATED FROM PYTHON SOURCE LINES 170-174 - -.. code-block:: default - - for i, j in itertools.combinations(range(5), 2): - val = umat[i*2, j*2 + 1, j*2 + 1, i*2].real - print(f"{orbitals[i]:<8} \t {orbitals[j]:<8} \t {val:.3f}") - - - - - -.. rst-class:: sphx-glr-script-out - - Out: - - .. code-block:: none - - 3z^2-r^2 xz 3.650 - 3z^2-r^2 yz 3.650 - 3z^2-r^2 x^2-y^2 2.900 - 3z^2-r^2 xy 2.900 - xz yz 3.150 - xz x^2-y^2 3.150 - xz xy 3.150 - yz x^2-y^2 3.150 - yz xy 3.150 - x^2-y^2 xy 3.900 - - - - -.. GENERATED FROM PYTHON SOURCE LINES 175-181 - -3. Inter-orbital exchange interactions -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -Exchange terms exist with the form -:math:`U_{\alpha\sigma,\beta\sigma^\prime,\alpha\sigma^\prime,\beta\sigma}`. -Expresions for these parameters are provided in column :math:`J` of -:ref:`table_2_orbital`. These come from terms like this in the matrix: - -.. GENERATED FROM PYTHON SOURCE LINES 181-185 - -.. code-block:: default - - for i, j in itertools.combinations(range(5), 2): - val = umat[i*2, j*2 + 1, i*2 + 1, j*2].real - print(f"{orbitals[i]:<8} \t {orbitals[j]:<8} \t {val:.3f}") - - - - - -.. rst-class:: sphx-glr-script-out - - Out: - - .. code-block:: none - - 3z^2-r^2 xz 0.300 - 3z^2-r^2 yz 0.300 - 3z^2-r^2 x^2-y^2 0.675 - 3z^2-r^2 xy 0.675 - xz yz 0.550 - xz x^2-y^2 0.550 - xz xy 0.550 - yz x^2-y^2 0.550 - yz xy 0.550 - x^2-y^2 xy 0.175 - - - - -.. GENERATED FROM PYTHON SOURCE LINES 186-193 - -4. Pair hopping term -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -Terms that swap pairs of electrons exist as -:math:`(1-\delta_{\sigma\sigma'})U_{\alpha\sigma,\alpha\bar\sigma,\beta\bar\sigma,\beta\sigma}` -and depend on exchange interactions column :math:`J` from -:ref:`table_2_orbital` -and here in the matrix. - -.. GENERATED FROM PYTHON SOURCE LINES 193-197 - -.. code-block:: default - - for i, j in itertools.combinations(range(5), 2): - val = umat[i*2, i*2 + 1, j*2 + 1, j*2].real - print(f"{orbitals[i]:<8} \t {orbitals[j]:<8} \t {val:.3f}") - - - - - -.. rst-class:: sphx-glr-script-out - - Out: - - .. code-block:: none - - 3z^2-r^2 xz 0.300 - 3z^2-r^2 yz 0.300 - 3z^2-r^2 x^2-y^2 0.675 - 3z^2-r^2 xy 0.675 - xz yz 0.550 - xz x^2-y^2 0.550 - xz xy 0.550 - yz x^2-y^2 0.550 - yz xy 0.550 - x^2-y^2 xy 0.175 - - - - -.. GENERATED FROM PYTHON SOURCE LINES 198-215 - -5. Three orbital -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -Another set of terms that one might not immediately anticipate involve three -orbitals like - - .. math:: - \begin{equation} - U_{\alpha\sigma, \gamma\sigma', \beta\sigma', \gamma\sigma} \\ - U_{\alpha\sigma, \gamma\sigma', \gamma\sigma', \beta\sigma} \\ - (1-\delta_{\sigma\sigma'}) - U_{\alpha\sigma, \beta\sigma', \gamma\sigma', \gamma\sigma} - \end{equation} - -for :math:`\alpha=3z^2-r^2, \beta=x^2-y^2, \gamma=xz/yz`. -These are needed to maintain the rotational symmetry of the interations. -See :ref:`table_3_orbital` for the expressions. We can print some of -these via: - -.. GENERATED FROM PYTHON SOURCE LINES 215-227 - -.. code-block:: default - - ijkl = [[0, 1, 3, 1], - [0, 2, 3, 2], - [1, 0, 3, 1], - [1, 1, 3, 0], - [2, 0, 3, 2], - [2, 2, 3, 0]] - - for i, j, k, l in ijkl: - val = umat[i*2, j*2 + 1, k*2 + 1, l*2].real - print(f"{orbitals[i]:<8} \t {orbitals[j]:<8} \t" - f"{orbitals[k]:<8} \t {orbitals[l]:<8} \t {val:.3f}") - - - - - -.. rst-class:: sphx-glr-script-out - - Out: - - .. code-block:: none - - 3z^2-r^2 xz x^2-y^2 xz 0.217 - 3z^2-r^2 yz x^2-y^2 yz -0.217 - xz 3z^2-r^2 x^2-y^2 xz -0.433 - xz xz x^2-y^2 3z^2-r^2 0.217 - yz 3z^2-r^2 x^2-y^2 yz 0.433 - yz yz x^2-y^2 3z^2-r^2 -0.217 - - - - -.. GENERATED FROM PYTHON SOURCE LINES 228-233 - -6. Four orbital -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -Futher multi-orbital terms include -:math:`U_{\alpha\sigma,\beta\sigma^\prime,\gamma\sigma^\prime,\delta\sigma}`. -We can find these here in the matrix: - -.. GENERATED FROM PYTHON SOURCE LINES 233-249 - -.. code-block:: default - - ijkl = [[0, 1, 2, 4], - [0, 1, 4, 2], - [0, 2, 1, 4], - [0, 2, 4, 1], - [0, 4, 1, 2], - [0, 4, 2, 1], - [3, 1, 4, 2], - [3, 2, 4, 1], - [3, 4, 1, 2], - [3, 4, 2, 1]] - - for i, j, k, l in ijkl: - val = umat[i*2, j*2 + 1, k*2 + 1, l*2].real - print(f"{orbitals[i]:<8} \t {orbitals[j]:<8} \t {orbitals[k]:<8}" - f"\t {orbitals[l]:<8} \t {val:.3f}") - - - - - -.. rst-class:: sphx-glr-script-out - - Out: - - .. code-block:: none - - 3z^2-r^2 xz yz xy -0.433 - 3z^2-r^2 xz xy yz 0.217 - 3z^2-r^2 yz xz xy -0.433 - 3z^2-r^2 yz xy xz 0.217 - 3z^2-r^2 xy xz yz 0.217 - 3z^2-r^2 xy yz xz 0.217 - x^2-y^2 xz xy yz -0.375 - x^2-y^2 yz xy xz 0.375 - x^2-y^2 xy xz yz -0.375 - x^2-y^2 xy yz xz 0.375 - - - - -.. GENERATED FROM PYTHON SOURCE LINES 250-256 - -Effects of multi-orbital terms ------------------------------------------------------------------------------- -To test the effects of the multi-orbital terms, let's plot the eigenenergy -spectra with and without multi-orbital terms switched on for system with and -without a cubic crystal field. We will use a :math:`d`-shell with two -electrons. - -.. GENERATED FROM PYTHON SOURCE LINES 256-292 - -.. code-block:: default - - ten_dqs = [0, 2, 4, 12] - - def diagonalize(ten_dq, umat): - emat = edrixs.cb_op(edrixs.cf_cubic_d(ten_dq), - edrixs.tmat_c2r('d', ispin=True)) - H = (edrixs.build_opers(4, umat, basis) - + edrixs.build_opers(2, emat, basis)) - e, v = scipy.linalg.eigh(H) - return e - e.min() - - basis = edrixs.get_fock_bin_by_N(10, 2) - umat_no_multiorbital = np.copy(umat) - B = F2/49 - 5*F4/441 - for val in [np.sqrt(3)*B/2, np.sqrt(3)*B, 3*B/2]: - umat_no_multiorbital[(np.abs(umat)- val) < 1e-6] = 0 - - fig, axs = plt.subplots(1, len(ten_dqs), figsize=(8, 3)) - - for cind, (ax, ten_dq) in enumerate(zip(axs, ten_dqs)): - ax.hlines(diagonalize(ten_dq, umat), xmin=0, xmax=1, - label='on', color=f'C{cind}') - ax.hlines(diagonalize(ten_dq, umat_no_multiorbital), - xmin=1.5, xmax=2.5, - label='off', - linestyle=':', color=f'C{cind}') - ax.set_title(f"$10D_q={ten_dq}$") - ax.set_ylim([-.5, 20]) - ax.set_xticks([]) - ax.legend() - - fig.suptitle("Eigenvalues with 3&4-orbital effects on/off") - fig.subplots_adjust(wspace=.3) - axs[0].set_ylabel('Eigenvalues (eV)') - fig.subplots_adjust(top=.8) - plt.show() - - - - -.. image-sg:: /auto_examples/images/sphx_glr_example_6_Coulomb_001.png - :alt: Eigenvalues with 3&4-orbital effects on/off, $10D_q=0$, $10D_q=2$, $10D_q=4$, $10D_q=12$ - :srcset: /auto_examples/images/sphx_glr_example_6_Coulomb_001.png - :class: sphx-glr-single-img - - - - - -.. GENERATED FROM PYTHON SOURCE LINES 293-382 - -On the left of the plot Coulomb interactions in spherical symmetry cause -substantial mxing between :math:`t_{2g}` and :math:`e_{g}` orbitals in the -eigenstates and 3 & 4 orbital orbital terms are crucial for obtaining the -the right eigenenergies. As :math:`10D_q` get large, this mixing is switched -off and the spectra start to become independent of whether the 3 & 4 orbital -orbital terms are included or not. - - - -.. _table_2_orbital: -.. table:: Table of 2 orbital interactions - - +-----------------------------+------------------+-----------------------+----------------+--------------------+ - |Orbitals :math:`\alpha,\beta`|:math:`U_0` Racah | :math:`U_0` Slater |:math:`J` Racah |:math:`J` Slater | - +=============================+==================+=======================+================+====================+ - |:math:`3z^2-r^2, xz` |:math:`A/2+B+C/2` |:math:`F_0/2+F_2-12F_4`| :math:`B/2+C/2`|:math:`F_2/2+15F_4` | - +-----------------------------+------------------+-----------------------+----------------+--------------------+ - |:math:`3z^2-r^2, yz` |:math:`A/2+B+C/2` |:math:`F_0/2+F_2-12F_4`| :math:`B/2+C/2`|:math:`F_2/2+15F_4` | - +-----------------------------+------------------+-----------------------+----------------+--------------------+ - |:math:`3z^2-r^2, x^2-y^2` |:math:`A/2-2B+C/2`|:math:`F_0/2-2F_2+3F_4`|:math:`2B+C/2` |:math:`2F_2+15F_4/2`| - +-----------------------------+------------------+-----------------------+----------------+--------------------+ - |:math:`3z^2-r^2, xy` |:math:`A/2-2B+C/2`|:math:`F_0/2-2F_2+3F_4`|:math:`2B+C/2` |:math:`2F_2+15F_4/2`| - +-----------------------------+------------------+-----------------------+----------------+--------------------+ - |:math:`xz, yz` |:math:`A/2-B+C/2` |:math:`F_0/2-F_2-12F_4`|:math:`3B/2+C/2`|:math:`3F_2/2+10F_4`| - +-----------------------------+------------------+-----------------------+----------------+--------------------+ - |:math:`xz, x^2-y^2` |:math:`A/2-B+C/2` |:math:`F_0/2-F_2-12F_4`|:math:`3B/2+C/2`|:math:`3F_2/2+10F_4`| - +-----------------------------+------------------+-----------------------+----------------+--------------------+ - |:math:`xz, xy` |:math:`A/2-B+C/2` |:math:`F_0/2-F_2-12F_4`|:math:`3B/2+C/2`|:math:`3F_2/2+10F_4`| - +-----------------------------+------------------+-----------------------+----------------+--------------------+ - |:math:`yz, x^2-y^2` |:math:`A/2-B+C/2` |:math:`F_0/2-F_2-12F_4`|:math:`3B/2+C/2`|:math:`3F_2/2+10F_4`| - +-----------------------------+------------------+-----------------------+----------------+--------------------+ - |:math:`yz, xy` |:math:`A/2-B+C/2` |:math:`F_0/2-F_2-12F_4`|:math:`3B/2+C/2`|:math:`3F_2/2+10F_4`| - +-----------------------------+------------------+-----------------------+----------------+--------------------+ - |:math:`x^2-y^2, xy` |:math:`A/2+2B+C/2`|:math:`F_0+4F_2-34F_4` | :math:`C/2` |:math:`35F_4/2` | - +-----------------------------+------------------+-----------------------+----------------+--------------------+ - - -.. _table_3_orbital: -.. table:: Table of 3 orbital interactions - - +-----------------------------+-------------+----------------------------------------------------+-----------------------------------------------------+ - |Orbitals :math:`\alpha,\beta,\gamma,\delta`|:math:`\langle\alpha\beta|\gamma\delta\rangle` Racah|:math:`\langle\alpha\beta|\gamma\delta\rangle` Slater| - +=============================+=============+====================================================+=====================================================+ - |:math:`3z^2-r^2, xz, x^2-y^2, xz` | :math:`\sqrt{3}B/2` | :math:`\sqrt{3}F_2/2-5\sqrt{3}F_4/2` | - +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ - |:math:`3z^2-r^2, yz, x^2-y^2, yz` | :math:`-\sqrt{3}B/2` | :math:`-\sqrt{3}F_2/2+5\sqrt{3}F_4/2` | - +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ - |:math:`xz, 3z^2-r^2, x^2-y^2, xz` | :math:`-\sqrt{3}B` | :math:`-\sqrt{3}F_2+5\sqrt{3}F_4` | - +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ - |:math:`xz, xz, x^2-y^2, 3z^2-r^2` | :math:`\sqrt{3}B/2` | :math:`\sqrt{3}F_2/2-5\sqrt{3}F_4/2` | - +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ - |:math:`yz, 3z^2-r^2, x^2-y^2, yz` | :math:`\sqrt{3}B` | :math:`\sqrt{3}F_2-5\sqrt{3}F_4` | - +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ - |:math:`yz, yz, x^2-y^2, 3z^2-r^2` | :math:`-\sqrt{3}B/2` | :math:`-\sqrt{3}F_2/2+5\sqrt{3}F_4/2` | - +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ - - -.. _table_4_orbital: -.. table:: Table of 4 orbital interactions - - +-----------------------------+-------------+----------------------------------------------------+-----------------------------------------------------+ - |Orbitals :math:`\alpha,\beta,\gamma,\delta`|:math:`\langle\alpha\beta|\gamma\delta\rangle` Racah|:math:`\langle\alpha\beta|\gamma\delta\rangle` Slater| - +=============================+=============+====================================================+=====================================================+ - |:math:`3z^2-r^2, xz, yz, xy` | :math:`-\sqrt{3}B` | :math:`-\sqrt{3}F_2+5\sqrt{3}F_4` | - +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ - |:math:`3z^2-r^2, xz, xy, yz` | :math:`\sqrt{3}B/2` | :math:`\sqrt{3}F_2/2-5\sqrt{3}F_4/2` | - +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ - |:math:`3z^2-r^2, yz, xz, xy` | :math:`-\sqrt{3}B` | :math:`-\sqrt{3}F_2+5\sqrt{3}F_4` | - +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ - |:math:`3z^2-r^2, yz, xy, xz` | :math:`\sqrt{3}B/2` | :math:`\sqrt{3}F_2/2-5\sqrt{3}F_4/2` | - +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ - |:math:`3z^2-r^2, xy, xz, yz` | :math:`\sqrt{3}B/2` | :math:`\sqrt{3}F_2/2-5\sqrt{3}F_4/2` | - +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ - |:math:`3z^2-r^2, xy, yz, xz` | :math:`\sqrt{3}B/2` | :math:`\sqrt{3}F_2/2-5\sqrt{3}F_4/2` | - +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ - |:math:`x^2-y^2 , xz, xy, yz` | :math:`-3B/2` | :math:`-3F_2/2+15F_4/2` | - +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ - |:math:`x^2-y^2 , yz, xy, xz` | :math:`3B/2` | :math:`3F_2/2-15F_4/2` | - +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ - |:math:`x^2-y^2 , xy, xz, yz` | :math:`-3B/2` | :math:`-3F_2/2+15F_4/2` | - +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ - |:math:`x^2-y^2 , xy, yz, xz` | :math:`3B/2` | :math:`3F_2/2-15F_4/2` | - +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ - - -.. rubric:: Footnotes - -.. [1] MSugano S, Tanabe Y and Kamimura H. 1970. Multiplets of - Transition-Metal Ions in Crystals. Academic Press, New York and London. - - -.. rst-class:: sphx-glr-timing - - **Total running time of the script:** ( 0 minutes 0.694 seconds) - - -.. _sphx_glr_download_auto_examples_example_6_Coulomb.py: - - -.. only :: html - - .. container:: sphx-glr-footer - :class: sphx-glr-footer-example - - - - .. container:: sphx-glr-download sphx-glr-download-python - - :download:`Download Python source code: example_6_Coulomb.py ` - - - - .. container:: sphx-glr-download sphx-glr-download-jupyter - - :download:`Download Jupyter notebook: example_6_Coulomb.ipynb ` - - -.. only:: html - - .. rst-class:: sphx-glr-signature - - `Gallery generated by Sphinx-Gallery `_ diff --git a/edrixs/_sources/auto_examples/example_6_Hubbard_dimer.rst.txt b/edrixs/_sources/auto_examples/example_6_Hubbard_dimer.rst.txt deleted file mode 100644 index e39b68637f..0000000000 --- a/edrixs/_sources/auto_examples/example_6_Hubbard_dimer.rst.txt +++ /dev/null @@ -1,363 +0,0 @@ - -.. DO NOT EDIT. -.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. -.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: -.. "auto_examples/example_6_Hubbard_dimer.py" -.. LINE NUMBERS ARE GIVEN BELOW. - -.. only:: html - - .. note:: - :class: sphx-glr-download-link-note - - :ref:`Go to the end ` - to download the full example code. - -.. rst-class:: sphx-glr-example-title - -.. _sphx_glr_auto_examples_example_6_Hubbard_dimer.py: - - -Hubbard Dimer -===================================== -This exercise will demonstrate how to handle hopping and multi-site problems within -edrixs using the example of a Hubbard dimer. We want to solve the equation - - .. math:: - \begin{equation} - \hat{H} = \sum_{i,j} \sum_{\sigma} t_{i,j} \hat{f}^{\dagger}_{i,\sigma} \hat{f}_{j, \sigma} - + U \sum_{i} \hat{n}_{i,\uparrow}\hat{n}_{i,\downarrow}, - \end{equation} - -which involves two sites labeled with indices :math:`i` or :math:`j` with two -electrons of spin :math:`\sigma\in{\uparrow,\downarrow}`. :math:`t_{i,j}` -is the hopping between sites, :math:`\hat{f}^{\dagger}_{i,\sigma}` is the -creation operators, and -:math:`\hat{n}^{\dagger}_{i,\sigma}=\hat{f}^{\dagger}_{i,\sigma}\hat{f}_{i,\sigma}` -is the number operator. The main task is to represent this Hamiltonian and -the related spin operator using the EDRIXS two-fermion and four-fermion form -where :math:`\alpha,\beta,\delta,\gamma` are the indices of the single -particle basis. - - .. math:: - \begin{equation} - \hat{H} = \sum_{\alpha,\beta} t_{\alpha,\beta} \hat{f}^{\dagger}_{\alpha} \hat{f}_{\beta} - + \sum_{\alpha,\beta,\gamma,\delta} U_{\alpha,\beta,\gamma,\delta} - \hat{f}^{\dagger}_{\alpha}\hat{f}^{\dagger}_{\beta}\hat{f}_{\gamma}\hat{f}_{\delta}. - \end{equation} - -.. GENERATED FROM PYTHON SOURCE LINES 34-39 - -Initialize matrices ------------------------------------------------------------------------------- -We start by noting that each of the two sites is like an :math:`l=0` -:math:`s`-orbital with two spin-orbitals each. We will include -two electron occupation and build the Fock basis. - -.. GENERATED FROM PYTHON SOURCE LINES 39-52 - -.. code-block:: Python - - import numpy as np - import matplotlib.pyplot as plt - import scipy - import edrixs - np.set_printoptions(precision=4) - - - ll = 0 - case = 's' - norb = 4 - noccu = 2 - basis = edrixs.get_fock_bin_by_N(norb, noccu) - - - - - - - - -.. GENERATED FROM PYTHON SOURCE LINES 53-69 - -Create function to populate and diagonalize matrices ------------------------------------------------------------------------------- -The Coulomb and hopping matrices :code:`umat` and :code:`emat` will be -represented by :math:`4\times4\times4\times4` and :math:`4\times4` matrices, -respectively. Note that we needed to specify -that these are, in general, complex, although -they happen to contain only real numbers in this case. We follow the convention -that these are ordered first by site and then by spin: -:math:`|0\uparrow>, |0\downarrow>, |1\uparrow>, |1\downarrow>`. -Consequently the :math:`2\times2` and :math:`2\times2\times2\times2` block -diagonal structures of the matrices will contain the on-site interactions. -The converse is true for the hopping between the sites. -From here let us generate a function to build and diagonalize the Hamiltonian. -We need to generate the Coulomb matrix for the on-site interactions and -apply it to the block diagonal. The hopping connects off-site indices with -the same spin. - -.. GENERATED FROM PYTHON SOURCE LINES 69-86 - -.. code-block:: Python - - def diagonalize(U, t, extra_emat=None): - """Diagonalize 2 site Hubbard Hamiltonian""" - umat = np.zeros((norb, norb, norb, norb), dtype=np.complex128) - emat = np.zeros((norb, norb), dtype=np.complex128) - U_mat_1site = edrixs.get_umat_slater('s', U) - umat[:2, :2, :2, :2,] = umat[2:, 2:, 2:, 2:] = U_mat_1site - emat[2, 0] = emat[3, 1] = emat[0, 2] = emat[1, 3] = t - - if extra_emat is not None: - emat = emat + extra_emat - - H = (edrixs.build_opers(2, emat, basis) - + edrixs.build_opers(4, umat, basis)) - - e, v = scipy.linalg.eigh(H) - return e, v - - - - - - - - -.. GENERATED FROM PYTHON SOURCE LINES 87-90 - -The large :math:`U` limit ------------------------------------------------------------------------------- -Let us see what happens with :math:`U \gg t`. - -.. GENERATED FROM PYTHON SOURCE LINES 90-94 - -.. code-block:: Python - - e, v = diagonalize(1000, 1) - print("Energies are") - print(e) - - - - - -.. rst-class:: sphx-glr-script-out - - .. code-block:: none - - Energies are - [ -0.004 0. 0. 0. 1000. 1000.004] - - - - -.. GENERATED FROM PYTHON SOURCE LINES 95-98 - -To analyze what is going on we can determine the spin expectation values -of the cluster. Building the operators follows the same form as the -Hamiltonian and the previous example. - -.. GENERATED FROM PYTHON SOURCE LINES 98-106 - -.. code-block:: Python - - spin_mom_one_site = edrixs.get_spin_momentum(ll) - spin_mom = np.zeros((3, norb, norb), dtype=np.complex128) - spin_mom[:, :2, :2] = spin_mom[:, 2:, 2:] = spin_mom_one_site - - opS = edrixs.build_opers(2, spin_mom, basis) - opS_squared = (np.dot(opS[0], opS[0]) + np.dot(opS[1], opS[1]) - + np.dot(opS[2], opS[2])) - - - - - - - - -.. GENERATED FROM PYTHON SOURCE LINES 107-110 - -This time let us include a tiny magnetic field along the :math:`z`-axis, so -that we have a well-defined measurement axis and print out the expectation -values. - -.. GENERATED FROM PYTHON SOURCE LINES 110-122 - -.. code-block:: Python - - zeeman = np.zeros((norb, norb), dtype=np.complex128) - zeeman[:2, :2] = zeeman[2:, 2:] = 1e-8*spin_mom_one_site[2] - e, v = diagonalize(1000, 1, extra_emat=zeeman) - - Ssq_exp = edrixs.cb_op(opS_squared, v).diagonal().real - Sz_exp = edrixs.cb_op(opS[2], v).diagonal().real - - header = "{:<10s}\t{:<6s}\t{:<6s}" - print(header.format("E", "S(S+1)", "")) - for i in range(len(e)): - print("{:<2f}\t{:.1f}\t{:.1f}".format(e[i], Ssq_exp[i], Sz_exp[i])) - - - - - -.. rst-class:: sphx-glr-script-out - - .. code-block:: none - - E S(S+1) - -0.004000 0.0 0.0 - -0.000000 2.0 -1.0 - -0.000000 2.0 0.0 - 0.000000 2.0 1.0 - 1000.000000 0.0 0.0 - 1000.004000 0.0 0.0 - - - - -.. GENERATED FROM PYTHON SOURCE LINES 123-127 - -For :math:`U \gg t` the two states with double occupancy acquire an energy of -approximately :math:`U`. The low energy states are a :math:`S=0` singlet and -and :math:`S=1` triplet, which are split by :math:`4t^2/U`, which is the -magnetic exchange term. - -.. GENERATED FROM PYTHON SOURCE LINES 129-132 - -:math:`U` dependence ------------------------------------------------------------------------------- -Let us plot the changes in energy with :math:`U`. - -.. GENERATED FROM PYTHON SOURCE LINES 132-143 - -.. code-block:: Python - - plt.figure() - - t = 1 - Us = np.linspace(0.01, 10, 50) - Es = np.array([diagonalize(U, t, extra_emat=zeeman)[0] for U in Us]) - - plt.plot(Us/t, Es/t) - plt.xlabel('U/t') - plt.ylabel('Eigenstate energies/t') - plt.show() - - - - -.. image-sg:: /auto_examples/images/sphx_glr_example_6_Hubbard_dimer_001.png - :alt: example 6 Hubbard dimer - :srcset: /auto_examples/images/sphx_glr_example_6_Hubbard_dimer_001.png - :class: sphx-glr-single-img - - - - - -.. GENERATED FROM PYTHON SOURCE LINES 144-146 - -To help interpret this, we can represent the eigenvectors in terms of a sum -of the single particle states. - -.. GENERATED FROM PYTHON SOURCE LINES 146-164 - -.. code-block:: Python - - - def get_single_particle_repesentations(v): - reps = [] - for i in range(6): - rep = sum([vec*weight for weight, vec - in zip(v[:, i], np.array(basis))]) - reps.append(rep) - - return np.array(reps) - - t = 1 - for U in [10000, 0.0001]: - e, v = diagonalize(U, t, extra_emat=zeeman) - repesentations = get_single_particle_repesentations(v) - print("For U={} t={} states are".format(U, t)) - print(repesentations.round(3).real) - print("\n") - - - - - -.. rst-class:: sphx-glr-script-out - - .. code-block:: none - - For U=10000 t=1 states are - [[-0.707 0.707 0.707 -0.707] - [ 0. 1. 0. 1. ] - [ 0.707 0.707 0.707 0.707] - [-1. 0. -1. 0. ] - [-0.707 -0.707 0.707 0.707] - [ 0.707 0.707 0.707 0.707]] - - - For U=0.0001 t=1 states are - [[-0. 1. 1. -0. ] - [ 0. 1. 0. 1. ] - [ 0.707 0.707 0.707 0.707] - [-1. 0. -1. 0. ] - [-0.707 -0.707 0.707 0.707] - [-1. -0. -0. -1. ]] - - - - - - -.. GENERATED FROM PYTHON SOURCE LINES 165-170 - -For :math:`U \gg t` the ground state maximizes its magnetic exchange -energy saving. In the :math:`U \ll t` condition the ground state maximizes -its kinetic energy saving. Since both states share the same parity, the -cross-over between them is smooth. This type of physics is at play in current -research on quantum materials [1]_ [2]_. - -.. GENERATED FROM PYTHON SOURCE LINES 172-176 - -.. rubric:: Footnotes - -.. [1] Y. Wang et al., `Phys. Rev. Lett. 122, 106401 (2019) `_. -.. [2] A. Revelli et al., `Science Advances 5, eaav4020 (2019) `_. - - -.. rst-class:: sphx-glr-timing - - **Total running time of the script:** (0 minutes 0.095 seconds) - - -.. _sphx_glr_download_auto_examples_example_6_Hubbard_dimer.py: - -.. only:: html - - .. container:: sphx-glr-footer sphx-glr-footer-example - - .. container:: sphx-glr-download sphx-glr-download-jupyter - - :download:`Download Jupyter notebook: example_6_Hubbard_dimer.ipynb ` - - .. container:: sphx-glr-download sphx-glr-download-python - - :download:`Download Python source code: example_6_Hubbard_dimer.py ` - - .. container:: sphx-glr-download sphx-glr-download-zip - - :download:`Download zipped: example_6_Hubbard_dimer.zip ` - - -.. only:: html - - .. rst-class:: sphx-glr-signature - - `Gallery generated by Sphinx-Gallery `_ diff --git a/edrixs/_sources/auto_examples/example_7_Coulomb.rst.txt b/edrixs/_sources/auto_examples/example_7_Coulomb.rst.txt deleted file mode 100644 index 3b800cd926..0000000000 --- a/edrixs/_sources/auto_examples/example_7_Coulomb.rst.txt +++ /dev/null @@ -1,632 +0,0 @@ - -.. DO NOT EDIT. -.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. -.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: -.. "auto_examples/example_7_Coulomb.py" -.. LINE NUMBERS ARE GIVEN BELOW. - -.. only:: html - - .. note:: - :class: sphx-glr-download-link-note - - Click :ref:`here ` - to download the full example code - -.. rst-class:: sphx-glr-example-title - -.. _sphx_glr_auto_examples_example_7_Coulomb.py: - - -Coulomb interactions -===================================== -In this example we provide more details on how Coulomb interactions are -implemented in multiplet calculations and EDRIXS in particular. We aim -to clarify the form of the matrices, how they are parametrized, -and how the breaking of spherical symmetry can switch on additional elements -that one might not anticipate. Our example is based on a :math:`d` atomic shell. - -.. GENERATED FROM PYTHON SOURCE LINES 13-50 - -Create matrix ------------------------------------------------------------------------------- -The Coulomb interaction between two particles can be written as - - .. math:: - \begin{equation} - \hat{H} = \frac{1}{2} - \int d\mathbf{r} \int d\mathbf{r}^\prime - \Sigma_{\sigma, \sigma^\prime} - |\hat{\psi}^\sigma(\mathbf{r})|^2 \frac{e^2}{R} - |\hat{\psi}^{\sigma^\prime}(\mathbf{r})|^2, - \end{equation} - -where :math:`\hat{\psi}^\sigma(\mathbf{r})` is the electron wavefunction, with -spin :math:`\sigma`, and :math:`R=|r-r^\prime|` is the electron separation. -Solving our problem in this form is difficult due to the need to symmeterize -the wavefunction to follow fermionic statistics. -Using second quantization, we can use operators to impose the required -particle exchange statistics and write the equation in terms of -a tensor :math:`U` - - .. math:: - \begin{equation} - \hat{H} = \sum_{\alpha,\beta,\gamma,\delta,\sigma,\sigma^\prime} - U_{\alpha\sigma,\beta\sigma^\prime,\gamma\sigma^\prime,\delta\sigma} - \hat{f}^{\dagger}_{\alpha\sigma} - \hat{f}^{\dagger}_{\beta\sigma^\prime} - \hat{f}_{t\sigma^\prime}\hat{f}_{\delta\sigma}, - \end{equation} - -where :math:`\alpha`, :math:`\beta`, :math:`\gamma`, :math:`\delta` are -orbital indices and :math:`\hat{f}^{\dagger}` -(:math:`\hat{f}`) are the creation (anihilation) operators. -For a :math:`d`-electron system, we have :math:`10` distinct spin-orbitals -(:math:`5` orbitals each with :math:`2` spins), which makes matrix the -:math:`10\times10\times10\times10` in total size. -In EDRIXS the matrix can be created as follows: - -.. GENERATED FROM PYTHON SOURCE LINES 50-58 - -.. code-block:: default - - import edrixs - import numpy as np - import scipy - import matplotlib.pyplot as plt - import itertools - - F0, F2, F4 = 6.94, 14.7, 4.41 - umat_chb = edrixs.get_umat_slater('d', F0, F2, F4) - - - - - - - -.. GENERATED FROM PYTHON SOURCE LINES 59-61 - -We stored this under variable :code:`umat_chb` where "cbh" stands for -complex harmonic basis, which is the default basis in EDRIXS. - -.. GENERATED FROM PYTHON SOURCE LINES 63-92 - -Parameterizing interactions ------------------------------------------------------------------------------- -EDRIXS parameterizes the interactions in :math:`U` via Slater integral -parameters :math:`F^{k}`. These relate to integrals of various spherical -Harmonics as well as Clebsch-Gordon coefficients, Gaunt coefficients, -and Wigner 3J symbols. Textbooks such as [1]_ can be used for further -reference. If you are interested in the details of how -EDRIXS does this (and you probably aren't) function :func:`.umat_slater`, -constructs the required matrix via Gaunt coeficents from -:func:`.get_gaunt`. Two alternative parameterizations are common. -The first are the Racah parameters, which are - - .. math:: - \begin{eqnarray} - A &=& F^0 - \frac{49}{441} F^4 \\ - B &=& \frac{1}{49}F^2 - \frac{5}{441}F^4 \\ - C &=& \frac{35}{441}F^4. - \end{eqnarray} - -or an alternative form for the Slater integrals - - .. math:: - \begin{eqnarray} - F_0 &=& F^0 \\ - F_2 &=& \frac{1}{49}F^2 \\ - F_4 &=& \frac{1}{441}F^4, - \end{eqnarray} - -which involves different normalization parameters. - -.. GENERATED FROM PYTHON SOURCE LINES 94-101 - -Basis transform ------------------------------------------------------------------------------- -If we want to use the real harmonic basis, we can use a tensor -transformation, which imposes the following orbital order -:math:`3z^2-r^2, xz, yz, x^2-y^2, xy`, each of which involves -:math:`\uparrow, \downarrow` spin pairs. Let's perform this transformation and -store a list of these orbitals. - -.. GENERATED FROM PYTHON SOURCE LINES 101-104 - -.. code-block:: default - - umat = edrixs.transform_utensor(umat_chb, edrixs.tmat_c2r('d', True)) - orbitals = ['3z^2-r^2', 'xz', 'yz', 'x^2-y^2', 'xy'] - - - - - - - - -.. GENERATED FROM PYTHON SOURCE LINES 105-158 - -Interactions ------------------------------------------------------------------------------- -Tensor :math:`U` is a series of matrix -elements - - .. math:: - \begin{equation} - \langle\psi_{\gamma,\delta}^{\bar{\sigma},\bar{\sigma}^\prime} - |\hat{H}| - \psi_{\alpha,\beta}^{\sigma,\sigma^\prime}\rangle - \end{equation} - -the combination of which defines the energetic cost of pairwise -electron-electron interactions between states :math:`\alpha,\sigma` -and :math:`\beta,\sigma^\prime`. In EDRIXS we follow the convention of -summing over all orbital pairs. Some other texts count each pair of -indices only once. The matrix elements here will consequently -be half the magnitude of those in other references. -We can express the interactions in terms of -the orbitals involved. It is common to distinguish "direct Coulomb" and -"exchange" interactions. The former come from electrons in the same orbital -and the later involve swapping orbital labels for electrons. We will use -:math:`U_0` and :math:`J` as a shorthand for distinguishing these. - -Before we describe the different types of interactions, we note that since -the Coulomb interaction is real, and due to the spin symmmetry properties -of the process :math:`U` always obeys - - .. math:: - \begin{equation} - U_{\alpha\sigma,\beta\sigma^\prime,\gamma\sigma^\prime,\delta\sigma} = - U_{\beta\sigma,\alpha\sigma^\prime,\delta\sigma^\prime,\gamma\sigma} = - U_{\delta\sigma,\gamma\sigma^\prime,\beta\sigma^\prime,\alpha\sigma} = - U_{\gamma\sigma,\delta\sigma^\prime,\alpha\sigma^\prime,\beta\sigma}. - \end{equation} - - -1. Intra orbital -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -The direct Coulomb energy cost to double-occupy an orbital comes from terms -like :math:`U_{\alpha\sigma,\alpha\bar\sigma,\alpha\bar\sigma,\alpha\sigma}`. -In this notation, we use :math:`\sigma^\prime` to denote that the matrix -element is summed over all pairs and :math:`\bar{\sigma}` to denote sums -over all opposite spin pairs. Due to rotational symmetry, all these -elements are the same and equal to - - .. math:: - \begin{eqnarray} - U_0 &=& \frac{A}{2} + 2B + \frac{3C}{2}\\ - &=& \frac{F_0}{2} + 2F_2 + 18F_4 - \end{eqnarray} - -Let's print these to demonstrate where these live in the array - -.. GENERATED FROM PYTHON SOURCE LINES 158-162 - -.. code-block:: default - - for i in range(0, 5): - val = umat[i*2, i*2 + 1, i*2 + 1, i*2].real - print(f"{orbitals[i]:<8} \t {val:.3f}") - - - - - -.. rst-class:: sphx-glr-script-out - - .. code-block:: none - - 3z^2-r^2 4.250 - xz 4.250 - yz 4.250 - x^2-y^2 4.250 - xy 4.250 - - - - -.. GENERATED FROM PYTHON SOURCE LINES 163-170 - -2. Inter orbital Coulomb interactions -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -Direct Coulomb repulsion between different orbitals depends on terms like -:math:`U_{\alpha\sigma,\beta\sigma^\prime,\beta\sigma^\prime,\alpha\sigma}`. -Expresions for these parameters are provided in column :math:`U` in -:ref:`table_2_orbital`. We can print the values from :code:`umat` -like this: - -.. GENERATED FROM PYTHON SOURCE LINES 170-174 - -.. code-block:: default - - for i, j in itertools.combinations(range(5), 2): - val = umat[i*2, j*2 + 1, j*2 + 1, i*2].real - print(f"{orbitals[i]:<8} \t {orbitals[j]:<8} \t {val:.3f}") - - - - - -.. rst-class:: sphx-glr-script-out - - .. code-block:: none - - 3z^2-r^2 xz 3.650 - 3z^2-r^2 yz 3.650 - 3z^2-r^2 x^2-y^2 2.900 - 3z^2-r^2 xy 2.900 - xz yz 3.150 - xz x^2-y^2 3.150 - xz xy 3.150 - yz x^2-y^2 3.150 - yz xy 3.150 - x^2-y^2 xy 3.900 - - - - -.. GENERATED FROM PYTHON SOURCE LINES 175-181 - -3. Inter-orbital exchange interactions -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -Exchange terms exist with the form -:math:`U_{\alpha\sigma,\beta\sigma^\prime,\alpha\sigma^\prime,\beta\sigma}`. -Expresions for these parameters are provided in column :math:`J` of -:ref:`table_2_orbital`. These come from terms like this in the matrix: - -.. GENERATED FROM PYTHON SOURCE LINES 181-185 - -.. code-block:: default - - for i, j in itertools.combinations(range(5), 2): - val = umat[i*2, j*2 + 1, i*2 + 1, j*2].real - print(f"{orbitals[i]:<8} \t {orbitals[j]:<8} \t {val:.3f}") - - - - - -.. rst-class:: sphx-glr-script-out - - .. code-block:: none - - 3z^2-r^2 xz 0.300 - 3z^2-r^2 yz 0.300 - 3z^2-r^2 x^2-y^2 0.675 - 3z^2-r^2 xy 0.675 - xz yz 0.550 - xz x^2-y^2 0.550 - xz xy 0.550 - yz x^2-y^2 0.550 - yz xy 0.550 - x^2-y^2 xy 0.175 - - - - -.. GENERATED FROM PYTHON SOURCE LINES 186-193 - -4. Pair hopping term -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -Terms that swap pairs of electrons exist as -:math:`(1-\delta_{\sigma\sigma'})U_{\alpha\sigma,\alpha\bar\sigma,\beta\bar\sigma,\beta\sigma}` -and depend on exchange interactions column :math:`J` from -:ref:`table_2_orbital` -and here in the matrix. - -.. GENERATED FROM PYTHON SOURCE LINES 193-197 - -.. code-block:: default - - for i, j in itertools.combinations(range(5), 2): - val = umat[i*2, i*2 + 1, j*2 + 1, j*2].real - print(f"{orbitals[i]:<8} \t {orbitals[j]:<8} \t {val:.3f}") - - - - - -.. rst-class:: sphx-glr-script-out - - .. code-block:: none - - 3z^2-r^2 xz 0.300 - 3z^2-r^2 yz 0.300 - 3z^2-r^2 x^2-y^2 0.675 - 3z^2-r^2 xy 0.675 - xz yz 0.550 - xz x^2-y^2 0.550 - xz xy 0.550 - yz x^2-y^2 0.550 - yz xy 0.550 - x^2-y^2 xy 0.175 - - - - -.. GENERATED FROM PYTHON SOURCE LINES 198-215 - -5. Three orbital -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -Another set of terms that one might not immediately anticipate involve three -orbitals like - - .. math:: - \begin{equation} - U_{\alpha\sigma, \gamma\sigma', \beta\sigma', \gamma\sigma} \\ - U_{\alpha\sigma, \gamma\sigma', \gamma\sigma', \beta\sigma} \\ - (1-\delta_{\sigma\sigma'}) - U_{\alpha\sigma, \beta\sigma', \gamma\sigma', \gamma\sigma} - \end{equation} - -for :math:`\alpha=3z^2-r^2, \beta=x^2-y^2, \gamma=xz/yz`. -These are needed to maintain the rotational symmetry of the interations. -See :ref:`table_3_orbital` for the expressions. We can print some of -these via: - -.. GENERATED FROM PYTHON SOURCE LINES 215-227 - -.. code-block:: default - - ijkl = [[0, 1, 3, 1], - [0, 2, 3, 2], - [1, 0, 3, 1], - [1, 1, 3, 0], - [2, 0, 3, 2], - [2, 2, 3, 0]] - - for i, j, k, l in ijkl: - val = umat[i*2, j*2 + 1, k*2 + 1, l*2].real - print(f"{orbitals[i]:<8} \t {orbitals[j]:<8} \t" - f"{orbitals[k]:<8} \t {orbitals[l]:<8} \t {val:.3f}") - - - - - -.. rst-class:: sphx-glr-script-out - - .. code-block:: none - - 3z^2-r^2 xz x^2-y^2 xz 0.217 - 3z^2-r^2 yz x^2-y^2 yz -0.217 - xz 3z^2-r^2 x^2-y^2 xz -0.433 - xz xz x^2-y^2 3z^2-r^2 0.217 - yz 3z^2-r^2 x^2-y^2 yz 0.433 - yz yz x^2-y^2 3z^2-r^2 -0.217 - - - - -.. GENERATED FROM PYTHON SOURCE LINES 228-233 - -6. Four orbital -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -Futher multi-orbital terms include -:math:`U_{\alpha\sigma,\beta\sigma^\prime,\gamma\sigma^\prime,\delta\sigma}`. -We can find these here in the matrix: - -.. GENERATED FROM PYTHON SOURCE LINES 233-249 - -.. code-block:: default - - ijkl = [[0, 1, 2, 4], - [0, 1, 4, 2], - [0, 2, 1, 4], - [0, 2, 4, 1], - [0, 4, 1, 2], - [0, 4, 2, 1], - [3, 1, 4, 2], - [3, 2, 4, 1], - [3, 4, 1, 2], - [3, 4, 2, 1]] - - for i, j, k, l in ijkl: - val = umat[i*2, j*2 + 1, k*2 + 1, l*2].real - print(f"{orbitals[i]:<8} \t {orbitals[j]:<8} \t {orbitals[k]:<8}" - f"\t {orbitals[l]:<8} \t {val:.3f}") - - - - - -.. rst-class:: sphx-glr-script-out - - .. code-block:: none - - 3z^2-r^2 xz yz xy -0.433 - 3z^2-r^2 xz xy yz 0.217 - 3z^2-r^2 yz xz xy -0.433 - 3z^2-r^2 yz xy xz 0.217 - 3z^2-r^2 xy xz yz 0.217 - 3z^2-r^2 xy yz xz 0.217 - x^2-y^2 xz xy yz -0.375 - x^2-y^2 yz xy xz 0.375 - x^2-y^2 xy xz yz -0.375 - x^2-y^2 xy yz xz 0.375 - - - - -.. GENERATED FROM PYTHON SOURCE LINES 250-256 - -Effects of multi-orbital terms ------------------------------------------------------------------------------- -To test the effects of the multi-orbital terms, let's plot the eigenenergy -spectra with and without multi-orbital terms switched on for system with and -without a cubic crystal field. We will use a :math:`d`-shell with two -electrons. - -.. GENERATED FROM PYTHON SOURCE LINES 256-292 - -.. code-block:: default - - ten_dqs = [0, 2, 4, 12] - - def diagonalize(ten_dq, umat): - emat = edrixs.cb_op(edrixs.cf_cubic_d(ten_dq), - edrixs.tmat_c2r('d', ispin=True)) - H = (edrixs.build_opers(4, umat, basis) - + edrixs.build_opers(2, emat, basis)) - e, v = scipy.linalg.eigh(H) - return e - e.min() - - basis = edrixs.get_fock_bin_by_N(10, 2) - umat_no_multiorbital = np.copy(umat) - B = F2/49 - 5*F4/441 - for val in [np.sqrt(3)*B/2, np.sqrt(3)*B, 3*B/2]: - umat_no_multiorbital[(np.abs(umat)- val) < 1e-6] = 0 - - fig, axs = plt.subplots(1, len(ten_dqs), figsize=(8, 3)) - - for cind, (ax, ten_dq) in enumerate(zip(axs, ten_dqs)): - ax.hlines(diagonalize(ten_dq, umat), xmin=0, xmax=1, - label='on', color=f'C{cind}') - ax.hlines(diagonalize(ten_dq, umat_no_multiorbital), - xmin=1.5, xmax=2.5, - label='off', - linestyle=':', color=f'C{cind}') - ax.set_title(f"$10D_q={ten_dq}$") - ax.set_ylim([-.5, 20]) - ax.set_xticks([]) - ax.legend() - - fig.suptitle("Eigenvalues with 3&4-orbital effects on/off") - fig.subplots_adjust(wspace=.3) - axs[0].set_ylabel('Eigenvalues (eV)') - fig.subplots_adjust(top=.8) - plt.show() - - - - -.. image-sg:: /auto_examples/images/sphx_glr_example_7_Coulomb_001.png - :alt: Eigenvalues with 3&4-orbital effects on/off, $10D_q=0$, $10D_q=2$, $10D_q=4$, $10D_q=12$ - :srcset: /auto_examples/images/sphx_glr_example_7_Coulomb_001.png - :class: sphx-glr-single-img - - - - - -.. GENERATED FROM PYTHON SOURCE LINES 293-382 - -On the left of the plot Coulomb interactions in spherical symmetry cause -substantial mxing between :math:`t_{2g}` and :math:`e_{g}` orbitals in the -eigenstates and 3 & 4 orbital orbital terms are crucial for obtaining the -the right eigenenergies. As :math:`10D_q` get large, this mixing is switched -off and the spectra start to become independent of whether the 3 & 4 orbital -orbital terms are included or not. - - - -.. _table_2_orbital: -.. table:: Table of 2 orbital interactions - - +-----------------------------+------------------+-----------------------+----------------+--------------------+ - |Orbitals :math:`\alpha,\beta`|:math:`U_0` Racah | :math:`U_0` Slater |:math:`J` Racah |:math:`J` Slater | - +=============================+==================+=======================+================+====================+ - |:math:`3z^2-r^2, xz` |:math:`A/2+B+C/2` |:math:`F_0/2+F_2-12F_4`| :math:`B/2+C/2`|:math:`F_2/2+15F_4` | - +-----------------------------+------------------+-----------------------+----------------+--------------------+ - |:math:`3z^2-r^2, yz` |:math:`A/2+B+C/2` |:math:`F_0/2+F_2-12F_4`| :math:`B/2+C/2`|:math:`F_2/2+15F_4` | - +-----------------------------+------------------+-----------------------+----------------+--------------------+ - |:math:`3z^2-r^2, x^2-y^2` |:math:`A/2-2B+C/2`|:math:`F_0/2-2F_2+3F_4`|:math:`2B+C/2` |:math:`2F_2+15F_4/2`| - +-----------------------------+------------------+-----------------------+----------------+--------------------+ - |:math:`3z^2-r^2, xy` |:math:`A/2-2B+C/2`|:math:`F_0/2-2F_2+3F_4`|:math:`2B+C/2` |:math:`2F_2+15F_4/2`| - +-----------------------------+------------------+-----------------------+----------------+--------------------+ - |:math:`xz, yz` |:math:`A/2-B+C/2` |:math:`F_0/2-F_2-12F_4`|:math:`3B/2+C/2`|:math:`3F_2/2+10F_4`| - +-----------------------------+------------------+-----------------------+----------------+--------------------+ - |:math:`xz, x^2-y^2` |:math:`A/2-B+C/2` |:math:`F_0/2-F_2-12F_4`|:math:`3B/2+C/2`|:math:`3F_2/2+10F_4`| - +-----------------------------+------------------+-----------------------+----------------+--------------------+ - |:math:`xz, xy` |:math:`A/2-B+C/2` |:math:`F_0/2-F_2-12F_4`|:math:`3B/2+C/2`|:math:`3F_2/2+10F_4`| - +-----------------------------+------------------+-----------------------+----------------+--------------------+ - |:math:`yz, x^2-y^2` |:math:`A/2-B+C/2` |:math:`F_0/2-F_2-12F_4`|:math:`3B/2+C/2`|:math:`3F_2/2+10F_4`| - +-----------------------------+------------------+-----------------------+----------------+--------------------+ - |:math:`yz, xy` |:math:`A/2-B+C/2` |:math:`F_0/2-F_2-12F_4`|:math:`3B/2+C/2`|:math:`3F_2/2+10F_4`| - +-----------------------------+------------------+-----------------------+----------------+--------------------+ - |:math:`x^2-y^2, xy` |:math:`A/2+2B+C/2`|:math:`F_0+4F_2-34F_4` | :math:`C/2` |:math:`35F_4/2` | - +-----------------------------+------------------+-----------------------+----------------+--------------------+ - - -.. _table_3_orbital: -.. table:: Table of 3 orbital interactions - - +-----------------------------+-------------+----------------------------------------------------+-----------------------------------------------------+ - |Orbitals :math:`\alpha,\beta,\gamma,\delta`|:math:`\langle\alpha\beta|\gamma\delta\rangle` Racah|:math:`\langle\alpha\beta|\gamma\delta\rangle` Slater| - +=============================+=============+====================================================+=====================================================+ - |:math:`3z^2-r^2, xz, x^2-y^2, xz` | :math:`\sqrt{3}B/2` | :math:`\sqrt{3}F_2/2-5\sqrt{3}F_4/2` | - +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ - |:math:`3z^2-r^2, yz, x^2-y^2, yz` | :math:`-\sqrt{3}B/2` | :math:`-\sqrt{3}F_2/2+5\sqrt{3}F_4/2` | - +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ - |:math:`xz, 3z^2-r^2, x^2-y^2, xz` | :math:`-\sqrt{3}B` | :math:`-\sqrt{3}F_2+5\sqrt{3}F_4` | - +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ - |:math:`xz, xz, x^2-y^2, 3z^2-r^2` | :math:`\sqrt{3}B/2` | :math:`\sqrt{3}F_2/2-5\sqrt{3}F_4/2` | - +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ - |:math:`yz, 3z^2-r^2, x^2-y^2, yz` | :math:`\sqrt{3}B` | :math:`\sqrt{3}F_2-5\sqrt{3}F_4` | - +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ - |:math:`yz, yz, x^2-y^2, 3z^2-r^2` | :math:`-\sqrt{3}B/2` | :math:`-\sqrt{3}F_2/2+5\sqrt{3}F_4/2` | - +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ - - -.. _table_4_orbital: -.. table:: Table of 4 orbital interactions - - +-----------------------------+-------------+----------------------------------------------------+-----------------------------------------------------+ - |Orbitals :math:`\alpha,\beta,\gamma,\delta`|:math:`\langle\alpha\beta|\gamma\delta\rangle` Racah|:math:`\langle\alpha\beta|\gamma\delta\rangle` Slater| - +=============================+=============+====================================================+=====================================================+ - |:math:`3z^2-r^2, xz, yz, xy` | :math:`-\sqrt{3}B` | :math:`-\sqrt{3}F_2+5\sqrt{3}F_4` | - +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ - |:math:`3z^2-r^2, xz, xy, yz` | :math:`\sqrt{3}B/2` | :math:`\sqrt{3}F_2/2-5\sqrt{3}F_4/2` | - +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ - |:math:`3z^2-r^2, yz, xz, xy` | :math:`-\sqrt{3}B` | :math:`-\sqrt{3}F_2+5\sqrt{3}F_4` | - +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ - |:math:`3z^2-r^2, yz, xy, xz` | :math:`\sqrt{3}B/2` | :math:`\sqrt{3}F_2/2-5\sqrt{3}F_4/2` | - +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ - |:math:`3z^2-r^2, xy, xz, yz` | :math:`\sqrt{3}B/2` | :math:`\sqrt{3}F_2/2-5\sqrt{3}F_4/2` | - +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ - |:math:`3z^2-r^2, xy, yz, xz` | :math:`\sqrt{3}B/2` | :math:`\sqrt{3}F_2/2-5\sqrt{3}F_4/2` | - +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ - |:math:`x^2-y^2 , xz, xy, yz` | :math:`-3B/2` | :math:`-3F_2/2+15F_4/2` | - +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ - |:math:`x^2-y^2 , yz, xy, xz` | :math:`3B/2` | :math:`3F_2/2-15F_4/2` | - +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ - |:math:`x^2-y^2 , xy, xz, yz` | :math:`-3B/2` | :math:`-3F_2/2+15F_4/2` | - +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ - |:math:`x^2-y^2 , xy, yz, xz` | :math:`3B/2` | :math:`3F_2/2-15F_4/2` | - +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ - - -.. rubric:: Footnotes - -.. [1] MSugano S, Tanabe Y and Kamimura H. 1970. Multiplets of - Transition-Metal Ions in Crystals. Academic Press, New York and London. - - -.. rst-class:: sphx-glr-timing - - **Total running time of the script:** ( 0 minutes 0.579 seconds) - - -.. _sphx_glr_download_auto_examples_example_7_Coulomb.py: - -.. only:: html - - .. container:: sphx-glr-footer sphx-glr-footer-example - - - .. container:: sphx-glr-download sphx-glr-download-python - - :download:`Download Python source code: example_7_Coulomb.py ` - - .. container:: sphx-glr-download sphx-glr-download-jupyter - - :download:`Download Jupyter notebook: example_7_Coulomb.ipynb ` - - -.. only:: html - - .. rst-class:: sphx-glr-signature - - `Gallery generated by Sphinx-Gallery `_ diff --git a/edrixs/_sources/auto_examples/example_7_Hunds_interactions.rst.txt b/edrixs/_sources/auto_examples/example_7_Hunds_interactions.rst.txt deleted file mode 100644 index f27f9b31a9..0000000000 --- a/edrixs/_sources/auto_examples/example_7_Hunds_interactions.rst.txt +++ /dev/null @@ -1,478 +0,0 @@ - -.. DO NOT EDIT. -.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. -.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: -.. "auto_examples/example_7_Hunds_interactions.py" -.. LINE NUMBERS ARE GIVEN BELOW. - -.. only:: html - - .. note:: - :class: sphx-glr-download-link-note - - :ref:`Go to the end ` - to download the full example code. - -.. rst-class:: sphx-glr-example-title - -.. _sphx_glr_auto_examples_example_7_Hunds_interactions.py: - - -Hund's Interactions in charge transfer insulators -================================================= -In this exercise we will solve a toy model relevant to cubic :math:`d^8` charge transfer insulators -such as NiO or NiPS\ :sub:`3`. We are interested in better understanding the interplay between the -Hund's interactions and the charge transfer energy in terms of the energy of the triplet-singlet -excitations of this model. These seem to act against each other in that the Hund's interactions -impose a energy cost for the triplet-singlet excitations whenever there are two holes on -the Ni :math:`d` orbitals. The charge transfer physics, on the other hand, will promote a -:math:`d^9\underline{L}` ground state in which the Hund's interactions are not active. - -The simplest model that captures this physics requires four Ni spin-orbitals, representing the Ni -:math:`e_g` manifold. We will represent the ligand states in the same way as the Anderson impurity -model in terms of one effective ligand spin-orbital per Ni spin-orbital. We assume these effective -orbitals have been constructed so that each Ni orbital only bonds to one sister orbital. For -simplicity, we will treat all Ni and all ligand orbitals as equivalent, even though a more -realistic model would account for the different Coulomb and hopping of the :math:`d_{3z^2-r^2}` -and :math:`d_{x^2-y^2}` orbitals. We therefore simply connect Ni and ligand orbitals via a constant -hopping :math:`t`. We also include the ligand energy parameter :math:`e_L`. - -The easiest way to implement the requried Coulomb interactions is to use the so-called Kanamori -Hamiltonian, which is a simplfied form for the interactions, which treats all orbitals as -equivalent. Daniel Khomskii's book provides a great explanation of this physics [1]_. We -parameterize the interactions via Coulomb repulsion parameter :math:`U` and Hund's exchange -:math:`J_H`. EDRIXS provides this functionality via the more general -:func:`.get_umat_kanamori` function. - -It's also easiest to consider this problem in hole langauge, which means our eight spin-orbitals -are populated by two fermions. - -.. GENERATED FROM PYTHON SOURCE LINES 34-38 - -Setup ------------------------------------------------------------------------------- -We start by loading the necessary modules, and defining the total number of -orbitals and electrons. - -.. GENERATED FROM PYTHON SOURCE LINES 38-46 - -.. code-block:: Python - - import edrixs - import scipy - import numpy as np - import matplotlib.pyplot as plt - - norb = 8 - noccu = 2 - - - - - - - - -.. GENERATED FROM PYTHON SOURCE LINES 47-55 - -Diagonalization ------------------------------------------------------------------------------- -Let's write a function to diagonalize our model in a similar way to -the :ref:`sphx_glr_auto_examples_example_6_Hubbard_dimer.py` example. -Within this function, we also create operators to count the number of -:math:`d` holes and operators to calculate expectation values for -:math:`S^2` and :math:`S_z`. For the latter to make sense, we also include a -small effective spin interaction along :math:`z`. - -.. GENERATED FROM PYTHON SOURCE LINES 55-103 - -.. code-block:: Python - - - - def diagonalize(U, JH, t, eL, n=1): - # Setup Coulomb matrix - umat = np.zeros((norb, norb, norb, norb), dtype=complex) - uNi = edrixs.get_umat_kanamori(norb//2, U, JH) - umat[:norb//2, :norb//2, :norb//2, :norb//2] = uNi - - # Setup hopping matrix - emat = np.zeros((norb, norb), dtype=complex) - ind = np.arange(norb//2) - emat[ind, ind + norb//2] = t - emat[ind+norb//2, ind] = np.conj(t) # conj is not needed, but is good practise. - ind = np.arange(norb//2, norb) - emat[ind, ind] += eL - - # Spin operator - spin_mom = np.zeros((3, norb, norb), dtype=complex) - spin_mom[:, :2, :2] = edrixs.get_spin_momentum(0) - spin_mom[:, 2:4, 2:4] = edrixs.get_spin_momentum(0) - spin_mom[:, 4:6, 4:6] = edrixs.get_spin_momentum(0) - spin_mom[:, 6:8, 6:8] = edrixs.get_spin_momentum(0) - - # add small effective field along z - emat += 1e-6*spin_mom[2] - - # Diagonalize - basis = edrixs.get_fock_bin_by_N(norb, noccu) - H = edrixs.build_opers(2, emat, basis) + edrixs.build_opers(4, umat, basis) - e, v = scipy.linalg.eigh(H) - e -= e[0] # Define ground state as zero energy - - # Operator for holes on Ni - basis = np.array(basis) - num_d_electrons = basis[:, :4].sum(1) - d0 = np.sum(np.abs(v[num_d_electrons == 0, :])**2, axis=0) - d1 = np.sum(np.abs(v[num_d_electrons == 1, :])**2, axis=0) - d2 = np.sum(np.abs(v[num_d_electrons == 2, :])**2, axis=0) - - # S^2 and Sz operators - opS = edrixs.build_opers(2, spin_mom, basis) - S_squared_op = np.dot(opS[0], opS[0]) + np.dot(opS[1], opS[1]) + np.dot(opS[2], opS[2]) - S_squared_exp = edrixs.cb_op(S_squared_op, v).diagonal().real - S_z_exp = edrixs.cb_op(opS[2], v).diagonal().real - - return e[:n], d0[:n], d1[:n], d2[:n], S_squared_exp[:n], S_z_exp[:n] - - - - - - - - - -.. GENERATED FROM PYTHON SOURCE LINES 104-110 - -The atomic limit ------------------------------------------------------------------------------- -For simplicity, let's start in the atomic limit with :math:`e_L \gg t \gg U` -where all holes are on nickel. In this case, there are six ways to distribute -two holes on the four Ni spin-orbitals. Let's examine the expectation values -of the :math:`S^2` and :math:`S_z` operators. - -.. GENERATED FROM PYTHON SOURCE LINES 110-125 - -.. code-block:: Python - - U = 10 - JH = 2 - t = 100 - eL = 1e10 - - e, d0, d1, d2, S_squared_exp, S_z_exp = diagonalize(U, JH, t, eL, n=6) - - print("Ground state\nE\t") - for i in range(3): - print(f"{e[i]:.2f}\t{S_squared_exp[i]:.2f}\t{S_z_exp[i]:.2f}") - - print("\nExcited state\nE\t") - for i in range(3, 6): - print(f"{e[i]:.2f}\t{S_squared_exp[i]:.2f}\t{S_z_exp[i]:.2f}") - - - - - -.. rst-class:: sphx-glr-script-out - - .. code-block:: none - - Ground state - E - 0.00 2.00 1.00 - 0.00 2.00 0.00 - 0.00 2.00 -1.00 - - Excited state - E - 4.00 0.00 0.00 - 4.00 0.00 0.00 - 8.00 0.00 0.00 - - - - -.. GENERATED FROM PYTHON SOURCE LINES 126-132 - -The ground state is a high-spin triplet. The fourth and fifth -states (the first excited state) are low-spin singlet excitons at -:math:`2 J_H`. These have one hole on each orbital in the antisymmetric -combination of :math:`|\uparrow\downarrow>-|\downarrow\uparrow>`. -The state at :math:`3 J_H` also has one hole on each orbital in the symmetric -:math:`|\uparrow\downarrow>+|\downarrow\uparrow>` configuration. - -.. GENERATED FROM PYTHON SOURCE LINES 134-140 - -Where are the holes for large hopping ------------------------------------------------------------------------------- -As discussed at the start, we are interested to see interplay between Hund's -and charge-transfer physics, which will obviously depend strongly on whether -the holes are on Ni or the ligand. Let's see what happens as :math:`e_L` is -reduced while observing the location of the ground state and exciton holes. - -.. GENERATED FROM PYTHON SOURCE LINES 140-164 - -.. code-block:: Python - - U = 10 - JH = 2 - t = 100 - - eLs = np.linspace(0, 1000, 30) - - fig, axs = plt.subplots(1, 2, figsize=(8, 4)) - - for ax, ind in zip(axs.ravel(), [0, 3]): - ds = np.array([diagonalize(U, JH, t, eL, n=6) - for eL in eLs]) - - ax.plot(eLs, ds[:, 1, ind], 'o', label='$d^0$') - ax.plot(eLs, ds[:, 2, ind], 's', label='$d^1$') - ax.plot(eLs, ds[:, 3, ind], '^', label='$d^2$') - ax.set_xlabel("Energy of ligands $e_L$") - ax.set_ylabel("Number of electrons") - ax.legend() - - axs[0].set_title("Location of ground state holes") - axs[1].set_title("Location of exciton holes") - - plt.tight_layout() - plt.show() - - - -.. image-sg:: /auto_examples/images/sphx_glr_example_7_Hunds_interactions_001.png - :alt: Location of ground state holes, Location of exciton holes - :srcset: /auto_examples/images/sphx_glr_example_7_Hunds_interactions_001.png - :class: sphx-glr-single-img - - - - - -.. GENERATED FROM PYTHON SOURCE LINES 165-171 - -For large :math:`e_L`, we see that both holes are on nickel as expected. In -the opposite limit of :math:`|e_L| \ll t` and :math:`U \ll t` the holes are -shared in the ratio 0.25:0.5:0.25 as there are two ways to have one hole on -Ni. In the limit of large :math:`e_L`, all holes move onto Ni. Since -:math:`t` is large, this applies equally to both the ground state and the -exciton. - -.. GENERATED FROM PYTHON SOURCE LINES 174-179 - -Connecton between atomic and charge transfer limits ------------------------------------------------------------------------------- -We now examine the quantum numbers during cross over between the two limits -with :math:`e_L`. Let's first look at the how :math:`` changes for the -ground state and exciton and then examine how the exciton energy changes. - -.. GENERATED FROM PYTHON SOURCE LINES 179-216 - -.. code-block:: Python - - - U = 10 - JH = 2 - t = 100 - - eLs = np.linspace(0, 1000, 30) - - info = np.array([diagonalize(U, JH, t, eL, n=6) - for eL in eLs]) - - fig, axs = plt.subplots(1, 2, figsize=(8, 4)) - - - axs[0].plot(eLs, info[:, 4, 0], label='Ground state') - axs[0].plot(eLs, info[:, 4, 3], label='Exciton') - axs[0].set_xlabel("Energy of ligands $e_L$") - axs[0].set_ylabel('$$') - axs[0].set_title('Quantum numbers') - axs[0].legend() - - axs[1].plot(eLs, info[:, 0, 3], '+', color='C0') - axs[1].set_xlabel("Energy of ligands $e_L$") - axs[1].set_ylabel('Exciton energy', color='C0') - axr = axs[1].twinx() - axr.plot(eLs, info[:, 3, 5], 'x', color='C1') - axr.set_ylabel('$d^2$ fraction', color='C1') - - for ax, color in zip([axs[1], axr], ['C0', 'C1']): - for tick in ax.get_yticklabels(): - tick.set_color(color) - - axs[1].set_ylim(0, 2*JH) - axr.set_ylim(0, 1) - - axs[1].set_title('Exciton energy vs. $d^2$ character') - plt.tight_layout() - plt.show() - - - -.. image-sg:: /auto_examples/images/sphx_glr_example_7_Hunds_interactions_002.png - :alt: Quantum numbers, Exciton energy vs. $d^2$ character - :srcset: /auto_examples/images/sphx_glr_example_7_Hunds_interactions_002.png - :class: sphx-glr-single-img - - - - - -.. GENERATED FROM PYTHON SOURCE LINES 217-229 - -In the left panel, we see that the two limits are adiabatically connected -as they preseve the same quantum numbers. This is because there is always -an appreciable double occupancy under conditions where the -:math:`d^9\underline{L}` character is maximized and this continues to favor -the high spin ground state. Other interactions such as strong tetragonal -crystal field would be needed to overcome the Hund's interactions and break -this paradigm. In the right panel, we see that the exciton energy simply -scales with the double occupancy. Overall, even though -Hund's interactions are irrelevant for the :math:`d^9\underline{L}` -electronic configuration, whenever :math:`t` is appreciable there is a -strong mixing with the :math:`d^8` component is always present, which -dominates the energy of the exciton. - -.. GENERATED FROM PYTHON SOURCE LINES 231-237 - -Charge transfer excitons ------------------------------------------------------------------------------- -Another limiting case of the model is where :math:`t` is smaller than the -Coulomb interactions. This, however, tends to produce -ground state and exciton configurations that correspond to those of distinct -atomic models. Let's look at the :math:`e_L` dependence in this case. - -.. GENERATED FROM PYTHON SOURCE LINES 237-265 - -.. code-block:: Python - - U = 10 - JH = 2 - t = .5 - eL = 7 - - eLs = np.linspace(0, 20, 30) - - fig, axs = plt.subplots(1, 2, figsize=(8, 4)) - - for ax, ind in zip(axs.ravel(), [0, 3]): - ds = np.array([diagonalize(U, JH, t, eL, n=6) - for eL in eLs]) - - ax.plot(eLs, ds[:, 1, ind], 'o', label='$d^0$') - ax.plot(eLs, ds[:, 2, ind], 's', label='$d^1$') - ax.plot(eLs, ds[:, 3, ind], '^', label='$d^2$') - ax.set_xlabel("Energy of ligands $e_L$") - ax.set_ylabel("Number of electrons") - ax.legend() - - axs[0].axvline(x=eL, linestyle=':', color='k') - axs[1].axvline(x=eL, linestyle=':', color='k') - - axs[0].set_title("Location of ground state holes") - axs[1].set_title("Location of exciton holes") - - plt.tight_layout() - plt.show() - - - -.. image-sg:: /auto_examples/images/sphx_glr_example_7_Hunds_interactions_003.png - :alt: Location of ground state holes, Location of exciton holes - :srcset: /auto_examples/images/sphx_glr_example_7_Hunds_interactions_003.png - :class: sphx-glr-single-img - - - - - -.. GENERATED FROM PYTHON SOURCE LINES 266-270 - -Around :math:`e_L = 7` the plot shows that the excition is primairly a -:math:`d^2 \rightarrow d^1` transition or a -:math:`d^8 \rightarrow d^{9}\underline{L}` transition in electron language. -Let's examine the energy and quantum numbers. - -.. GENERATED FROM PYTHON SOURCE LINES 270-281 - -.. code-block:: Python - - - e, d0, d1, d2, S_squared_exp, S_z_exp = diagonalize(U, JH, t, eL, n=6) - - print("Ground state\nE\t") - for i in range(3): - print(f"{e[i]:.2f}\t{S_squared_exp[i]:.2f}\t{S_z_exp[i]:.2f}") - - print("\nExcited state\nE\t") - for i in range(3, 6): - print(f"{e[i]:.2f}\t{S_squared_exp[i]:.2f}\t{S_z_exp[i]:.2f}") - - - - - -.. rst-class:: sphx-glr-script-out - - .. code-block:: none - - Ground state - E - 0.00 2.00 -1.00 - 0.00 2.00 -0.00 - 0.00 2.00 1.00 - - Excited state - E - 2.74 0.00 0.00 - 2.74 0.00 0.00 - 2.99 0.00 0.00 - - - - -.. GENERATED FROM PYTHON SOURCE LINES 282-284 - -We once again see the same quantum numbers, despite the differences in mixing -in the ground state and exciton. - -.. GENERATED FROM PYTHON SOURCE LINES 287-290 - -.. rubric:: Footnotes - -.. [1] D. Khomskii, Transition Metal Compounds, Cambridge University Press (2014) - - -.. rst-class:: sphx-glr-timing - - **Total running time of the script:** (0 minutes 1.206 seconds) - - -.. _sphx_glr_download_auto_examples_example_7_Hunds_interactions.py: - -.. only:: html - - .. container:: sphx-glr-footer sphx-glr-footer-example - - .. container:: sphx-glr-download sphx-glr-download-jupyter - - :download:`Download Jupyter notebook: example_7_Hunds_interactions.ipynb ` - - .. container:: sphx-glr-download sphx-glr-download-python - - :download:`Download Python source code: example_7_Hunds_interactions.py ` - - .. container:: sphx-glr-download sphx-glr-download-zip - - :download:`Download zipped: example_7_Hunds_interactions.zip ` - - -.. only:: html - - .. rst-class:: sphx-glr-signature - - `Gallery generated by Sphinx-Gallery `_ diff --git a/edrixs/_sources/auto_examples/example_7_transitions.rst.txt b/edrixs/_sources/auto_examples/example_7_transitions.rst.txt deleted file mode 100644 index c57976e1eb..0000000000 --- a/edrixs/_sources/auto_examples/example_7_transitions.rst.txt +++ /dev/null @@ -1,359 +0,0 @@ - -.. DO NOT EDIT. -.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. -.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: -.. "auto_examples/example_7_transitions.py" -.. LINE NUMBERS ARE GIVEN BELOW. - -.. only:: html - - .. note:: - :class: sphx-glr-download-link-note - - :ref:`Go to the end ` - to download the full example code. - -.. rst-class:: sphx-glr-example-title - -.. _sphx_glr_auto_examples_example_7_transitions.py: - - -X-ray transitions -================================================================================ -This example explains how to calculate x-ray transition amplitudes between -specific orbital and spin states. We take the case of a cuprate which has one -hole in the :math:`d_{x^2-y^2}` orbital and a spin ordering direction along the -in-plane diagaonal direction and compute the angular dependence of spin-flip -and non-spin-flip processes. - -This case was chosen because the eigenvectors in question are simple enough -for us to write them out more-or-less by hand, so this example helps the reader -to understand what happens under the hood in more complex cases. - -Some of the code here is credited to Yao Shen who used this approach for the -analysis of a low valence nickelate material [1]_. The task performed repeats -analysis done by many researchers e.g. Luuk Ament et al [2]_ as well as -several other groups. - -.. GENERATED FROM PYTHON SOURCE LINES 20-25 - -.. code-block:: Python - - import edrixs - import numpy as np - import matplotlib.pyplot as plt - import scipy - - - - - - - - -.. GENERATED FROM PYTHON SOURCE LINES 26-42 - -Eigenvectors ------------------------------------------------------------------------------- -Let us start by determining the eigenvectors involved in the transitions. -The spin direction can be set using a vector -:math:`\vec{B}` to represent a magnetic field in terms of generalized spin -operator :math:`\tilde{\sigma}=\vec{B}\cdot\sigma` based on the Pauli matrices -:math:`\sigma`. Let's put the spin along the :math:`[1, 1, 0]` direction -and formuate the problem in the hole basis. -For one particle, we know that the Hamiltonian will be diagonal in the real -harmonic basis. -We can generate the required eigenvectors by making a diagonal -matrix, transforming it to the required -complex harmonic basis (as is standard for EDRIXS) and diagonalizing it. -As long as the crystal field splitting is much larger than the magnetic -field, the eigenvectors will be independent of the exact values of both -these parameters. - -.. GENERATED FROM PYTHON SOURCE LINES 42-56 - -.. code-block:: Python - - - B = 1e-3*np.array([1, 1, 0]) - cf_splitting = 1e3 - zeeman = sum(s*b for s, b in zip(edrixs.get_spin_momentum(2), B)) - dd_levels = np.array([energy for dd_level in cf_splitting*np.arange(5) - for energy in [dd_level]*2], dtype=complex) - emat_rhb = np.diag(dd_levels) - emat = edrixs.cb_op(emat_rhb, edrixs.tmat_r2c('d', ispin=True)) + zeeman - _, eigenvectors = np.linalg.eigh(emat) - - def get_eigenvector(orbital_index, spin_index): - return eigenvectors[:, 2*orbital_index + spin_index] - - - - - - - - - -.. GENERATED FROM PYTHON SOURCE LINES 57-63 - -Let's examine the :math:`d_{x^2-y^2}` orbital first. Recall from the -:ref:`sphx_glr_auto_examples_example_1_crystal_field.py` -example that edrixs uses the standard orbital order of -:math:`d_{3z^2-r^2}, d_{xz}, d_{yz}, d_{x^2-y^2}, d_{xy}`. So we want -:code:`orbital_index = 3` element. Using this, we can build spin-up and -down -eigenvectors. - -.. GENERATED FROM PYTHON SOURCE LINES 63-68 - -.. code-block:: Python - - orbital_index = 3 - - groundstate_vector = get_eigenvector(orbital_index, 0) - excitedstate_vector = get_eigenvector(orbital_index, 1) - - - - - - - - -.. GENERATED FROM PYTHON SOURCE LINES 69-96 - -Transition operators and scattering matrix ------------------------------------------------------------------------------- -Here we are considering the :math:`L_3`-edge. This means -a :math:`2p_{3/2} \rightarrow 3d` -absoprtion transition and a :math:`2p_{3/2} \rightarrow 3d` -emission transition. We can read the relevant matrix from the edrixs database, -keeping in mind that there are in fact three operations for -:math:`x, y,` & :math:`z` directions. Note that edrixs provides operators -in electron notation. If we define :math:`D` as the transition operator in -electron language, :math:`D^\dagger` is the operator in the hole language -we are using for this example. -The angular dependence of a RIXS transition can be conveniently described -using the scattering matrix, which is a :math:`3\times3` element object that -specifies the transition amplitude for each incoming and outgoing x-ray -polarization. Correspondingly, we have - - .. math:: - \begin{equation} - \mathcal{F}=\sum_n\langle f|D|n\rangle\langle n|D^{\dagger}|g\rangle - \end{equation}. - -In matrix form this is - - .. math:: - \begin{equation} - \mathcal{F}(m,n)=\{f^{\dagger} \cdot D(m)\} \cdot \{D^{\dagger}(n) \cdot g\} - \end{equation}. - -.. GENERATED FROM PYTHON SOURCE LINES 96-107 - -.. code-block:: Python - - - D_Tmat = edrixs.get_trans_oper('dp32') - - def get_F(vector_i, vector_f): - F = np.zeros((3, 3), dtype=complex) - for i in range(3): - for j in range(3): - F[i, j] = np.dot(np.dot(np.conj(vector_f.T), D_Tmat[i]), - np.dot(np.conj(D_Tmat[j].T), vector_i)) - return F - - - - - - - - -.. GENERATED FROM PYTHON SOURCE LINES 108-111 - -Using this function, we can obtain non-spin-flip (NSF) and spin-flip (SF) -scattering matrices by choosing whether we return to the ground state or -whether we access the excited state with the spin flipped. - -.. GENERATED FROM PYTHON SOURCE LINES 111-114 - -.. code-block:: Python - - F_NSF = get_F(groundstate_vector, groundstate_vector) - F_SF = get_F(groundstate_vector, excitedstate_vector) - - - - - - - - -.. GENERATED FROM PYTHON SOURCE LINES 115-122 - -Angular dependence ------------------------------------------------------------------------------- -Let's consider the common case of fixing the total scattering angle at -:code:`two_theta = 90` and choosing a series of incident angles :code:`thins`. -Since the detector does not resolve polarization, we need to add both outgoing -polarizations. It is then convenient to use function :func:`.dipole_polvec_rixs` -to obtain the incoming and outgoing polarization vectors. - -.. GENERATED FROM PYTHON SOURCE LINES 122-137 - -.. code-block:: Python - - thins = np.linspace(0, 90) - two_theta = 90 - phi = 0 - - - def get_I(thin, alpha, F): - intensity = 0 - for beta in [0, np.pi/2]: - thout = two_theta - thin - ei, ef = edrixs.dipole_polvec_rixs(thin*np.pi/180, thout*np.pi/180, - phi*np.pi/180, alpha, beta) - intensity += np.abs(np.dot(ef, np.dot(F, ei)))**2 - return intensity - - - - - - - - - -.. GENERATED FROM PYTHON SOURCE LINES 138-143 - -Plot ------------------------------------------------------------------------------- -We now run through a few configurations specified in terms of incoming -polarization angle :math:`\alpha` (defined in radians w.r.t. the scattering -plane), :math:`F`, plotting label, and plotting color. - -.. GENERATED FROM PYTHON SOURCE LINES 143-159 - -.. code-block:: Python - - fig, ax = plt.subplots() - - config = [[0, F_NSF, r'$\pi$ NSF', 'C0'], - [np.pi/2, F_NSF, r'$\sigma$ NSF', 'C1'], - [0, F_SF, r'$\pi$ SF', 'C2'], - [np.pi/2, F_SF, r'$\sigma$ SF', 'C3']] - - for alpha, F, label, color in config: - Is = np.array([get_I(thin, alpha, F) for thin in thins]) - ax.plot(thins, Is, label=label, color=color) - - ax.legend() - ax.set_xlabel(r'Theta ($^\circ$)') - ax.set_ylabel('Relative intensity') - plt.show() - - - - -.. image-sg:: /auto_examples/images/sphx_glr_example_7_transitions_001.png - :alt: example 7 transitions - :srcset: /auto_examples/images/sphx_glr_example_7_transitions_001.png - :class: sphx-glr-single-img - - - - - -.. GENERATED FROM PYTHON SOURCE LINES 160-164 - -Run through orbitals ------------------------------------------------------------------------------- -For completeness, let's look at transitions from :math:`x^2-y^2` to all other -orbitals. - -.. GENERATED FROM PYTHON SOURCE LINES 164-191 - -.. code-block:: Python - - fig, axs = plt.subplots(5, 1, figsize=(7, 7), - sharex=True, sharey=True) - - orbitals = ['$d_{3z^2-r^2}$', '$d_{xz}$', '$d_{yz}$', - '$d_{x^2-y^2}$', '$d_{xy}$'] - orbital_order = [4, 1, 2, 0, 3] - - plot_index = 0 - for ax, orbital_index in zip(axs, orbital_order): - for spin_index, spin_label in zip([0, 1], ['NSF', 'SF']): - excitedstate_vector = get_eigenvector(orbital_index, spin_index) - F = get_F(groundstate_vector, excitedstate_vector) - for alpha, pol_label in zip([0, np.pi/2], [r'$\pi$', r'$\sigma$']): - Is = np.array([get_I(thin, alpha, F) for thin in thins]) - ax.plot(thins, Is*10, label=f'{pol_label} {spin_label}', - color=f'C{plot_index%4}') - plot_index += 1 - ax.legend(title=orbitals[orbital_index], bbox_to_anchor=(1.1, 1), - loc="upper left", fontsize=8) - - - axs[-1].set_xlabel(r'$\theta$ ($^\circ$)') - axs[2].set_ylabel('Scattering intensity') - - fig.subplots_adjust(hspace=0, left=.3, right=.6) - plt.show() - - - - -.. image-sg:: /auto_examples/images/sphx_glr_example_7_transitions_002.png - :alt: example 7 transitions - :srcset: /auto_examples/images/sphx_glr_example_7_transitions_002.png - :class: sphx-glr-single-img - - - - - -.. GENERATED FROM PYTHON SOURCE LINES 192-198 - -.. rubric:: Footnotes - -.. [1] Yao Shen et al., - `arXiv:2110.08937 (2022) `_. -.. [2] Luuk J. P. Ament et al., - `Phys. Rev. Lett. 103, 117003 (2009) `_ - - -.. rst-class:: sphx-glr-timing - - **Total running time of the script:** (0 minutes 0.514 seconds) - - -.. _sphx_glr_download_auto_examples_example_7_transitions.py: - -.. only:: html - - .. container:: sphx-glr-footer sphx-glr-footer-example - - .. container:: sphx-glr-download sphx-glr-download-jupyter - - :download:`Download Jupyter notebook: example_7_transitions.ipynb ` - - .. container:: sphx-glr-download sphx-glr-download-python - - :download:`Download Python source code: example_7_transitions.py ` - - .. container:: sphx-glr-download sphx-glr-download-zip - - :download:`Download zipped: example_7_transitions.zip ` - - -.. only:: html - - .. rst-class:: sphx-glr-signature - - `Gallery generated by Sphinx-Gallery `_ diff --git a/edrixs/_sources/auto_examples/example_8_Coulomb.rst.txt b/edrixs/_sources/auto_examples/example_8_Coulomb.rst.txt deleted file mode 100644 index a0832e5714..0000000000 --- a/edrixs/_sources/auto_examples/example_8_Coulomb.rst.txt +++ /dev/null @@ -1,635 +0,0 @@ - -.. DO NOT EDIT. -.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. -.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: -.. "auto_examples/example_8_Coulomb.py" -.. LINE NUMBERS ARE GIVEN BELOW. - -.. only:: html - - .. note:: - :class: sphx-glr-download-link-note - - :ref:`Go to the end ` - to download the full example code. - -.. rst-class:: sphx-glr-example-title - -.. _sphx_glr_auto_examples_example_8_Coulomb.py: - - -Coulomb interactions -===================================== -In this example we provide more details on how Coulomb interactions are -implemented in multiplet calculations and EDRIXS in particular. We aim -to clarify the form of the matrices, how they are parametrized, -and how the breaking of spherical symmetry can switch on additional elements -that one might not anticipate. Our example is based on a :math:`d` atomic shell. - -.. GENERATED FROM PYTHON SOURCE LINES 13-50 - -Create matrix ------------------------------------------------------------------------------- -The Coulomb interaction between two particles can be written as - - .. math:: - \begin{equation} - \hat{H} = \frac{1}{2} - \int d\mathbf{r} \int d\mathbf{r}^\prime - \Sigma_{\sigma, \sigma^\prime} - |\hat{\psi}^\sigma(\mathbf{r})|^2 \frac{e^2}{R} - |\hat{\psi}^{\sigma^\prime}(\mathbf{r^\prime})|^2, - \end{equation} - -where :math:`\hat{\psi}^\sigma(\mathbf{r})` is the electron wavefunction, with -spin :math:`\sigma`, and :math:`R=|r-r^\prime|` is the electron separation. -Solving our problem in this form is difficult due to the need to symmeterize -the wavefunction to follow fermionic statistics. -Using second quantization, we can use operators to impose the required -particle exchange statistics and write the equation in terms of -a tensor :math:`U` - - .. math:: - \begin{equation} - \hat{H} = \sum_{\alpha,\beta,\gamma,\delta,\sigma,\sigma^\prime} - U_{\alpha\sigma,\beta\sigma^\prime,\gamma\sigma^\prime,\delta\sigma} - \hat{f}^{\dagger}_{\alpha\sigma} - \hat{f}^{\dagger}_{\beta\sigma^\prime} - \hat{f}_{\gamma\sigma^\prime}\hat{f}_{\delta\sigma}, - \end{equation} - -where :math:`\alpha`, :math:`\beta`, :math:`\gamma`, :math:`\delta` are -orbital indices and :math:`\hat{f}^{\dagger}` -(:math:`\hat{f}`) are the creation (anihilation) operators. -For a :math:`d`-electron system, we have :math:`10` distinct spin-orbitals -(:math:`5` orbitals each with :math:`2` spins), which makes matrix the -:math:`10\times10\times10\times10` in total size. -In EDRIXS the matrix can be created as follows: - -.. GENERATED FROM PYTHON SOURCE LINES 50-58 - -.. code-block:: Python - - import edrixs - import numpy as np - import scipy - import matplotlib.pyplot as plt - import itertools - - F0, F2, F4 = 6.94, 14.7, 4.41 - umat_chb = edrixs.get_umat_slater('d', F0, F2, F4) - - - - - - - -.. GENERATED FROM PYTHON SOURCE LINES 59-61 - -We stored this under variable :code:`umat_chb` where "cbh" stands for -complex harmonic basis, which is the default basis in EDRIXS. - -.. GENERATED FROM PYTHON SOURCE LINES 63-92 - -Parameterizing interactions ------------------------------------------------------------------------------- -EDRIXS parameterizes the interactions in :math:`U` via Slater integral -parameters :math:`F^{k}`. These relate to integrals of various spherical -Harmonics as well as Clebsch-Gordon coefficients, Gaunt coefficients, -and Wigner 3J symbols. Textbooks such as [1]_ can be used for further -reference. If you are interested in the details of how -EDRIXS does this (and you probably aren't) function :func:`.umat_slater`, -constructs the required matrix via Gaunt coeficents from -:func:`.get_gaunt`. Two alternative parameterizations are common. -The first are the Racah parameters, which are - - .. math:: - \begin{eqnarray} - A &=& F^0 - \frac{49}{441} F^4 \\ - B &=& \frac{1}{49}F^2 - \frac{5}{441}F^4 \\ - C &=& \frac{35}{441}F^4. - \end{eqnarray} - -or an alternative form for the Slater integrals - - .. math:: - \begin{eqnarray} - F_0 &=& F^0 \\ - F_2 &=& \frac{1}{49}F^2 \\ - F_4 &=& \frac{1}{441}F^4, - \end{eqnarray} - -which involves different normalization parameters. - -.. GENERATED FROM PYTHON SOURCE LINES 94-101 - -Basis transform ------------------------------------------------------------------------------- -If we want to use the real harmonic basis, we can use a tensor -transformation, which imposes the following orbital order -:math:`3z^2-r^2, xz, yz, x^2-y^2, xy`, each of which involves -:math:`\uparrow, \downarrow` spin pairs. Let's perform this transformation and -store a list of these orbitals. - -.. GENERATED FROM PYTHON SOURCE LINES 101-104 - -.. code-block:: Python - - umat = edrixs.transform_utensor(umat_chb, edrixs.tmat_c2r('d', True)) - orbitals = ['3z^2-r^2', 'xz', 'yz', 'x^2-y^2', 'xy'] - - - - - - - - -.. GENERATED FROM PYTHON SOURCE LINES 105-158 - -Interactions ------------------------------------------------------------------------------- -Tensor :math:`U` is a series of matrix -elements - - .. math:: - \begin{equation} - \langle\psi_{\gamma,\delta}^{\bar{\sigma},\bar{\sigma}^\prime} - |\hat{H}| - \psi_{\alpha,\beta}^{\sigma,\sigma^\prime}\rangle - \end{equation} - -the combination of which defines the energetic cost of pairwise -electron-electron interactions between states :math:`\alpha,\sigma` -and :math:`\beta,\sigma^\prime`. In EDRIXS we follow the convention of -summing over all orbital pairs. Some other texts count each pair of -indices only once. The matrix elements here will consequently -be half the magnitude of those in other references. -We can express the interactions in terms of -the orbitals involved. It is common to distinguish "direct Coulomb" and -"exchange" interactions. The former come from electrons in the same orbital -and the later involve swapping orbital labels for electrons. We will use -:math:`U_0` and :math:`J` as a shorthand for distinguishing these. - -Before we describe the different types of interactions, we note that since -the Coulomb interaction is real, and due to the spin symmmetry properties -of the process :math:`U` always obeys - - .. math:: - \begin{equation} - U_{\alpha\sigma,\beta\sigma^\prime,\gamma\sigma^\prime,\delta\sigma} = - U_{\beta\sigma,\alpha\sigma^\prime,\delta\sigma^\prime,\gamma\sigma} = - U_{\delta\sigma,\gamma\sigma^\prime,\beta\sigma^\prime,\alpha\sigma} = - U_{\gamma\sigma,\delta\sigma^\prime,\alpha\sigma^\prime,\beta\sigma}. - \end{equation} - - -1. Intra orbital -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -The direct Coulomb energy cost to double-occupy an orbital comes from terms -like :math:`U_{\alpha\sigma,\alpha\bar\sigma,\alpha\bar\sigma,\alpha\sigma}`. -In this notation, we use :math:`\sigma^\prime` to denote that the matrix -element is summed over all pairs and :math:`\bar{\sigma}` to denote sums -over all opposite spin pairs. Due to rotational symmetry, all these -elements are the same and equal to - - .. math:: - \begin{eqnarray} - U_0 &=& \frac{A}{2} + 2B + \frac{3C}{2}\\ - &=& \frac{F_0}{2} + 2F_2 + 18F_4 - \end{eqnarray} - -Let's print these to demonstrate where these live in the array - -.. GENERATED FROM PYTHON SOURCE LINES 158-162 - -.. code-block:: Python - - for i in range(0, 5): - val = umat[i*2, i*2 + 1, i*2 + 1, i*2].real - print(f"{orbitals[i]:<8} \t {val:.3f}") - - - - - -.. rst-class:: sphx-glr-script-out - - .. code-block:: none - - 3z^2-r^2 4.250 - xz 4.250 - yz 4.250 - x^2-y^2 4.250 - xy 4.250 - - - - -.. GENERATED FROM PYTHON SOURCE LINES 163-170 - -2. Inter orbital Coulomb interactions -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -Direct Coulomb repulsion between different orbitals depends on terms like -:math:`U_{\alpha\sigma,\beta\sigma^\prime,\beta\sigma^\prime,\alpha\sigma}`. -Expresions for these parameters are provided in column :math:`U` in -:ref:`table_2_orbital`. We can print the values from :code:`umat` -like this: - -.. GENERATED FROM PYTHON SOURCE LINES 170-174 - -.. code-block:: Python - - for i, j in itertools.combinations(range(5), 2): - val = umat[i*2, j*2 + 1, j*2 + 1, i*2].real - print(f"{orbitals[i]:<8} \t {orbitals[j]:<8} \t {val:.3f}") - - - - - -.. rst-class:: sphx-glr-script-out - - .. code-block:: none - - 3z^2-r^2 xz 3.650 - 3z^2-r^2 yz 3.650 - 3z^2-r^2 x^2-y^2 2.900 - 3z^2-r^2 xy 2.900 - xz yz 3.150 - xz x^2-y^2 3.150 - xz xy 3.150 - yz x^2-y^2 3.150 - yz xy 3.150 - x^2-y^2 xy 3.900 - - - - -.. GENERATED FROM PYTHON SOURCE LINES 175-181 - -3. Inter-orbital exchange interactions -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -Exchange terms exist with the form -:math:`U_{\alpha\sigma,\beta\sigma^\prime,\alpha\sigma^\prime,\beta\sigma}`. -Expresions for these parameters are provided in column :math:`J` of -:ref:`table_2_orbital`. These come from terms like this in the matrix: - -.. GENERATED FROM PYTHON SOURCE LINES 181-185 - -.. code-block:: Python - - for i, j in itertools.combinations(range(5), 2): - val = umat[i*2, j*2 + 1, i*2 + 1, j*2].real - print(f"{orbitals[i]:<8} \t {orbitals[j]:<8} \t {val:.3f}") - - - - - -.. rst-class:: sphx-glr-script-out - - .. code-block:: none - - 3z^2-r^2 xz 0.300 - 3z^2-r^2 yz 0.300 - 3z^2-r^2 x^2-y^2 0.675 - 3z^2-r^2 xy 0.675 - xz yz 0.550 - xz x^2-y^2 0.550 - xz xy 0.550 - yz x^2-y^2 0.550 - yz xy 0.550 - x^2-y^2 xy 0.175 - - - - -.. GENERATED FROM PYTHON SOURCE LINES 186-193 - -4. Pair hopping term -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -Terms that swap pairs of electrons exist as -:math:`(1-\delta_{\sigma\sigma'})U_{\alpha\sigma,\alpha\bar\sigma,\beta\bar\sigma,\beta\sigma}` -and depend on exchange interactions column :math:`J` from -:ref:`table_2_orbital` -and here in the matrix. - -.. GENERATED FROM PYTHON SOURCE LINES 193-197 - -.. code-block:: Python - - for i, j in itertools.combinations(range(5), 2): - val = umat[i*2, i*2 + 1, j*2 + 1, j*2].real - print(f"{orbitals[i]:<8} \t {orbitals[j]:<8} \t {val:.3f}") - - - - - -.. rst-class:: sphx-glr-script-out - - .. code-block:: none - - 3z^2-r^2 xz 0.300 - 3z^2-r^2 yz 0.300 - 3z^2-r^2 x^2-y^2 0.675 - 3z^2-r^2 xy 0.675 - xz yz 0.550 - xz x^2-y^2 0.550 - xz xy 0.550 - yz x^2-y^2 0.550 - yz xy 0.550 - x^2-y^2 xy 0.175 - - - - -.. GENERATED FROM PYTHON SOURCE LINES 198-215 - -5. Three orbital -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -Another set of terms that one might not immediately anticipate involve three -orbitals like - - .. math:: - \begin{equation} - U_{\alpha\sigma, \gamma\sigma', \beta\sigma', \gamma\sigma} \\ - U_{\alpha\sigma, \gamma\sigma', \gamma\sigma', \beta\sigma} \\ - (1-\delta_{\sigma\sigma'}) - U_{\alpha\sigma, \beta\sigma', \gamma\sigma', \gamma\sigma} - \end{equation} - -for :math:`\alpha=3z^2-r^2, \beta=x^2-y^2, \gamma=xz/yz`. -These are needed to maintain the rotational symmetry of the interations. -See :ref:`table_3_orbital` for the expressions. We can print some of -these via: - -.. GENERATED FROM PYTHON SOURCE LINES 215-227 - -.. code-block:: Python - - ijkl = [[0, 1, 3, 1], - [0, 2, 3, 2], - [1, 0, 3, 1], - [1, 1, 3, 0], - [2, 0, 3, 2], - [2, 2, 3, 0]] - - for i, j, k, l in ijkl: - val = umat[i*2, j*2 + 1, k*2 + 1, l*2].real - print(f"{orbitals[i]:<8} \t {orbitals[j]:<8} \t" - f"{orbitals[k]:<8} \t {orbitals[l]:<8} \t {val:.3f}") - - - - - -.. rst-class:: sphx-glr-script-out - - .. code-block:: none - - 3z^2-r^2 xz x^2-y^2 xz 0.217 - 3z^2-r^2 yz x^2-y^2 yz -0.217 - xz 3z^2-r^2 x^2-y^2 xz -0.433 - xz xz x^2-y^2 3z^2-r^2 0.217 - yz 3z^2-r^2 x^2-y^2 yz 0.433 - yz yz x^2-y^2 3z^2-r^2 -0.217 - - - - -.. GENERATED FROM PYTHON SOURCE LINES 228-233 - -6. Four orbital -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -Futher multi-orbital terms include -:math:`U_{\alpha\sigma,\beta\sigma^\prime,\gamma\sigma^\prime,\delta\sigma}`. -We can find these here in the matrix: - -.. GENERATED FROM PYTHON SOURCE LINES 233-249 - -.. code-block:: Python - - ijkl = [[0, 1, 2, 4], - [0, 1, 4, 2], - [0, 2, 1, 4], - [0, 2, 4, 1], - [0, 4, 1, 2], - [0, 4, 2, 1], - [3, 1, 4, 2], - [3, 2, 4, 1], - [3, 4, 1, 2], - [3, 4, 2, 1]] - - for i, j, k, l in ijkl: - val = umat[i*2, j*2 + 1, k*2 + 1, l*2].real - print(f"{orbitals[i]:<8} \t {orbitals[j]:<8} \t {orbitals[k]:<8}" - f"\t {orbitals[l]:<8} \t {val:.3f}") - - - - - -.. rst-class:: sphx-glr-script-out - - .. code-block:: none - - 3z^2-r^2 xz yz xy -0.433 - 3z^2-r^2 xz xy yz 0.217 - 3z^2-r^2 yz xz xy -0.433 - 3z^2-r^2 yz xy xz 0.217 - 3z^2-r^2 xy xz yz 0.217 - 3z^2-r^2 xy yz xz 0.217 - x^2-y^2 xz xy yz -0.375 - x^2-y^2 yz xy xz 0.375 - x^2-y^2 xy xz yz -0.375 - x^2-y^2 xy yz xz 0.375 - - - - -.. GENERATED FROM PYTHON SOURCE LINES 250-256 - -Effects of multi-orbital terms ------------------------------------------------------------------------------- -To test the effects of the multi-orbital terms, let's plot the eigenenergy -spectra with and without multi-orbital terms switched on for system with and -without a cubic crystal field. We will use a :math:`d`-shell with two -electrons. - -.. GENERATED FROM PYTHON SOURCE LINES 256-292 - -.. code-block:: Python - - ten_dqs = [0, 2, 4, 12] - - def diagonalize(ten_dq, umat): - emat = edrixs.cb_op(edrixs.cf_cubic_d(ten_dq), - edrixs.tmat_c2r('d', ispin=True)) - H = (edrixs.build_opers(4, umat, basis) - + edrixs.build_opers(2, emat, basis)) - e, v = scipy.linalg.eigh(H) - return e - e.min() - - basis = edrixs.get_fock_bin_by_N(10, 2) - umat_no_multiorbital = np.copy(umat) - B = F2/49 - 5*F4/441 - for val in [np.sqrt(3)*B/2, np.sqrt(3)*B, 3*B/2]: - umat_no_multiorbital[(np.abs(umat)- val) < 1e-6] = 0 - - fig, axs = plt.subplots(1, len(ten_dqs), figsize=(8, 3)) - - for cind, (ax, ten_dq) in enumerate(zip(axs, ten_dqs)): - ax.hlines(diagonalize(ten_dq, umat), xmin=0, xmax=1, - label='on', color=f'C{cind}') - ax.hlines(diagonalize(ten_dq, umat_no_multiorbital), - xmin=1.5, xmax=2.5, - label='off', - linestyle=':', color=f'C{cind}') - ax.set_title(f"$10D_q={ten_dq}$") - ax.set_ylim([-.5, 20]) - ax.set_xticks([]) - ax.legend() - - fig.suptitle("Eigenvalues with 3&4-orbital effects on/off") - fig.subplots_adjust(wspace=.3) - axs[0].set_ylabel('Eigenvalues (eV)') - fig.subplots_adjust(top=.8) - plt.show() - - - - -.. image-sg:: /auto_examples/images/sphx_glr_example_8_Coulomb_001.png - :alt: Eigenvalues with 3&4-orbital effects on/off, $10D_q=0$, $10D_q=2$, $10D_q=4$, $10D_q=12$ - :srcset: /auto_examples/images/sphx_glr_example_8_Coulomb_001.png - :class: sphx-glr-single-img - - - - - -.. GENERATED FROM PYTHON SOURCE LINES 293-382 - -On the left of the plot Coulomb interactions in spherical symmetry cause -substantial mxing between :math:`t_{2g}` and :math:`e_{g}` orbitals in the -eigenstates and 3 & 4 orbital orbital terms are crucial for obtaining the -the right eigenenergies. As :math:`10D_q` get large, this mixing is switched -off and the spectra start to become independent of whether the 3 & 4 orbital -orbital terms are included or not. - - - -.. _table_2_orbital: -.. table:: Table of 2 orbital interactions - - +-----------------------------+------------------+-----------------------+----------------+--------------------+ - |Orbitals :math:`\alpha,\beta`|:math:`U_0` Racah | :math:`U_0` Slater |:math:`J` Racah |:math:`J` Slater | - +=============================+==================+=======================+================+====================+ - |:math:`3z^2-r^2, xz` |:math:`A/2+B+C/2` |:math:`F_0/2+F_2-12F_4`| :math:`B/2+C/2`|:math:`F_2/2+15F_4` | - +-----------------------------+------------------+-----------------------+----------------+--------------------+ - |:math:`3z^2-r^2, yz` |:math:`A/2+B+C/2` |:math:`F_0/2+F_2-12F_4`| :math:`B/2+C/2`|:math:`F_2/2+15F_4` | - +-----------------------------+------------------+-----------------------+----------------+--------------------+ - |:math:`3z^2-r^2, x^2-y^2` |:math:`A/2-2B+C/2`|:math:`F_0/2-2F_2+3F_4`|:math:`2B+C/2` |:math:`2F_2+15F_4/2`| - +-----------------------------+------------------+-----------------------+----------------+--------------------+ - |:math:`3z^2-r^2, xy` |:math:`A/2-2B+C/2`|:math:`F_0/2-2F_2+3F_4`|:math:`2B+C/2` |:math:`2F_2+15F_4/2`| - +-----------------------------+------------------+-----------------------+----------------+--------------------+ - |:math:`xz, yz` |:math:`A/2-B+C/2` |:math:`F_0/2-F_2-12F_4`|:math:`3B/2+C/2`|:math:`3F_2/2+10F_4`| - +-----------------------------+------------------+-----------------------+----------------+--------------------+ - |:math:`xz, x^2-y^2` |:math:`A/2-B+C/2` |:math:`F_0/2-F_2-12F_4`|:math:`3B/2+C/2`|:math:`3F_2/2+10F_4`| - +-----------------------------+------------------+-----------------------+----------------+--------------------+ - |:math:`xz, xy` |:math:`A/2-B+C/2` |:math:`F_0/2-F_2-12F_4`|:math:`3B/2+C/2`|:math:`3F_2/2+10F_4`| - +-----------------------------+------------------+-----------------------+----------------+--------------------+ - |:math:`yz, x^2-y^2` |:math:`A/2-B+C/2` |:math:`F_0/2-F_2-12F_4`|:math:`3B/2+C/2`|:math:`3F_2/2+10F_4`| - +-----------------------------+------------------+-----------------------+----------------+--------------------+ - |:math:`yz, xy` |:math:`A/2-B+C/2` |:math:`F_0/2-F_2-12F_4`|:math:`3B/2+C/2`|:math:`3F_2/2+10F_4`| - +-----------------------------+------------------+-----------------------+----------------+--------------------+ - |:math:`x^2-y^2, xy` |:math:`A/2+2B+C/2`|:math:`F_0+4F_2-34F_4` | :math:`C/2` |:math:`35F_4/2` | - +-----------------------------+------------------+-----------------------+----------------+--------------------+ - - -.. _table_3_orbital: -.. table:: Table of 3 orbital interactions - - +-----------------------------+-------------+----------------------------------------------------+-----------------------------------------------------+ - |Orbitals :math:`\alpha,\beta,\gamma,\delta`|:math:`\langle\alpha\beta|\gamma\delta\rangle` Racah|:math:`\langle\alpha\beta|\gamma\delta\rangle` Slater| - +=============================+=============+====================================================+=====================================================+ - |:math:`3z^2-r^2, xz, x^2-y^2, xz` | :math:`\sqrt{3}B/2` | :math:`\sqrt{3}F_2/2-5\sqrt{3}F_4/2` | - +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ - |:math:`3z^2-r^2, yz, x^2-y^2, yz` | :math:`-\sqrt{3}B/2` | :math:`-\sqrt{3}F_2/2+5\sqrt{3}F_4/2` | - +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ - |:math:`xz, 3z^2-r^2, x^2-y^2, xz` | :math:`-\sqrt{3}B` | :math:`-\sqrt{3}F_2+5\sqrt{3}F_4` | - +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ - |:math:`xz, xz, x^2-y^2, 3z^2-r^2` | :math:`\sqrt{3}B/2` | :math:`\sqrt{3}F_2/2-5\sqrt{3}F_4/2` | - +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ - |:math:`yz, 3z^2-r^2, x^2-y^2, yz` | :math:`\sqrt{3}B` | :math:`\sqrt{3}F_2-5\sqrt{3}F_4` | - +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ - |:math:`yz, yz, x^2-y^2, 3z^2-r^2` | :math:`-\sqrt{3}B/2` | :math:`-\sqrt{3}F_2/2+5\sqrt{3}F_4/2` | - +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ - - -.. _table_4_orbital: -.. table:: Table of 4 orbital interactions - - +-----------------------------+-------------+----------------------------------------------------+-----------------------------------------------------+ - |Orbitals :math:`\alpha,\beta,\gamma,\delta`|:math:`\langle\alpha\beta|\gamma\delta\rangle` Racah|:math:`\langle\alpha\beta|\gamma\delta\rangle` Slater| - +=============================+=============+====================================================+=====================================================+ - |:math:`3z^2-r^2, xz, yz, xy` | :math:`-\sqrt{3}B` | :math:`-\sqrt{3}F_2+5\sqrt{3}F_4` | - +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ - |:math:`3z^2-r^2, xz, xy, yz` | :math:`\sqrt{3}B/2` | :math:`\sqrt{3}F_2/2-5\sqrt{3}F_4/2` | - +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ - |:math:`3z^2-r^2, yz, xz, xy` | :math:`-\sqrt{3}B` | :math:`-\sqrt{3}F_2+5\sqrt{3}F_4` | - +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ - |:math:`3z^2-r^2, yz, xy, xz` | :math:`\sqrt{3}B/2` | :math:`\sqrt{3}F_2/2-5\sqrt{3}F_4/2` | - +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ - |:math:`3z^2-r^2, xy, xz, yz` | :math:`\sqrt{3}B/2` | :math:`\sqrt{3}F_2/2-5\sqrt{3}F_4/2` | - +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ - |:math:`3z^2-r^2, xy, yz, xz` | :math:`\sqrt{3}B/2` | :math:`\sqrt{3}F_2/2-5\sqrt{3}F_4/2` | - +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ - |:math:`x^2-y^2 , xz, xy, yz` | :math:`-3B/2` | :math:`-3F_2/2+15F_4/2` | - +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ - |:math:`x^2-y^2 , yz, xy, xz` | :math:`3B/2` | :math:`3F_2/2-15F_4/2` | - +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ - |:math:`x^2-y^2 , xy, xz, yz` | :math:`-3B/2` | :math:`-3F_2/2+15F_4/2` | - +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ - |:math:`x^2-y^2 , xy, yz, xz` | :math:`3B/2` | :math:`3F_2/2-15F_4/2` | - +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ - - -.. rubric:: Footnotes - -.. [1] MSugano S, Tanabe Y and Kamimura H. 1970. Multiplets of - Transition-Metal Ions in Crystals. Academic Press, New York and London. - - -.. rst-class:: sphx-glr-timing - - **Total running time of the script:** (0 minutes 0.360 seconds) - - -.. _sphx_glr_download_auto_examples_example_8_Coulomb.py: - -.. only:: html - - .. container:: sphx-glr-footer sphx-glr-footer-example - - .. container:: sphx-glr-download sphx-glr-download-jupyter - - :download:`Download Jupyter notebook: example_8_Coulomb.ipynb ` - - .. container:: sphx-glr-download sphx-glr-download-python - - :download:`Download Python source code: example_8_Coulomb.py ` - - .. container:: sphx-glr-download sphx-glr-download-zip - - :download:`Download zipped: example_8_Coulomb.zip ` - - -.. only:: html - - .. rst-class:: sphx-glr-signature - - `Gallery generated by Sphinx-Gallery `_ diff --git a/edrixs/_sources/auto_examples/example_8_Hunds_interactions.rst.txt b/edrixs/_sources/auto_examples/example_8_Hunds_interactions.rst.txt deleted file mode 100644 index 1e13dfccc0..0000000000 --- a/edrixs/_sources/auto_examples/example_8_Hunds_interactions.rst.txt +++ /dev/null @@ -1,478 +0,0 @@ - -.. DO NOT EDIT. -.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. -.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: -.. "auto_examples/example_8_Hunds_interactions.py" -.. LINE NUMBERS ARE GIVEN BELOW. - -.. only:: html - - .. note:: - :class: sphx-glr-download-link-note - - :ref:`Go to the end ` - to download the full example code. - -.. rst-class:: sphx-glr-example-title - -.. _sphx_glr_auto_examples_example_8_Hunds_interactions.py: - - -Hund's Interactions in charge transfer insulators -================================================= -In this exercise we will solve a toy model relevant to cubic :math:`d^8` charge transfer insulators -such as NiO or NiPS\ :sub:`3`. We are interested in better understanding the interplay between the -Hund's interactions and the charge transfer energy in terms of the energy of the triplet-singlet -excitations of this model. These seem to act against each other in that the Hund's interactions -impose a energy cost for the triplet-singlet excitations whenever there are two holes on -the Ni :math:`d` orbitals. The charge transfer physics, on the other hand, will promote a -:math:`d^9\underline{L}` ground state in which the Hund's interactions are not active. - -The simplest model that captures this physics requires four Ni spin-orbitals, representing the Ni -:math:`e_g` manifold. We will represent the ligand states in the same way as the Anderson impurity -model in terms of one effective ligand spin-orbital per Ni spin-orbital. We assume these effective -orbitals have been constructed so that each Ni orbital only bonds to one sister orbital. For -simplicity, we will treat all Ni and all ligand orbitals as equivalent, even though a more -realistic model would account for the different Coulomb and hopping of the :math:`d_{3z^2-r^2}` -and :math:`d_{x^2-y^2}` orbitals. We therefore simply connect Ni and ligand orbitals via a constant -hopping :math:`t`. We also include the ligand energy parameter :math:`e_L`. - -The easiest way to implement the requried Coulomb interactions is to use the so-called Kanamori -Hamiltonian, which is a simplfied form for the interactions, which treats all orbitals as -equivalent. Daniel Khomskii's book provides a great explanation of this physics [1]_. We -parameterize the interactions via Coulomb repulsion parameter :math:`U` and Hund's exchange -:math:`J_H`. EDRIXS provides this functionality via the more general -:func:`.get_umat_kanamori` function. - -It's also easiest to consider this problem in hole langauge, which means our eight spin-orbitals -are populated by two fermions. - -.. GENERATED FROM PYTHON SOURCE LINES 34-38 - -Setup ------------------------------------------------------------------------------- -We start by loading the necessary modules, and defining the total number of -orbitals and electrons. - -.. GENERATED FROM PYTHON SOURCE LINES 38-46 - -.. code-block:: Python - - import edrixs - import scipy - import numpy as np - import matplotlib.pyplot as plt - - norb = 8 - noccu = 2 - - - - - - - - -.. GENERATED FROM PYTHON SOURCE LINES 47-55 - -Diagonalization ------------------------------------------------------------------------------- -Let's write a function to diagonalize our model in a similar way to -the :ref:`sphx_glr_auto_examples_example_6_Hubbard_dimer.py` example. -Within this function, we also create operators to count the number of -:math:`d` holes and operators to calculate expectation values for -:math:`S^2` and :math:`S_z`. For the latter to make sense, we also include a -small effective spin interaction along :math:`z`. - -.. GENERATED FROM PYTHON SOURCE LINES 55-103 - -.. code-block:: Python - - - - def diagonalize(U, JH, t, eL, n=1): - # Setup Coulomb matrix - umat = np.zeros((norb, norb, norb, norb), dtype=complex) - uNi = edrixs.get_umat_kanamori(norb//2, U, JH) - umat[:norb//2, :norb//2, :norb//2, :norb//2] = uNi - - # Setup hopping matrix - emat = np.zeros((norb, norb), dtype=complex) - ind = np.arange(norb//2) - emat[ind, ind + norb//2] = t - emat[ind+norb//2, ind] = np.conj(t) # conj is not needed, but is good practise. - ind = np.arange(norb//2, norb) - emat[ind, ind] += eL - - # Spin operator - spin_mom = np.zeros((3, norb, norb), dtype=complex) - spin_mom[:, :2, :2] = edrixs.get_spin_momentum(0) - spin_mom[:, 2:4, 2:4] = edrixs.get_spin_momentum(0) - spin_mom[:, 4:6, 4:6] = edrixs.get_spin_momentum(0) - spin_mom[:, 6:8, 6:8] = edrixs.get_spin_momentum(0) - - # add small effective field along z - emat += 1e-6*spin_mom[2] - - # Diagonalize - basis = edrixs.get_fock_bin_by_N(norb, noccu) - H = edrixs.build_opers(2, emat, basis) + edrixs.build_opers(4, umat, basis) - e, v = scipy.linalg.eigh(H) - e -= e[0] # Define ground state as zero energy - - # Operator for holes on Ni - basis = np.array(basis) - num_d_electrons = basis[:, :4].sum(1) - d0 = np.sum(np.abs(v[num_d_electrons == 0, :])**2, axis=0) - d1 = np.sum(np.abs(v[num_d_electrons == 1, :])**2, axis=0) - d2 = np.sum(np.abs(v[num_d_electrons == 2, :])**2, axis=0) - - # S^2 and Sz operators - opS = edrixs.build_opers(2, spin_mom, basis) - S_squared_op = np.dot(opS[0], opS[0]) + np.dot(opS[1], opS[1]) + np.dot(opS[2], opS[2]) - S_squared_exp = edrixs.cb_op(S_squared_op, v).diagonal().real - S_z_exp = edrixs.cb_op(opS[2], v).diagonal().real - - return e[:n], d0[:n], d1[:n], d2[:n], S_squared_exp[:n], S_z_exp[:n] - - - - - - - - - -.. GENERATED FROM PYTHON SOURCE LINES 104-110 - -The atomic limit ------------------------------------------------------------------------------- -For simplicity, let's start in the atomic limit with :math:`e_L \gg t \gg U` -where all holes are on nickel. In this case, there are six ways to distribute -two holes on the four Ni spin-orbitals. Let's examine the expectation values -of the :math:`S^2` and :math:`S_z` operators. - -.. GENERATED FROM PYTHON SOURCE LINES 110-125 - -.. code-block:: Python - - U = 10 - JH = 2 - t = 100 - eL = 1e10 - - e, d0, d1, d2, S_squared_exp, S_z_exp = diagonalize(U, JH, t, eL, n=6) - - print("Ground state\nE\t") - for i in range(3): - print(f"{e[i]:.2f}\t{S_squared_exp[i]:.2f}\t{S_z_exp[i]:.2f}") - - print("\nExcited state\nE\t") - for i in range(3, 6): - print(f"{e[i]:.2f}\t{S_squared_exp[i]:.2f}\t{S_z_exp[i]:.2f}") - - - - - -.. rst-class:: sphx-glr-script-out - - .. code-block:: none - - Ground state - E - 0.00 2.00 1.00 - 0.00 2.00 0.00 - 0.00 2.00 -1.00 - - Excited state - E - 4.00 0.00 0.00 - 4.00 0.00 0.00 - 8.00 0.00 0.00 - - - - -.. GENERATED FROM PYTHON SOURCE LINES 126-132 - -The ground state is a high-spin triplet. The fourth and fifth -states (the first excited state) are low-spin singlet excitons at -:math:`2 J_H`. These have one hole on each orbital in the antisymmetric -combination of :math:`|\uparrow\downarrow>-|\downarrow\uparrow>`. -The state at :math:`3 J_H` also has one hole on each orbital in the symmetric -:math:`|\uparrow\downarrow>+|\downarrow\uparrow>` configuration. - -.. GENERATED FROM PYTHON SOURCE LINES 134-140 - -Where are the holes for large hopping ------------------------------------------------------------------------------- -As discussed at the start, we are interested to see interplay between Hund's -and charge-transfer physics, which will obviously depend strongly on whether -the holes are on Ni or the ligand. Let's see what happens as :math:`e_L` is -reduced while observing the location of the ground state and exciton holes. - -.. GENERATED FROM PYTHON SOURCE LINES 140-164 - -.. code-block:: Python - - U = 10 - JH = 2 - t = 100 - - eLs = np.linspace(0, 1000, 30) - - fig, axs = plt.subplots(1, 2, figsize=(8, 4)) - - for ax, ind in zip(axs.ravel(), [0, 3]): - ds = np.array([diagonalize(U, JH, t, eL, n=6) - for eL in eLs]) - - ax.plot(eLs, ds[:, 1, ind], 'o', label='$d^0$') - ax.plot(eLs, ds[:, 2, ind], 's', label='$d^1$') - ax.plot(eLs, ds[:, 3, ind], '^', label='$d^2$') - ax.set_xlabel("Energy of ligands $e_L$") - ax.set_ylabel("Number of electrons") - ax.legend() - - axs[0].set_title("Location of ground state holes") - axs[1].set_title("Location of exciton holes") - - plt.tight_layout() - plt.show() - - - -.. image-sg:: /auto_examples/images/sphx_glr_example_8_Hunds_interactions_001.png - :alt: Location of ground state holes, Location of exciton holes - :srcset: /auto_examples/images/sphx_glr_example_8_Hunds_interactions_001.png - :class: sphx-glr-single-img - - - - - -.. GENERATED FROM PYTHON SOURCE LINES 165-171 - -For large :math:`e_L`, we see that both holes are on nickel as expected. In -the opposite limit of :math:`|e_L| \ll t` and :math:`U \ll t` the holes are -shared in the ratio 0.25:0.5:0.25 as there are two ways to have one hole on -Ni. In the limit of large :math:`e_L`, all holes move onto Ni. Since -:math:`t` is large, this applies equally to both the ground state and the -exciton. - -.. GENERATED FROM PYTHON SOURCE LINES 174-179 - -Connecton between atomic and charge transfer limits ------------------------------------------------------------------------------- -We now examine the quantum numbers during cross over between the two limits -with :math:`e_L`. Let's first look at the how :math:`` changes for the -ground state and exciton and then examine how the exciton energy changes. - -.. GENERATED FROM PYTHON SOURCE LINES 179-216 - -.. code-block:: Python - - - U = 10 - JH = 2 - t = 100 - - eLs = np.linspace(0, 1000, 30) - - info = np.array([diagonalize(U, JH, t, eL, n=6) - for eL in eLs]) - - fig, axs = plt.subplots(1, 2, figsize=(8, 4)) - - - axs[0].plot(eLs, info[:, 4, 0], label='Ground state') - axs[0].plot(eLs, info[:, 4, 3], label='Exciton') - axs[0].set_xlabel("Energy of ligands $e_L$") - axs[0].set_ylabel('$$') - axs[0].set_title('Quantum numbers') - axs[0].legend() - - axs[1].plot(eLs, info[:, 0, 3], '+', color='C0') - axs[1].set_xlabel("Energy of ligands $e_L$") - axs[1].set_ylabel('Exciton energy', color='C0') - axr = axs[1].twinx() - axr.plot(eLs, info[:, 3, 5], 'x', color='C1') - axr.set_ylabel('$d^2$ fraction', color='C1') - - for ax, color in zip([axs[1], axr], ['C0', 'C1']): - for tick in ax.get_yticklabels(): - tick.set_color(color) - - axs[1].set_ylim(0, 2*JH) - axr.set_ylim(0, 1) - - axs[1].set_title('Exciton energy vs. $d^2$ character') - plt.tight_layout() - plt.show() - - - -.. image-sg:: /auto_examples/images/sphx_glr_example_8_Hunds_interactions_002.png - :alt: Quantum numbers, Exciton energy vs. $d^2$ character - :srcset: /auto_examples/images/sphx_glr_example_8_Hunds_interactions_002.png - :class: sphx-glr-single-img - - - - - -.. GENERATED FROM PYTHON SOURCE LINES 217-229 - -In the left panel, we see that the two limits are adiabatically connected -as they preseve the same quantum numbers. This is because there is always -an appreciable double occupancy under conditions where the -:math:`d^9\underline{L}` character is maximized and this continues to favor -the high spin ground state. Other interactions such as strong tetragonal -crystal field would be needed to overcome the Hund's interactions and break -this paradigm. In the right panel, we see that the exciton energy simply -scales with the double occupancy. Overall, even though -Hund's interactions are irrelevant for the :math:`d^9\underline{L}` -electronic configuration, whenever :math:`t` is appreciable there is a -strong mixing with the :math:`d^8` component is always present, which -dominates the energy of the exciton. - -.. GENERATED FROM PYTHON SOURCE LINES 231-237 - -Charge transfer excitons ------------------------------------------------------------------------------- -Another limiting case of the model is where :math:`t` is smaller than the -Coulomb interactions. This, however, tends to produce -ground state and exciton configurations that correspond to those of distinct -atomic models. Let's look at the :math:`e_L` dependence in this case. - -.. GENERATED FROM PYTHON SOURCE LINES 237-265 - -.. code-block:: Python - - U = 10 - JH = 2 - t = .5 - eL = 7 - - eLs = np.linspace(0, 20, 30) - - fig, axs = plt.subplots(1, 2, figsize=(8, 4)) - - for ax, ind in zip(axs.ravel(), [0, 3]): - ds = np.array([diagonalize(U, JH, t, eL, n=6) - for eL in eLs]) - - ax.plot(eLs, ds[:, 1, ind], 'o', label='$d^0$') - ax.plot(eLs, ds[:, 2, ind], 's', label='$d^1$') - ax.plot(eLs, ds[:, 3, ind], '^', label='$d^2$') - ax.set_xlabel("Energy of ligands $e_L$") - ax.set_ylabel("Number of electrons") - ax.legend() - - axs[0].axvline(x=eL, linestyle=':', color='k') - axs[1].axvline(x=eL, linestyle=':', color='k') - - axs[0].set_title("Location of ground state holes") - axs[1].set_title("Location of exciton holes") - - plt.tight_layout() - plt.show() - - - -.. image-sg:: /auto_examples/images/sphx_glr_example_8_Hunds_interactions_003.png - :alt: Location of ground state holes, Location of exciton holes - :srcset: /auto_examples/images/sphx_glr_example_8_Hunds_interactions_003.png - :class: sphx-glr-single-img - - - - - -.. GENERATED FROM PYTHON SOURCE LINES 266-270 - -Around :math:`e_L = 7` the plot shows that the excition is primairly a -:math:`d^2 \rightarrow d^1` transition or a -:math:`d^8 \rightarrow d^{9}\underline{L}` transition in electron language. -Let's examine the energy and quantum numbers. - -.. GENERATED FROM PYTHON SOURCE LINES 270-281 - -.. code-block:: Python - - - e, d0, d1, d2, S_squared_exp, S_z_exp = diagonalize(U, JH, t, eL, n=6) - - print("Ground state\nE\t") - for i in range(3): - print(f"{e[i]:.2f}\t{S_squared_exp[i]:.2f}\t{S_z_exp[i]:.2f}") - - print("\nExcited state\nE\t") - for i in range(3, 6): - print(f"{e[i]:.2f}\t{S_squared_exp[i]:.2f}\t{S_z_exp[i]:.2f}") - - - - - -.. rst-class:: sphx-glr-script-out - - .. code-block:: none - - Ground state - E - 0.00 2.00 -1.00 - 0.00 2.00 -0.00 - 0.00 2.00 1.00 - - Excited state - E - 2.74 0.00 0.00 - 2.74 0.00 0.00 - 2.99 0.00 0.00 - - - - -.. GENERATED FROM PYTHON SOURCE LINES 282-284 - -We once again see the same quantum numbers, despite the differences in mixing -in the ground state and exciton. - -.. GENERATED FROM PYTHON SOURCE LINES 287-290 - -.. rubric:: Footnotes - -.. [1] D. Khomskii, Transition Metal Compounds, Cambridge University Press (2014) - - -.. rst-class:: sphx-glr-timing - - **Total running time of the script:** (0 minutes 1.236 seconds) - - -.. _sphx_glr_download_auto_examples_example_8_Hunds_interactions.py: - -.. only:: html - - .. container:: sphx-glr-footer sphx-glr-footer-example - - .. container:: sphx-glr-download sphx-glr-download-jupyter - - :download:`Download Jupyter notebook: example_8_Hunds_interactions.ipynb ` - - .. container:: sphx-glr-download sphx-glr-download-python - - :download:`Download Python source code: example_8_Hunds_interactions.py ` - - .. container:: sphx-glr-download sphx-glr-download-zip - - :download:`Download zipped: example_8_Hunds_interactions.zip ` - - -.. only:: html - - .. rst-class:: sphx-glr-signature - - `Gallery generated by Sphinx-Gallery `_ diff --git a/edrixs/_sources/auto_examples/example_9_Coulomb.rst.txt b/edrixs/_sources/auto_examples/example_9_Coulomb.rst.txt deleted file mode 100644 index d5a697fbc8..0000000000 --- a/edrixs/_sources/auto_examples/example_9_Coulomb.rst.txt +++ /dev/null @@ -1,635 +0,0 @@ - -.. DO NOT EDIT. -.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. -.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: -.. "auto_examples/example_9_Coulomb.py" -.. LINE NUMBERS ARE GIVEN BELOW. - -.. only:: html - - .. note:: - :class: sphx-glr-download-link-note - - :ref:`Go to the end ` - to download the full example code. - -.. rst-class:: sphx-glr-example-title - -.. _sphx_glr_auto_examples_example_9_Coulomb.py: - - -Coulomb interactions -===================================== -In this example we provide more details on how Coulomb interactions are -implemented in multiplet calculations and EDRIXS in particular. We aim -to clarify the form of the matrices, how they are parametrized, -and how the breaking of spherical symmetry can switch on additional elements -that one might not anticipate. Our example is based on a :math:`d` atomic shell. - -.. GENERATED FROM PYTHON SOURCE LINES 13-50 - -Create matrix ------------------------------------------------------------------------------- -The Coulomb interaction between two particles can be written as - - .. math:: - \begin{equation} - \hat{H} = \frac{1}{2} - \int d\mathbf{r} \int d\mathbf{r}^\prime - \Sigma_{\sigma, \sigma^\prime} - |\hat{\psi}^\sigma(\mathbf{r})|^2 \frac{e^2}{R} - |\hat{\psi}^{\sigma^\prime}(\mathbf{r^\prime})|^2, - \end{equation} - -where :math:`\hat{\psi}^\sigma(\mathbf{r})` is the electron wavefunction, with -spin :math:`\sigma`, and :math:`R=|r-r^\prime|` is the electron separation. -Solving our problem in this form is difficult due to the need to symmeterize -the wavefunction to follow fermionic statistics. -Using second quantization, we can use operators to impose the required -particle exchange statistics and write the equation in terms of -a tensor :math:`U` - - .. math:: - \begin{equation} - \hat{H} = \sum_{\alpha,\beta,\gamma,\delta,\sigma,\sigma^\prime} - U_{\alpha\sigma,\beta\sigma^\prime,\gamma\sigma^\prime,\delta\sigma} - \hat{f}^{\dagger}_{\alpha\sigma} - \hat{f}^{\dagger}_{\beta\sigma^\prime} - \hat{f}_{\gamma\sigma^\prime}\hat{f}_{\delta\sigma}, - \end{equation} - -where :math:`\alpha`, :math:`\beta`, :math:`\gamma`, :math:`\delta` are -orbital indices and :math:`\hat{f}^{\dagger}` -(:math:`\hat{f}`) are the creation (anihilation) operators. -For a :math:`d`-electron system, we have :math:`10` distinct spin-orbitals -(:math:`5` orbitals each with :math:`2` spins), which makes matrix the -:math:`10\times10\times10\times10` in total size. -In EDRIXS the matrix can be created as follows: - -.. GENERATED FROM PYTHON SOURCE LINES 50-58 - -.. code-block:: Python - - import edrixs - import numpy as np - import scipy - import matplotlib.pyplot as plt - import itertools - - F0, F2, F4 = 6.94, 14.7, 4.41 - umat_chb = edrixs.get_umat_slater('d', F0, F2, F4) - - - - - - - -.. GENERATED FROM PYTHON SOURCE LINES 59-61 - -We stored this under variable :code:`umat_chb` where "cbh" stands for -complex harmonic basis, which is the default basis in EDRIXS. - -.. GENERATED FROM PYTHON SOURCE LINES 63-92 - -Parameterizing interactions ------------------------------------------------------------------------------- -EDRIXS parameterizes the interactions in :math:`U` via Slater integral -parameters :math:`F^{k}`. These relate to integrals of various spherical -Harmonics as well as Clebsch-Gordon coefficients, Gaunt coefficients, -and Wigner 3J symbols. Textbooks such as [1]_ can be used for further -reference. If you are interested in the details of how -EDRIXS does this (and you probably aren't) function :func:`.umat_slater`, -constructs the required matrix via Gaunt coeficents from -:func:`.get_gaunt`. Two alternative parameterizations are common. -The first are the Racah parameters, which are - - .. math:: - \begin{eqnarray} - A &=& F^0 - \frac{49}{441} F^4 \\ - B &=& \frac{1}{49}F^2 - \frac{5}{441}F^4 \\ - C &=& \frac{35}{441}F^4. - \end{eqnarray} - -or an alternative form for the Slater integrals - - .. math:: - \begin{eqnarray} - F_0 &=& F^0 \\ - F_2 &=& \frac{1}{49}F^2 \\ - F_4 &=& \frac{1}{441}F^4, - \end{eqnarray} - -which involves different normalization parameters. - -.. GENERATED FROM PYTHON SOURCE LINES 94-101 - -Basis transform ------------------------------------------------------------------------------- -If we want to use the real harmonic basis, we can use a tensor -transformation, which imposes the following orbital order -:math:`3z^2-r^2, xz, yz, x^2-y^2, xy`, each of which involves -:math:`\uparrow, \downarrow` spin pairs. Let's perform this transformation and -store a list of these orbitals. - -.. GENERATED FROM PYTHON SOURCE LINES 101-104 - -.. code-block:: Python - - umat = edrixs.transform_utensor(umat_chb, edrixs.tmat_c2r('d', True)) - orbitals = ['3z^2-r^2', 'xz', 'yz', 'x^2-y^2', 'xy'] - - - - - - - - -.. GENERATED FROM PYTHON SOURCE LINES 105-158 - -Interactions ------------------------------------------------------------------------------- -Tensor :math:`U` is a series of matrix -elements - - .. math:: - \begin{equation} - \langle\psi_{\gamma,\delta}^{\bar{\sigma},\bar{\sigma}^\prime} - |\hat{H}| - \psi_{\alpha,\beta}^{\sigma,\sigma^\prime}\rangle - \end{equation} - -the combination of which defines the energetic cost of pairwise -electron-electron interactions between states :math:`\alpha,\sigma` -and :math:`\beta,\sigma^\prime`. In EDRIXS we follow the convention of -summing over all orbital pairs. Some other texts count each pair of -indices only once. The matrix elements here will consequently -be half the magnitude of those in other references. -We can express the interactions in terms of -the orbitals involved. It is common to distinguish "direct Coulomb" and -"exchange" interactions. The former come from electrons in the same orbital -and the later involve swapping orbital labels for electrons. We will use -:math:`U_0` and :math:`J` as a shorthand for distinguishing these. - -Before we describe the different types of interactions, we note that since -the Coulomb interaction is real, and due to the spin symmmetry properties -of the process :math:`U` always obeys - - .. math:: - \begin{equation} - U_{\alpha\sigma,\beta\sigma^\prime,\gamma\sigma^\prime,\delta\sigma} = - U_{\beta\sigma,\alpha\sigma^\prime,\delta\sigma^\prime,\gamma\sigma} = - U_{\delta\sigma,\gamma\sigma^\prime,\beta\sigma^\prime,\alpha\sigma} = - U_{\gamma\sigma,\delta\sigma^\prime,\alpha\sigma^\prime,\beta\sigma}. - \end{equation} - - -1. Intra orbital -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -The direct Coulomb energy cost to double-occupy an orbital comes from terms -like :math:`U_{\alpha\sigma,\alpha\bar\sigma,\alpha\bar\sigma,\alpha\sigma}`. -In this notation, we use :math:`\sigma^\prime` to denote that the matrix -element is summed over all pairs and :math:`\bar{\sigma}` to denote sums -over all opposite spin pairs. Due to rotational symmetry, all these -elements are the same and equal to - - .. math:: - \begin{eqnarray} - U_0 &=& \frac{A}{2} + 2B + \frac{3C}{2}\\ - &=& \frac{F_0}{2} + 2F_2 + 18F_4 - \end{eqnarray} - -Let's print these to demonstrate where these live in the array - -.. GENERATED FROM PYTHON SOURCE LINES 158-162 - -.. code-block:: Python - - for i in range(0, 5): - val = umat[i*2, i*2 + 1, i*2 + 1, i*2].real - print(f"{orbitals[i]:<8} \t {val:.3f}") - - - - - -.. rst-class:: sphx-glr-script-out - - .. code-block:: none - - 3z^2-r^2 4.250 - xz 4.250 - yz 4.250 - x^2-y^2 4.250 - xy 4.250 - - - - -.. GENERATED FROM PYTHON SOURCE LINES 163-170 - -2. Inter orbital Coulomb interactions -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -Direct Coulomb repulsion between different orbitals depends on terms like -:math:`U_{\alpha\sigma,\beta\sigma^\prime,\beta\sigma^\prime,\alpha\sigma}`. -Expresions for these parameters are provided in column :math:`U` in -:ref:`table_2_orbital`. We can print the values from :code:`umat` -like this: - -.. GENERATED FROM PYTHON SOURCE LINES 170-174 - -.. code-block:: Python - - for i, j in itertools.combinations(range(5), 2): - val = umat[i*2, j*2 + 1, j*2 + 1, i*2].real - print(f"{orbitals[i]:<8} \t {orbitals[j]:<8} \t {val:.3f}") - - - - - -.. rst-class:: sphx-glr-script-out - - .. code-block:: none - - 3z^2-r^2 xz 3.650 - 3z^2-r^2 yz 3.650 - 3z^2-r^2 x^2-y^2 2.900 - 3z^2-r^2 xy 2.900 - xz yz 3.150 - xz x^2-y^2 3.150 - xz xy 3.150 - yz x^2-y^2 3.150 - yz xy 3.150 - x^2-y^2 xy 3.900 - - - - -.. GENERATED FROM PYTHON SOURCE LINES 175-181 - -3. Inter-orbital exchange interactions -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -Exchange terms exist with the form -:math:`U_{\alpha\sigma,\beta\sigma^\prime,\alpha\sigma^\prime,\beta\sigma}`. -Expresions for these parameters are provided in column :math:`J` of -:ref:`table_2_orbital`. These come from terms like this in the matrix: - -.. GENERATED FROM PYTHON SOURCE LINES 181-185 - -.. code-block:: Python - - for i, j in itertools.combinations(range(5), 2): - val = umat[i*2, j*2 + 1, i*2 + 1, j*2].real - print(f"{orbitals[i]:<8} \t {orbitals[j]:<8} \t {val:.3f}") - - - - - -.. rst-class:: sphx-glr-script-out - - .. code-block:: none - - 3z^2-r^2 xz 0.300 - 3z^2-r^2 yz 0.300 - 3z^2-r^2 x^2-y^2 0.675 - 3z^2-r^2 xy 0.675 - xz yz 0.550 - xz x^2-y^2 0.550 - xz xy 0.550 - yz x^2-y^2 0.550 - yz xy 0.550 - x^2-y^2 xy 0.175 - - - - -.. GENERATED FROM PYTHON SOURCE LINES 186-193 - -4. Pair hopping term -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -Terms that swap pairs of electrons exist as -:math:`(1-\delta_{\sigma\sigma'})U_{\alpha\sigma,\alpha\bar\sigma,\beta\bar\sigma,\beta\sigma}` -and depend on exchange interactions column :math:`J` from -:ref:`table_2_orbital` -and here in the matrix. - -.. GENERATED FROM PYTHON SOURCE LINES 193-197 - -.. code-block:: Python - - for i, j in itertools.combinations(range(5), 2): - val = umat[i*2, i*2 + 1, j*2 + 1, j*2].real - print(f"{orbitals[i]:<8} \t {orbitals[j]:<8} \t {val:.3f}") - - - - - -.. rst-class:: sphx-glr-script-out - - .. code-block:: none - - 3z^2-r^2 xz 0.300 - 3z^2-r^2 yz 0.300 - 3z^2-r^2 x^2-y^2 0.675 - 3z^2-r^2 xy 0.675 - xz yz 0.550 - xz x^2-y^2 0.550 - xz xy 0.550 - yz x^2-y^2 0.550 - yz xy 0.550 - x^2-y^2 xy 0.175 - - - - -.. GENERATED FROM PYTHON SOURCE LINES 198-215 - -5. Three orbital -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -Another set of terms that one might not immediately anticipate involve three -orbitals like - - .. math:: - \begin{equation} - U_{\alpha\sigma, \gamma\sigma', \beta\sigma', \gamma\sigma} \\ - U_{\alpha\sigma, \gamma\sigma', \gamma\sigma', \beta\sigma} \\ - (1-\delta_{\sigma\sigma'}) - U_{\alpha\sigma, \beta\sigma', \gamma\sigma', \gamma\sigma} - \end{equation} - -for :math:`\alpha=3z^2-r^2, \beta=x^2-y^2, \gamma=xz/yz`. -These are needed to maintain the rotational symmetry of the interations. -See :ref:`table_3_orbital` for the expressions. We can print some of -these via: - -.. GENERATED FROM PYTHON SOURCE LINES 215-227 - -.. code-block:: Python - - ijkl = [[0, 1, 3, 1], - [0, 2, 3, 2], - [1, 0, 3, 1], - [1, 1, 3, 0], - [2, 0, 3, 2], - [2, 2, 3, 0]] - - for i, j, k, l in ijkl: - val = umat[i*2, j*2 + 1, k*2 + 1, l*2].real - print(f"{orbitals[i]:<8} \t {orbitals[j]:<8} \t" - f"{orbitals[k]:<8} \t {orbitals[l]:<8} \t {val:.3f}") - - - - - -.. rst-class:: sphx-glr-script-out - - .. code-block:: none - - 3z^2-r^2 xz x^2-y^2 xz 0.217 - 3z^2-r^2 yz x^2-y^2 yz -0.217 - xz 3z^2-r^2 x^2-y^2 xz -0.433 - xz xz x^2-y^2 3z^2-r^2 0.217 - yz 3z^2-r^2 x^2-y^2 yz 0.433 - yz yz x^2-y^2 3z^2-r^2 -0.217 - - - - -.. GENERATED FROM PYTHON SOURCE LINES 228-233 - -6. Four orbital -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -Futher multi-orbital terms include -:math:`U_{\alpha\sigma,\beta\sigma^\prime,\gamma\sigma^\prime,\delta\sigma}`. -We can find these here in the matrix: - -.. GENERATED FROM PYTHON SOURCE LINES 233-249 - -.. code-block:: Python - - ijkl = [[0, 1, 2, 4], - [0, 1, 4, 2], - [0, 2, 1, 4], - [0, 2, 4, 1], - [0, 4, 1, 2], - [0, 4, 2, 1], - [3, 1, 4, 2], - [3, 2, 4, 1], - [3, 4, 1, 2], - [3, 4, 2, 1]] - - for i, j, k, l in ijkl: - val = umat[i*2, j*2 + 1, k*2 + 1, l*2].real - print(f"{orbitals[i]:<8} \t {orbitals[j]:<8} \t {orbitals[k]:<8}" - f"\t {orbitals[l]:<8} \t {val:.3f}") - - - - - -.. rst-class:: sphx-glr-script-out - - .. code-block:: none - - 3z^2-r^2 xz yz xy -0.433 - 3z^2-r^2 xz xy yz 0.217 - 3z^2-r^2 yz xz xy -0.433 - 3z^2-r^2 yz xy xz 0.217 - 3z^2-r^2 xy xz yz 0.217 - 3z^2-r^2 xy yz xz 0.217 - x^2-y^2 xz xy yz -0.375 - x^2-y^2 yz xy xz 0.375 - x^2-y^2 xy xz yz -0.375 - x^2-y^2 xy yz xz 0.375 - - - - -.. GENERATED FROM PYTHON SOURCE LINES 250-256 - -Effects of multi-orbital terms ------------------------------------------------------------------------------- -To test the effects of the multi-orbital terms, let's plot the eigenenergy -spectra with and without multi-orbital terms switched on for system with and -without a cubic crystal field. We will use a :math:`d`-shell with two -electrons. - -.. GENERATED FROM PYTHON SOURCE LINES 256-292 - -.. code-block:: Python - - ten_dqs = [0, 2, 4, 12] - - def diagonalize(ten_dq, umat): - emat = edrixs.cb_op(edrixs.cf_cubic_d(ten_dq), - edrixs.tmat_c2r('d', ispin=True)) - H = (edrixs.build_opers(4, umat, basis) - + edrixs.build_opers(2, emat, basis)) - e, v = scipy.linalg.eigh(H) - return e - e.min() - - basis = edrixs.get_fock_bin_by_N(10, 2) - umat_no_multiorbital = np.copy(umat) - B = F2/49 - 5*F4/441 - for val in [np.sqrt(3)*B/2, np.sqrt(3)*B, 3*B/2]: - umat_no_multiorbital[(np.abs(umat)- val) < 1e-6] = 0 - - fig, axs = plt.subplots(1, len(ten_dqs), figsize=(8, 3)) - - for cind, (ax, ten_dq) in enumerate(zip(axs, ten_dqs)): - ax.hlines(diagonalize(ten_dq, umat), xmin=0, xmax=1, - label='on', color=f'C{cind}') - ax.hlines(diagonalize(ten_dq, umat_no_multiorbital), - xmin=1.5, xmax=2.5, - label='off', - linestyle=':', color=f'C{cind}') - ax.set_title(f"$10D_q={ten_dq}$") - ax.set_ylim([-.5, 20]) - ax.set_xticks([]) - ax.legend() - - fig.suptitle("Eigenvalues with 3&4-orbital effects on/off") - fig.subplots_adjust(wspace=.3) - axs[0].set_ylabel('Eigenvalues (eV)') - fig.subplots_adjust(top=.8) - plt.show() - - - - -.. image-sg:: /auto_examples/images/sphx_glr_example_9_Coulomb_001.png - :alt: Eigenvalues with 3&4-orbital effects on/off, $10D_q=0$, $10D_q=2$, $10D_q=4$, $10D_q=12$ - :srcset: /auto_examples/images/sphx_glr_example_9_Coulomb_001.png - :class: sphx-glr-single-img - - - - - -.. GENERATED FROM PYTHON SOURCE LINES 293-382 - -On the left of the plot Coulomb interactions in spherical symmetry cause -substantial mxing between :math:`t_{2g}` and :math:`e_{g}` orbitals in the -eigenstates and 3 & 4 orbital orbital terms are crucial for obtaining the -the right eigenenergies. As :math:`10D_q` get large, this mixing is switched -off and the spectra start to become independent of whether the 3 & 4 orbital -orbital terms are included or not. - - - -.. _table_2_orbital: -.. table:: Table of 2 orbital interactions - - +-----------------------------+------------------+-----------------------+----------------+--------------------+ - |Orbitals :math:`\alpha,\beta`|:math:`U_0` Racah | :math:`U_0` Slater |:math:`J` Racah |:math:`J` Slater | - +=============================+==================+=======================+================+====================+ - |:math:`3z^2-r^2, xz` |:math:`A/2+B+C/2` |:math:`F_0/2+F_2-12F_4`| :math:`B/2+C/2`|:math:`F_2/2+15F_4` | - +-----------------------------+------------------+-----------------------+----------------+--------------------+ - |:math:`3z^2-r^2, yz` |:math:`A/2+B+C/2` |:math:`F_0/2+F_2-12F_4`| :math:`B/2+C/2`|:math:`F_2/2+15F_4` | - +-----------------------------+------------------+-----------------------+----------------+--------------------+ - |:math:`3z^2-r^2, x^2-y^2` |:math:`A/2-2B+C/2`|:math:`F_0/2-2F_2+3F_4`|:math:`2B+C/2` |:math:`2F_2+15F_4/2`| - +-----------------------------+------------------+-----------------------+----------------+--------------------+ - |:math:`3z^2-r^2, xy` |:math:`A/2-2B+C/2`|:math:`F_0/2-2F_2+3F_4`|:math:`2B+C/2` |:math:`2F_2+15F_4/2`| - +-----------------------------+------------------+-----------------------+----------------+--------------------+ - |:math:`xz, yz` |:math:`A/2-B+C/2` |:math:`F_0/2-F_2-12F_4`|:math:`3B/2+C/2`|:math:`3F_2/2+10F_4`| - +-----------------------------+------------------+-----------------------+----------------+--------------------+ - |:math:`xz, x^2-y^2` |:math:`A/2-B+C/2` |:math:`F_0/2-F_2-12F_4`|:math:`3B/2+C/2`|:math:`3F_2/2+10F_4`| - +-----------------------------+------------------+-----------------------+----------------+--------------------+ - |:math:`xz, xy` |:math:`A/2-B+C/2` |:math:`F_0/2-F_2-12F_4`|:math:`3B/2+C/2`|:math:`3F_2/2+10F_4`| - +-----------------------------+------------------+-----------------------+----------------+--------------------+ - |:math:`yz, x^2-y^2` |:math:`A/2-B+C/2` |:math:`F_0/2-F_2-12F_4`|:math:`3B/2+C/2`|:math:`3F_2/2+10F_4`| - +-----------------------------+------------------+-----------------------+----------------+--------------------+ - |:math:`yz, xy` |:math:`A/2-B+C/2` |:math:`F_0/2-F_2-12F_4`|:math:`3B/2+C/2`|:math:`3F_2/2+10F_4`| - +-----------------------------+------------------+-----------------------+----------------+--------------------+ - |:math:`x^2-y^2, xy` |:math:`A/2+2B+C/2`|:math:`F_0+4F_2-34F_4` | :math:`C/2` |:math:`35F_4/2` | - +-----------------------------+------------------+-----------------------+----------------+--------------------+ - - -.. _table_3_orbital: -.. table:: Table of 3 orbital interactions - - +-----------------------------+-------------+----------------------------------------------------+-----------------------------------------------------+ - |Orbitals :math:`\alpha,\beta,\gamma,\delta`|:math:`\langle\alpha\beta|\gamma\delta\rangle` Racah|:math:`\langle\alpha\beta|\gamma\delta\rangle` Slater| - +=============================+=============+====================================================+=====================================================+ - |:math:`3z^2-r^2, xz, x^2-y^2, xz` | :math:`\sqrt{3}B/2` | :math:`\sqrt{3}F_2/2-5\sqrt{3}F_4/2` | - +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ - |:math:`3z^2-r^2, yz, x^2-y^2, yz` | :math:`-\sqrt{3}B/2` | :math:`-\sqrt{3}F_2/2+5\sqrt{3}F_4/2` | - +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ - |:math:`xz, 3z^2-r^2, x^2-y^2, xz` | :math:`-\sqrt{3}B` | :math:`-\sqrt{3}F_2+5\sqrt{3}F_4` | - +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ - |:math:`xz, xz, x^2-y^2, 3z^2-r^2` | :math:`\sqrt{3}B/2` | :math:`\sqrt{3}F_2/2-5\sqrt{3}F_4/2` | - +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ - |:math:`yz, 3z^2-r^2, x^2-y^2, yz` | :math:`\sqrt{3}B` | :math:`\sqrt{3}F_2-5\sqrt{3}F_4` | - +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ - |:math:`yz, yz, x^2-y^2, 3z^2-r^2` | :math:`-\sqrt{3}B/2` | :math:`-\sqrt{3}F_2/2+5\sqrt{3}F_4/2` | - +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ - - -.. _table_4_orbital: -.. table:: Table of 4 orbital interactions - - +-----------------------------+-------------+----------------------------------------------------+-----------------------------------------------------+ - |Orbitals :math:`\alpha,\beta,\gamma,\delta`|:math:`\langle\alpha\beta|\gamma\delta\rangle` Racah|:math:`\langle\alpha\beta|\gamma\delta\rangle` Slater| - +=============================+=============+====================================================+=====================================================+ - |:math:`3z^2-r^2, xz, yz, xy` | :math:`-\sqrt{3}B` | :math:`-\sqrt{3}F_2+5\sqrt{3}F_4` | - +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ - |:math:`3z^2-r^2, xz, xy, yz` | :math:`\sqrt{3}B/2` | :math:`\sqrt{3}F_2/2-5\sqrt{3}F_4/2` | - +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ - |:math:`3z^2-r^2, yz, xz, xy` | :math:`-\sqrt{3}B` | :math:`-\sqrt{3}F_2+5\sqrt{3}F_4` | - +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ - |:math:`3z^2-r^2, yz, xy, xz` | :math:`\sqrt{3}B/2` | :math:`\sqrt{3}F_2/2-5\sqrt{3}F_4/2` | - +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ - |:math:`3z^2-r^2, xy, xz, yz` | :math:`\sqrt{3}B/2` | :math:`\sqrt{3}F_2/2-5\sqrt{3}F_4/2` | - +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ - |:math:`3z^2-r^2, xy, yz, xz` | :math:`\sqrt{3}B/2` | :math:`\sqrt{3}F_2/2-5\sqrt{3}F_4/2` | - +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ - |:math:`x^2-y^2 , xz, xy, yz` | :math:`-3B/2` | :math:`-3F_2/2+15F_4/2` | - +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ - |:math:`x^2-y^2 , yz, xy, xz` | :math:`3B/2` | :math:`3F_2/2-15F_4/2` | - +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ - |:math:`x^2-y^2 , xy, xz, yz` | :math:`-3B/2` | :math:`-3F_2/2+15F_4/2` | - +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ - |:math:`x^2-y^2 , xy, yz, xz` | :math:`3B/2` | :math:`3F_2/2-15F_4/2` | - +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ - - -.. rubric:: Footnotes - -.. [1] MSugano S, Tanabe Y and Kamimura H. 1970. Multiplets of - Transition-Metal Ions in Crystals. Academic Press, New York and London. - - -.. rst-class:: sphx-glr-timing - - **Total running time of the script:** (0 minutes 0.368 seconds) - - -.. _sphx_glr_download_auto_examples_example_9_Coulomb.py: - -.. only:: html - - .. container:: sphx-glr-footer sphx-glr-footer-example - - .. container:: sphx-glr-download sphx-glr-download-jupyter - - :download:`Download Jupyter notebook: example_9_Coulomb.ipynb ` - - .. container:: sphx-glr-download sphx-glr-download-python - - :download:`Download Python source code: example_9_Coulomb.py ` - - .. container:: sphx-glr-download sphx-glr-download-zip - - :download:`Download zipped: example_9_Coulomb.zip ` - - -.. only:: html - - .. rst-class:: sphx-glr-signature - - `Gallery generated by Sphinx-Gallery `_ diff --git a/edrixs/_sources/auto_examples/index.rst.txt b/edrixs/_sources/auto_examples/index.rst.txt deleted file mode 100644 index 2878202d35..0000000000 --- a/edrixs/_sources/auto_examples/index.rst.txt +++ /dev/null @@ -1,225 +0,0 @@ -:orphan: - -Pedagogical examples -==================== - -Below are some examples illustrating the concepts behind and usage of EDRIXS. - - - -.. raw:: html - -
- -.. thumbnail-parent-div-open - -.. raw:: html - -
- -.. only:: html - - .. image:: /auto_examples/images/thumb/sphx_glr_example_0_ed_calculator_thumb.png - :alt: - - :ref:`sphx_glr_auto_examples_example_0_ed_calculator.py` - -.. raw:: html - -
Exact diagonalization
-
- - -.. raw:: html - -
- -.. only:: html - - .. image:: /auto_examples/images/thumb/sphx_glr_example_1_crystal_field_thumb.png - :alt: - - :ref:`sphx_glr_auto_examples_example_1_crystal_field.py` - -.. raw:: html - -
Crystal fields
-
- - -.. raw:: html - -
- -.. only:: html - - .. image:: /auto_examples/images/thumb/sphx_glr_example_2_single_atom_RIXS_thumb.png - :alt: - - :ref:`sphx_glr_auto_examples_example_2_single_atom_RIXS.py` - -.. raw:: html - -
RIXS calculations for an atomic model
-
- - -.. raw:: html - -
- -.. only:: html - - .. image:: /auto_examples/images/thumb/sphx_glr_example_3_AIM_XAS_thumb.png - :alt: - - :ref:`sphx_glr_auto_examples_example_3_AIM_XAS.py` - -.. raw:: html - -
Anderson impurity model for NiO XAS
-
- - -.. raw:: html - -
- -.. only:: html - - .. image:: /auto_examples/images/thumb/sphx_glr_example_4_GS_analysis_thumb.png - :alt: - - :ref:`sphx_glr_auto_examples_example_4_GS_analysis.py` - -.. raw:: html - -
Ground state analysis for NiO
-
- - -.. raw:: html - -
- -.. only:: html - - .. image:: /auto_examples/images/thumb/sphx_glr_example_5_charge_transfer_thumb.png - :alt: - - :ref:`sphx_glr_auto_examples_example_5_charge_transfer.py` - -.. raw:: html - -
Charge-transfer energy for NiO
-
- - -.. raw:: html - -
- -.. only:: html - - .. image:: /auto_examples/images/thumb/sphx_glr_example_6_Hubbard_dimer_thumb.png - :alt: - - :ref:`sphx_glr_auto_examples_example_6_Hubbard_dimer.py` - -.. raw:: html - -
Hubbard Dimer
-
- - -.. raw:: html - -
- -.. only:: html - - .. image:: /auto_examples/images/thumb/sphx_glr_example_7_transitions_thumb.png - :alt: - - :ref:`sphx_glr_auto_examples_example_7_transitions.py` - -.. raw:: html - -
X-ray transitions
-
- - -.. raw:: html - -
- -.. only:: html - - .. image:: /auto_examples/images/thumb/sphx_glr_example_8_Hunds_interactions_thumb.png - :alt: - - :ref:`sphx_glr_auto_examples_example_8_Hunds_interactions.py` - -.. raw:: html - -
Hund's Interactions in charge transfer insulators
-
- - -.. raw:: html - -
- -.. only:: html - - .. image:: /auto_examples/images/thumb/sphx_glr_example_9_Coulomb_thumb.png - :alt: - - :ref:`sphx_glr_auto_examples_example_9_Coulomb.py` - -.. raw:: html - -
Coulomb interactions
-
- - -.. thumbnail-parent-div-close - -.. raw:: html - -
- - -.. toctree:: - :hidden: - - /auto_examples/example_0_ed_calculator - /auto_examples/example_1_crystal_field - /auto_examples/example_2_single_atom_RIXS - /auto_examples/example_3_AIM_XAS - /auto_examples/example_4_GS_analysis - /auto_examples/example_5_charge_transfer - /auto_examples/example_6_Hubbard_dimer - /auto_examples/example_7_transitions - /auto_examples/example_8_Hunds_interactions - /auto_examples/example_9_Coulomb - - -.. only:: html - - .. container:: sphx-glr-footer sphx-glr-footer-gallery - - .. container:: sphx-glr-download sphx-glr-download-python - - :download:`Download all examples in Python source code: auto_examples_python.zip ` - - .. container:: sphx-glr-download sphx-glr-download-jupyter - - :download:`Download all examples in Jupyter notebooks: auto_examples_jupyter.zip ` - - -.. only:: html - - .. rst-class:: sphx-glr-signature - - `Gallery generated by Sphinx-Gallery `_ diff --git a/edrixs/_sources/auto_examples/sg_execution_times.rst.txt b/edrixs/_sources/auto_examples/sg_execution_times.rst.txt deleted file mode 100644 index 1e468729aa..0000000000 --- a/edrixs/_sources/auto_examples/sg_execution_times.rst.txt +++ /dev/null @@ -1,64 +0,0 @@ - -:orphan: - -.. _sphx_glr_auto_examples_sg_execution_times: - - -Computation times -================= -**00:07.932** total execution time for 10 files **from auto_examples**: - -.. container:: - - .. raw:: html - - - - - - - - .. list-table:: - :header-rows: 1 - :class: table table-striped sg-datatable - - * - Example - - Time - - Mem (MB) - * - :ref:`sphx_glr_auto_examples_example_2_single_atom_RIXS.py` (``example_2_single_atom_RIXS.py``) - - 00:03.985 - - 0.0 - * - :ref:`sphx_glr_auto_examples_example_8_Hunds_interactions.py` (``example_8_Hunds_interactions.py``) - - 00:01.236 - - 0.0 - * - :ref:`sphx_glr_auto_examples_example_3_AIM_XAS.py` (``example_3_AIM_XAS.py``) - - 00:00.548 - - 0.0 - * - :ref:`sphx_glr_auto_examples_example_7_transitions.py` (``example_7_transitions.py``) - - 00:00.514 - - 0.0 - * - :ref:`sphx_glr_auto_examples_example_5_charge_transfer.py` (``example_5_charge_transfer.py``) - - 00:00.478 - - 0.0 - * - :ref:`sphx_glr_auto_examples_example_4_GS_analysis.py` (``example_4_GS_analysis.py``) - - 00:00.421 - - 0.0 - * - :ref:`sphx_glr_auto_examples_example_9_Coulomb.py` (``example_9_Coulomb.py``) - - 00:00.368 - - 0.0 - * - :ref:`sphx_glr_auto_examples_example_1_crystal_field.py` (``example_1_crystal_field.py``) - - 00:00.149 - - 0.0 - * - :ref:`sphx_glr_auto_examples_example_0_ed_calculator.py` (``example_0_ed_calculator.py``) - - 00:00.138 - - 0.0 - * - :ref:`sphx_glr_auto_examples_example_6_Hubbard_dimer.py` (``example_6_Hubbard_dimer.py``) - - 00:00.095 - - 0.0 diff --git a/edrixs/_sources/index.rst.txt b/edrixs/_sources/index.rst.txt deleted file mode 100644 index a1ec052fbc..0000000000 --- a/edrixs/_sources/index.rst.txt +++ /dev/null @@ -1,15 +0,0 @@ -.. Packaging Scientific Python documentation master file, created by - sphinx-quickstart on Thu Jun 28 12:35:56 2018. - You can adapt this file completely to your liking, but it should at least - contain the root `toctree` directive. - -edrixs Documentation -==================== - -.. toctree:: - :maxdepth: 2 - - user/index - auto_examples/index - reference/index - release-history diff --git a/edrixs/_sources/reference/angular_momentum.rst.txt b/edrixs/_sources/reference/angular_momentum.rst.txt deleted file mode 100644 index bfaf9b4144..0000000000 --- a/edrixs/_sources/reference/angular_momentum.rst.txt +++ /dev/null @@ -1,4 +0,0 @@ -angular_momentum -================ -.. automodule:: edrixs.angular_momentum - :members: diff --git a/edrixs/_sources/reference/basis_transform.rst.txt b/edrixs/_sources/reference/basis_transform.rst.txt deleted file mode 100644 index 11a9c20a4c..0000000000 --- a/edrixs/_sources/reference/basis_transform.rst.txt +++ /dev/null @@ -1,4 +0,0 @@ -basis_transform -================ -.. automodule:: edrixs.basis_transform - :members: diff --git a/edrixs/_sources/reference/coulomb_utensor.rst.txt b/edrixs/_sources/reference/coulomb_utensor.rst.txt deleted file mode 100644 index ee5152e6c9..0000000000 --- a/edrixs/_sources/reference/coulomb_utensor.rst.txt +++ /dev/null @@ -1,9 +0,0 @@ -.. EDRIXS documentation master file, created by - sphinx-quickstart on Sun Sep 16 11:58:12 2018. - You can adapt this file completely to your liking, but it should at least - contain the root `toctree` directive. - -coulomb_utensor -=============== -.. automodule:: edrixs.coulomb_utensor - :members: diff --git a/edrixs/_sources/reference/fit_hyb.rst.txt b/edrixs/_sources/reference/fit_hyb.rst.txt deleted file mode 100644 index 474cdf4be0..0000000000 --- a/edrixs/_sources/reference/fit_hyb.rst.txt +++ /dev/null @@ -1,4 +0,0 @@ -fit_hyb -======= -.. automodule:: edrixs.fit_hyb - :members: diff --git a/edrixs/_sources/reference/fock_basis.rst.txt b/edrixs/_sources/reference/fock_basis.rst.txt deleted file mode 100644 index cf5b3b0630..0000000000 --- a/edrixs/_sources/reference/fock_basis.rst.txt +++ /dev/null @@ -1,4 +0,0 @@ -fock_basis -========== -.. automodule:: edrixs.fock_basis - :members: diff --git a/edrixs/_sources/reference/index.rst.txt b/edrixs/_sources/reference/index.rst.txt deleted file mode 100644 index 24e7580eca..0000000000 --- a/edrixs/_sources/reference/index.rst.txt +++ /dev/null @@ -1,39 +0,0 @@ -.. _reference: - -################ -edrixs Reference -################ - -:Release: |version| -:Date: |today| - -.. module:: edrixs - -This reference manual details functions, modules, and objects -included in EDRIXS, describing what they are and what they do. - -.. toctree:: - :maxdepth: 2 - :caption: Contents: - - angular_momentum - basis_transform - coulomb_utensor - fit_hyb - fock_basis - iostream - manybody_operator - photon_transition - plot_spectrum - rixs_utils - soc - solvers - utils - wannier_ham - -Indices and tables -================== - -* :ref:`genindex` -* :ref:`modindex` -* :ref:`search` diff --git a/edrixs/_sources/reference/iostream.rst.txt b/edrixs/_sources/reference/iostream.rst.txt deleted file mode 100644 index 59daa01a86..0000000000 --- a/edrixs/_sources/reference/iostream.rst.txt +++ /dev/null @@ -1,4 +0,0 @@ -iostream -========== -.. automodule:: edrixs.iostream - :members: diff --git a/edrixs/_sources/reference/manybody_operator.rst.txt b/edrixs/_sources/reference/manybody_operator.rst.txt deleted file mode 100644 index e4beb489f3..0000000000 --- a/edrixs/_sources/reference/manybody_operator.rst.txt +++ /dev/null @@ -1,4 +0,0 @@ -manybody_operator -================= -.. automodule:: edrixs.manybody_operator - :members: diff --git a/edrixs/_sources/reference/photon_transition.rst.txt b/edrixs/_sources/reference/photon_transition.rst.txt deleted file mode 100644 index 5e6417c647..0000000000 --- a/edrixs/_sources/reference/photon_transition.rst.txt +++ /dev/null @@ -1,4 +0,0 @@ -photon_transition -================= -.. automodule:: edrixs.photon_transition - :members: diff --git a/edrixs/_sources/reference/plot_spectrum.rst.txt b/edrixs/_sources/reference/plot_spectrum.rst.txt deleted file mode 100644 index 237d4061ee..0000000000 --- a/edrixs/_sources/reference/plot_spectrum.rst.txt +++ /dev/null @@ -1,4 +0,0 @@ -plot_spectrum -============= -.. automodule:: edrixs.plot_spectrum - :members: diff --git a/edrixs/_sources/reference/rixs_utils.rst.txt b/edrixs/_sources/reference/rixs_utils.rst.txt deleted file mode 100644 index 1471d801f7..0000000000 --- a/edrixs/_sources/reference/rixs_utils.rst.txt +++ /dev/null @@ -1,4 +0,0 @@ -rixs_utils -========== -.. automodule:: edrixs.rixs_utils - :members: diff --git a/edrixs/_sources/reference/soc.rst.txt b/edrixs/_sources/reference/soc.rst.txt deleted file mode 100644 index f65ed14734..0000000000 --- a/edrixs/_sources/reference/soc.rst.txt +++ /dev/null @@ -1,4 +0,0 @@ -soc -=== -.. automodule:: edrixs.soc - :members: diff --git a/edrixs/_sources/reference/solvers.rst.txt b/edrixs/_sources/reference/solvers.rst.txt deleted file mode 100644 index 1141b02701..0000000000 --- a/edrixs/_sources/reference/solvers.rst.txt +++ /dev/null @@ -1,4 +0,0 @@ -solvers -======= -.. automodule:: edrixs.solvers - :members: diff --git a/edrixs/_sources/reference/utils.rst.txt b/edrixs/_sources/reference/utils.rst.txt deleted file mode 100644 index 27a61adb02..0000000000 --- a/edrixs/_sources/reference/utils.rst.txt +++ /dev/null @@ -1,6 +0,0 @@ -.. _utils: - -utils -===== -.. automodule:: edrixs.utils - :members: diff --git a/edrixs/_sources/reference/wannier_ham.rst.txt b/edrixs/_sources/reference/wannier_ham.rst.txt deleted file mode 100644 index 999a5cb220..0000000000 --- a/edrixs/_sources/reference/wannier_ham.rst.txt +++ /dev/null @@ -1,4 +0,0 @@ -wannier_ham -=========== -.. automodule:: edrixs.wannier_ham - :members: diff --git a/edrixs/_sources/release-history.rst.txt b/edrixs/_sources/release-history.rst.txt deleted file mode 100644 index 53707fbb43..0000000000 --- a/edrixs/_sources/release-history.rst.txt +++ /dev/null @@ -1,6 +0,0 @@ -=============== -Release History -=============== - -Initial Release (YYYY-MM-DD) ----------------------------- diff --git a/edrixs/_sources/sg_execution_times.rst.txt b/edrixs/_sources/sg_execution_times.rst.txt deleted file mode 100644 index f2a554f2d3..0000000000 --- a/edrixs/_sources/sg_execution_times.rst.txt +++ /dev/null @@ -1,64 +0,0 @@ - -:orphan: - -.. _sphx_glr_sg_execution_times: - - -Computation times -================= -**00:07.932** total execution time for 10 files **from all galleries**: - -.. container:: - - .. raw:: html - - - - - - - - .. list-table:: - :header-rows: 1 - :class: table table-striped sg-datatable - - * - Example - - Time - - Mem (MB) - * - :ref:`sphx_glr_auto_examples_example_2_single_atom_RIXS.py` (``../../examples/sphinx/example_2_single_atom_RIXS.py``) - - 00:03.985 - - 0.0 - * - :ref:`sphx_glr_auto_examples_example_8_Hunds_interactions.py` (``../../examples/sphinx/example_8_Hunds_interactions.py``) - - 00:01.236 - - 0.0 - * - :ref:`sphx_glr_auto_examples_example_3_AIM_XAS.py` (``../../examples/sphinx/example_3_AIM_XAS.py``) - - 00:00.548 - - 0.0 - * - :ref:`sphx_glr_auto_examples_example_7_transitions.py` (``../../examples/sphinx/example_7_transitions.py``) - - 00:00.514 - - 0.0 - * - :ref:`sphx_glr_auto_examples_example_5_charge_transfer.py` (``../../examples/sphinx/example_5_charge_transfer.py``) - - 00:00.478 - - 0.0 - * - :ref:`sphx_glr_auto_examples_example_4_GS_analysis.py` (``../../examples/sphinx/example_4_GS_analysis.py``) - - 00:00.421 - - 0.0 - * - :ref:`sphx_glr_auto_examples_example_9_Coulomb.py` (``../../examples/sphinx/example_9_Coulomb.py``) - - 00:00.368 - - 0.0 - * - :ref:`sphx_glr_auto_examples_example_1_crystal_field.py` (``../../examples/sphinx/example_1_crystal_field.py``) - - 00:00.149 - - 0.0 - * - :ref:`sphx_glr_auto_examples_example_0_ed_calculator.py` (``../../examples/sphinx/example_0_ed_calculator.py``) - - 00:00.138 - - 0.0 - * - :ref:`sphx_glr_auto_examples_example_6_Hubbard_dimer.py` (``../../examples/sphinx/example_6_Hubbard_dimer.py``) - - 00:00.095 - - 0.0 diff --git a/edrixs/_sources/user/basics.ed.rst.txt b/edrixs/_sources/user/basics.ed.rst.txt deleted file mode 100644 index 4f1bd0bff1..0000000000 --- a/edrixs/_sources/user/basics.ed.rst.txt +++ /dev/null @@ -1,5 +0,0 @@ -.. _basics.ed: - -******** -ED -******** diff --git a/edrixs/_sources/user/basics.rixs.rst.txt b/edrixs/_sources/user/basics.rixs.rst.txt deleted file mode 100644 index 89a68f8f81..0000000000 --- a/edrixs/_sources/user/basics.rixs.rst.txt +++ /dev/null @@ -1,5 +0,0 @@ -.. _basics.rixs: - -******** -RIXS -******** diff --git a/edrixs/_sources/user/basics.rst.txt b/edrixs/_sources/user/basics.rst.txt deleted file mode 100644 index 417e9beb0c..0000000000 --- a/edrixs/_sources/user/basics.rst.txt +++ /dev/null @@ -1,10 +0,0 @@ -************* -edrixs basics -************* - -.. toctree:: - :maxdepth: 1 - - basics.ed - basics.xas - basics.rixs diff --git a/edrixs/_sources/user/basics.xas.rst.txt b/edrixs/_sources/user/basics.xas.rst.txt deleted file mode 100644 index 99a7b58cf3..0000000000 --- a/edrixs/_sources/user/basics.xas.rst.txt +++ /dev/null @@ -1,5 +0,0 @@ -.. _basics.xas: - -******** -XAS -******** diff --git a/edrixs/_sources/user/examples.rst.txt b/edrixs/_sources/user/examples.rst.txt deleted file mode 100644 index 9104250738..0000000000 --- a/edrixs/_sources/user/examples.rst.txt +++ /dev/null @@ -1,69 +0,0 @@ -.. _examples: - -=================== -Examples -=================== - -Here we outline the examples available. - - -Examples from our CPC paper ---------------------------- - -The original examples from our Computer Physics Communications paper [1]_ are -kept in our -`GitHub repository `_. -These are scripts that compute XAS and RIXS for particular physical -models including: - -* Single atom model for `Ni `_. See Ref. [1]_ section 5.1. -* A two-site `Ir-Ir cluster `_ model. - See Ref. [1]_ section 5.2. This allows the modeling of dimer excitations in RIXS where electrons transition between atoms. - We used this model in our manuscript on Ba5AlIr2O11. [2]_ -* `Anderson impurity model `_. - See Ref. [1]_ section 5.3. This addresses Ba2YOsO6. - - -Extended examples ------------------ - -Additional examples are available -`here `_. -Thus far we have addressed. - -* Single atom and Anderson impurity models for - `Ba2YOsO6 `_. -* Two-site cluster for - `Ba3InIr2O9 `_. -* Single atom model for - `La2NiO4 `_ - taken from Ref. [3]_ extended to show how spin direction can influence XAS and RIXS spectra. -* Single atom model for - `LaNiO3 heterostructures `_ - taken from Ref. [4]_. -* Heavy fermion example for `plutonium O-edge `_ -* Single atom model for - `pyrochlore iridates `_ - taking into account differently oriented octahedra. -* Single atom model for O-edge RIXS on - `URu2Si2 from `_ - including both 5f and 5d orbitals from Ref. [5]_. -* Single atom model for - `L-edge RIXS on URu2Si2 `_ - from Ref. [6]_. - - -.. rubric:: Footnotes - -.. [1] Y. Wang et al., - `Comput. Phys. Commun 243, 151-165 (2019) `_ -.. [2] Y. Wang et al., - `Phys. Rev. Lett. 122, 106401 (2019) `_. -.. [3] G. Fabbris et al., - `Phys. Rev. Lett. 118, 156402 (2017) `_ -.. [4] G. Fabbris et al., - `Phys. Rev. Lett. 117, 147401 (2016) `_ -.. [5] L. Andrew Wray et al., - `Phys. Rev. Lett. 114, 236401 (2015) `_ -.. [6] Y. Wang et al., - `Phys. Rev. B 96, 085146 (2017) `_ diff --git a/edrixs/_sources/user/index.rst.txt b/edrixs/_sources/user/index.rst.txt deleted file mode 100644 index 7c92009988..0000000000 --- a/edrixs/_sources/user/index.rst.txt +++ /dev/null @@ -1,20 +0,0 @@ -.. _user: - -################# -edrixs User Guide -################# - -:Release: |version| -:Date: |today| - -.. toctree:: - :maxdepth: 1 - - whatisedrixs - installation - usedocker - quickstart - basics - pythontips - examples - papers diff --git a/edrixs/_sources/user/installation.rst.txt b/edrixs/_sources/user/installation.rst.txt deleted file mode 100644 index 9ad5bfaec8..0000000000 --- a/edrixs/_sources/user/installation.rst.txt +++ /dev/null @@ -1,366 +0,0 @@ -************ -Installation -************ -For Linux users we suggest :ref:`installing with anaconda `. For Windows and macOS machines, we suggest using the :ref:`docker instructions `, which are relatively straightforward. If desired, you can also compile the code from the source for Linux. - - -.. _AnacondaInstall: - -Install and use edrixs via Anaconda -==================================================== -A conda package has been built for Linux. To use edrixs via Anaconda, you need first to install `Anaconda `_ in your system. -We recommend installing edrixs into a separate environment, for example, called ``edrixs_env``, together with any other packages you might want to use like this:: - - conda create --name edrixs_env -c conda-forge python=3.10 edrixs matplotlib - -We endeavor to keep the conda-forge release up to date, but note that these builds will usually not correspond to the latest version of edrixs, which is available in the `master branch of edrixs `_. - -edrixs will also run on `Google Colaboratory `_, but does not come installed as default. Installing it requires a you to install conda and then edrixs, which can be done by executing a cell:: - - !pip install -q condacolab - import condacolab - condacolab.install() - !conda install -c conda-forge edrixs - -from within a notebook cell. - -Requirements -============ -Several tools and libraries are required to build and install edrixs, - - * Fortran compiler: gfortran and ifort are supported - * MPI environment: openmpi and mpich are tested - * MPI Fortran and C compilers: mpif90, mpicc - * BLAS and LAPACK libraries: `OpenBLAS `_ with gfortran and MKL with ifort - * ARPACK library: `arpack-ng `_ with mpi enabled - * Only Python3 is supported - * numpy, scipy, sympy, matplotlib, sphinx, numpydoc - * mpi4py with the same MPI implementation libraries (``openmpi`` or ``mpich``) as building edrixs - -Build from source -================= -We will show how to build edrixs from source on Ubuntu Linux 20.04 and macOS Mojave (OSX 10.14) as examples. -We will use gcc, gfortran, openmpi and OpenBLAS in these examples. -Building edrixs on other versions of Linux or macOS, or with Intel's ifort+MKL will be similar. - -Ubuntu Linux 20.04 ------------------- -Install compilers and tools:: - - sudo apt-get update - sudo apt-get install build-essential gfortran gcc - sudo apt-get install git wget - sudo apt-get install python3 libpython3-dev python3-pip python3-venv - -Create and activate a python virtual environment for edrixs:: - - python3 -m venv VIRTUAL_ENV - source VIRTUAL_ENV/bin/activate - -where ``VIRTUAL_ENV`` should be replaced by the directory where you wish to install edrixs. - -Alternatively create and activate a conda environment for edrixs:: - - conda create --name edrixs_env python=3.8 - conda activate edrixs_env - -We will assume ``python`` and ``pip`` are pointing to the activated environment from now on. -Check we are using the expected python and pip:: - - which python - which pip - python --version - -Fetch the latest version of ``pip``:: - - pip install --upgrade pip - -openmpi, OpenBLAS, ARPACK can be installed by ``apt-get``, but their versions are old and may not work properly. -However, they can also be compiled from source easily. In the following, we will show both ways, but we always recommend to build newer ones from source. - -openmpi can be installed by:: - - sudo apt-get install libopenmpi-dev - -or from newer version of source, for example v3.1.4:: - - wget https://download.open-mpi.org/release/open-mpi/v3.1/openmpi-3.1.4.tar.bz2 - tar -xjf openmpi-3.1.4.tar.bz2 - cd openmpi-3.1.4 - ./configure CC=gcc CXX=g++ FC=gfortran - make - sudo make install - -the compiling process will take a while. - -OpenBLAS can be installed by:: - - sudo apt-get install libopenblas-dev - -or from a newer version of source:: - - wget https://github.com/xianyi/OpenBLAS/archive/v0.3.6.tar.gz - tar -xzf v0.3.6.tar.gz - cd OpenBLAS-0.3.6 - make CC=gcc FC=gfortran - sudo make PREFIX=/usr/local install - -ARPACK can be installed by:: - - sudo apt-get install libarpack2-dev libparpack2-dev - -or from a newer version of source:: - - wget https://github.com/opencollab/arpack-ng/archive/3.6.3.tar.gz - tar -xzf 3.6.3.tar.gz - cd arpack-ng-3.6.3 - ./bootstrap - ./configure --enable-mpi --with-blas="-L/usr/local/lib/ -lopenblas" FC=gfortran F77=gfortran MPIFC=mpif90 MPIF77=mpif90 - make - sudo make install - -mpi4py can be installed by:: - - export MPICC=/usr/local/bin/mpicc - sudo pip install --no-cache-dir mpi4py - -or from source:: - - wget https://github.com/mpi4py/mpi4py/archive/3.0.1.tar.gz - tar xzf 3.0.1.tar.gz - cd mpi4py-3.0.1 - -edit mpi.cfg to set MPI paths as following:: - - [mpi] - mpi_dir = /usr/local - mpicc = %(mpi_dir)s/bin/mpicc - mpicxx = %(mpi_dir)s/bin/mpicxx - include_dirs = %(mpi_dir)s/include - libraries = mpi - library_dirs = %(mpi_dir)s/lib - runtime_library_dirs = %(mpi_dir)s/lib - -and comment all other contents. Then, build and install by:: - - python setup.py build - sudo pip install . - -Check whether the MPI paths are correct by:: - - python - >>> import mpi4py - >>> mpi4py.get_config() - {'mpicc': '/usr/local/bin/mpicc', - 'mpicxx': '/usr/local/bin/mpicxx', - 'include_dirs': '/usr/local/include', - 'libraries': 'mpi', - 'library_dirs': '/usr/local/lib', - 'runtime_library_dirs': '/usr/local/lib'} - -Now, we are ready to build edrixs:: - - git clone https://github.com/NSLS-II/edrixs.git - cd edrixs - pip install -v . - -Start to play with edrixs by:: - - python - >>> import edrixs - >>> edrixs.some_functions(...) - -or go to ``examples`` directory to run some examples:: - - cd examples/more/ED/14orb - ./get_inputs.py - mpirun -np 2 ed.x - mpirun -np 2 ./run_fedsolver.py - cd ../../RIXS/LaNiO3_thin - mpirun -np 2 ./run_rixs_fsolver.py - -if no errors, the installation is successful. - -macOS Mojave (OSX 10.14) ------------------------- -Install newest Xcode through App store. - -Use MacPorts -~~~~~~~~~~~~ -Download and install `MacPorts `_. -Update MacPorts by:: - - sudo port -v selfupdate - -Install gcc8, arpack, openblas and openmpi:: - - sudo port -v install gcc8 - sudo port select gcc mp-gcc8 - sudo port -v install openmpi-default +gcc8 - sudo port -v install openblas +gcc8 - sudo port -v install arpack +openblas +openmpi - sudo port select --set mpi openmpi-mp-fortran - -Install Python, pip, numpy, scipy, sympy, matplotlib:: - - sudo port -v install python37 py37-pip - sudo port -v install py37-numpy +gcc8 +openblas - sudo port -v install py37-scipy +gcc8 +openblas - sudo port -v install py37-sympy - sudo port -v install py37-matplotlib - -**Notes:** - -* DO NOT use pip to install numpy because it will use ``clang`` as default compiler, which has a strange bug when using ``f2py`` with ``mpif90`` compiler. If you cannot solve this issue by ``sudo port install py37-numpy +gcc8``, you can compile numpy from its source with ``gcc`` compiler. Always use gcc to compile numpy if you want to build it from source. - -* You can also try ``gcc9`` if it is already available, but be sure to change all ``gcc8`` to ``gcc9`` in the above commands. - -We will assume ``python`` pointing to ``python3.7`` and ``pip`` pointing to ``pip3.7`` from now on. If this is not the case, you can make links explicitly. -Check we are using the expected python and pip:: - - which python - python --version - which pip - pip --version - -Add the following two lines into ``~/.bash_profile``:: - - export PATH="/opt/local/bin:/opt/local/sbin:$PATH" - export PATH=/opt/local/Library/Frameworks/Python.framework/Versions/3.7/bin:$PATH - -Close current terminal and open a new one. - -Install mpi4py:: - - export MPICC=/opt/local/bin/mpicc - sudo pip install --no-cache-dir mpi4py - -Please be sure to check whether the MPI paths of mpi4py are correct by:: - - python - >>> import mpi4py - >>> mpi4py.get_config() - {'mpicc': '/opt/local/bin/mpicc'} - -Now, we are ready to build edrixs:: - - git clone https://github.com/NSLS-II/edrixs.git - cd edrixs - make -C src F90=mpif90 LIBS="-L/opt/local/lib -lopenblas -lparpack -larpack" - make -C src install - python setup.py config_fc --f77exec=mpif90 --f90exec=mpif90 build_ext --libraries=openblas,parpack,arpack --library-dirs=/opt/local/lib - sudo pip install . - -You can add ``edrixs/bin`` to the environment variable ``PATH`` in ~/.bash_profile. - -Go to ``examples`` directory to run some examples:: - - cd examples/more/ED/14orb - ./get_inputs.py - mpirun -np 2 ../../../../src/ed.x - mpirun -np 2 ./run_fedsolver.py - cd ../../RIXS/LaNiO3_thin - mpirun -np 2 ./run_rixs_fsolver.py - -if no errors, the installation is successful. - -All done, enjoy! - -Use Homebrew -~~~~~~~~~~~~~ -Install Homebrew:: - - /usr/bin/ruby -e "$(curl -fsSL https://raw.githubusercontent.com/Homebrew/install/master/install)" - -Add following line to ``~/.bash_profile``:: - - export PATH="/usr/local/bin:$PATH" - -Install gcc9:: - - brew install gcc@9 - -Install openblas and arpack:: - - brew install openblas - brew install arpack - -openmpi has been automatically installed when installing arpack. - -Install python3.7:: - - brew install python - -We will assume ``python`` pointing to ``python3.7`` and ``pip`` pointing to ``pip3.7`` from now on. If this is not the case, you can make link explicitly. -Check we are using the expected python and pip:: - - which python - python --version - which pip - pip --version - -Make links if gcc, g++ and gfortran are not pointing to gcc-9, g++-9, gfortran-9, for example:: - - ln -s /usr/local/Cellar/gcc/9.1.0/bin/gcc-9 /usr/local/bin/gcc - ln -s /usr/local/Cellar/gcc/9.1.0/bin/g++-9 /usr/local/bin/g++ - ln -s /usr/local/Cellar/gcc/9.1.0/bin/gfortran-9 /usr/local/bin/gfortran - -DO NOT install numpy through ``pip`` because it uses ``clang`` as default compiler, which will cause problems. -We will build numpy from source with gcc:: - - wget https://github.com/numpy/numpy/archive/v1.16.3.tar.gz - tar xzf v1.16.3.tar.gz - cd numpy-1.16.3 - export CC=gcc CXX=g++ - python setup.py build - pip install . - -You might need to do ``brew install wget`` if it is not already installed. -If you have BLIS or MKL installed, you will need to tell numpy to compile with -openblas. Create a file in the numpy directory called site.cfg and put the -following text in it:: - - [openblas] - libraries = openblas - library_dirs = /usr/local/Cellar/openblas/0.3.9/lib - include_dirs = /usr/local/Cellar/openblas/0.3.9/include - runtime_library_dirs = /usr/local/Cellar/openblas/0.3.9/lib - -Now we are ready to install scipy, sympy, matplotlib:: - - pip install scipy sympy matplotlib - export MPICC=/usr/local/bin/mpicc - pip install --no-cache-dir mpi4py - -Please be sure to check whether the MPI paths of mpi4py are correct by:: - - python - >>> import mpi4py - >>> mpi4py.get_config() - {'mpicc': '/usr/local/bin/mpicc'} - -Now, we are ready to build edrixs:: - - git clone https://github.com/NSLS-II/edrixs.git - cd edrixs - make -C src F90=mpif90 LIBS="-L/usr/local/opt/openblas/lib -lopenblas -L/usr/local/lib -lparpack -larpack" - make -C src install - python setup.py config_fc --f77exec=mpif90 --f90exec=mpif90 build_ext --libraries=openblas,parpack,arpack --library-dirs=/usr/local/lib:/usr/local/opt/openblas/lib - pip install . - -You can add ``edrixs/bin`` to the environment variable ``PATH`` in ``~/.bash_profile``. - -Go to ``examples`` directory to run some examples:: - - cd examples/more/ED/14orb - ./get_inputs.py - mpirun -np 2 ../../../../src/ed.x - mpirun -np 2 ./run_fedsolver.py - cd ../../RIXS/LaNiO3_thin - mpirun -np 2 ./run_rixs_fsolver.py - -if no errors, the installation is successful. - -All done, enjoy! - -.. [#] To change your default pip you need to add a line to your ``~/.bashrc`` on linux or to your ``~/.bash_profile`` on macOS. This should be ``alias pip='/usr/bin/pip3'`` where the path is determined by calling ``which pip3`` from your terminal. diff --git a/edrixs/_sources/user/papers.rst.txt b/edrixs/_sources/user/papers.rst.txt deleted file mode 100644 index 6f91c1eb0f..0000000000 --- a/edrixs/_sources/user/papers.rst.txt +++ /dev/null @@ -1,54 +0,0 @@ -**************** -EDRIXS papers -**************** - -How to cite ------------ -If you are using the EDRIXS code to do some studies and would like to publish your great works, it would be really appreciated if you can cite the following paper: - -* EDRIXS: An open source toolkit for simulating spectra of resonant inelastic x-ray scattering, Y.L. Wang, G. Fabbris, M.P.M. Dean and G. Kotliar, `Computer Physics Communications,243, 151 (2019) `_ - -Papers using EDRIXS -------------------- - -* Strong Orbital Polarization in a Cobaltate-Titanate Oxide Heterostructure, Sangjae Lee, Alex Taekyung Lee, Alexandru B. Georgescu, Gilberto Fabbris, Myung-Geun Han, Yimei Zhu, John W. Freeland, Ankit S. Disa, Yichen Jia, Mark P. M. Dean, Frederick J. Walker, Sohrab Ismail-Beigi, and Charles H. Ahn, `Phys. Rev. Lett. 123, 117201 (2019) `_ - -* Large Polarons as Key Quasiparticles in SrTiO\ :sub:`3` and SrTiO\ :sub:`3`\ -Based Heterostructures, Andrey Geondzhian, Alessia Sambri, Gabriella M. De Luca, Roberto Di Capua, Emiliano Di Gennaro, Davide Betto, Matteo Rossi, Ying Ying Peng, Roberto Fumagalli, Nicholas B. Brookes, Lucio Braicovich, Keith Gilmore, Giacomo Ghiringhelli, and Marco Salluzzo, `Phys. Rev. Lett. 125, 126401 (2020) `_ - -* Enhanced hybridization in the electronic ground state of the intercalated honeycomb iridate Ag\ :sub:`3`\ LiIr\ :sub:`2`\ O\ :sub:`6`\ , A. de la Torre, B. Zager, F. Bahrami, M. DiScala, J. R. Chamorro, M. H. Upton, G. Fabbris, D. Haskel, D. Casa, T. M. McQueen, F. Tafti, and K. W. Plumb, `Phys. Rev. B 104, L100416 (2021) `_ - -* Probing Physical Oxidation State by Resonant X-ray Emission Spectroscopy: Applications to Iron Model Complexes and Nitrogenase, Rebeca G. Castillo, Anselm W. Hahn, Benjamin E. Van Kuiken, Justin T. Henthorn, Jeremy McGale, and Serena DeBeer, `Angew. Chem. Int. Ed. 60, 10112–1012 (2021) `_ - -* Computing Local Multipoint Correlators Using the Numerical Renormalization Group, Seung-Sup B. Lee, Fabian B. Kugler, and Jan von Delft, `Phys. Rev. X 11, 041007 (2021) `_ - -* Role of Oxygen States in the Low Valence Nickelate La\ :sub:`4`\ Ni\ :sub:`3`\ O\ :sub:`8`\ , Y. Shen, J. Sears, G. Fabbris, J. Li, J. Pelliciari, I. Jarrige, Xi He, I. Božović, M. Mitrano, Junjie Zhang, J. F. Mitchell, A. S. Botana, V. Bisogni, M. R. Norman, S. Johnston, and M. P. M. Dean, `Phys. Rev. X 12, 011055 (2022) `_ - -* Role of disorder in electronic and magnetic properties of Ag\ :sub:`3`\ LiIr\ :sub:`2`\ O\ :sub:`6`\ , Ying Li and Roser Valentí, `Phys. Rev. B 105, 115123 (2022) `_ - -* Excited-state exchange interaction in NiO determined by high-resolution resonant inelastic x-ray scattering at the Ni M\ :sub:`2,3`\ edges Chun-Yu Liu, Kari Ruotsalainen, Karl Bauer, Régis Decker, Annette Pietzsch, and Alexander Föhlisch, `Phys. Rev. B 106, 035104 (2022) `_ - -* Electronic ground state of two nonmagnetic pentavalent honeycomb iridates, A. de la Torre, B. Zager, J. R. Chamorro, M. H. Upton, G. Fabbris, D. Haskel, D. Casa, T. M. McQueen, and K. W. Plumb, `Phys. Rev. Materials 6, 084406 (2022) `_ - -* Emergence of Spinons in Layered Trimer Iridate Ba\ :sub:`4`\ Ir\ :sub:`3`\ O\ :sub:`10` , Y. Shen, J. Sears, G. Fabbris, A. Weichselbaum, W. Yin, H. Zhao, D. G. Mazzone, H. Miao, M. H. Upton, D. Casa, R. Acevedo-Esteves, C. Nelson, A. M. Barbour, C. Mazzoli, G. Cao, and M. P. M. Dean, `Phys. Rev. Lett. 129, 207201 (2022) `_ - -* Site-specific electronic and magnetic excitations of the skyrmion material Cu\ :sub:`2`\ OSeO\ :sub:`3`\ , Yanhong Gu, Yilin Wang, Jiaqi Lin, Jonathan Pelliciari, Jiemin Li, Myung-Geun Han, Marcus Schmidt, Gabriel Kotliar, Claudio Mazzoli, Mark P. M. Dean, and Valentina Bisogni, `Communications Physics 5, 156 (2022) `_ - -* Electronic structure of the frustrated diamond lattice magnet NiRh\ :sub:`2`\ O\ :sub:`4`\ , B. Zager, J. R. Chamorro, L. Ge, F. Bahrami, V. Bisogni, J. Pelliciari, J. Li, G. Fabbris, T. M. McQueen, M. Mourigal, and K. W. Plumb, `Phys. Rev. B 106, 045134 (2022) `_ - -* Electronic Character of Charge Order in Square-Planar Low-Valence Nickelates, Y. Shen, J. Sears, G. Fabbris, J. Li, J. Pelliciari, M. Mitrano, W. He, Junjie Zhang, J. F. Mitchell, V. Bisogni, M. R. Norman, S. Johnston, and M. P. M. Dean, `Phys. Rev. X 13, 011021 (2023) `_ - -* Resonant inelastic X-ray scattering in topological semimetal FeSi, Yao Shen, Anirudh Chandrasekaran, Jennifer Sears, Tiantian Zhang, Xin Han, Youguo Shi, Jiemin Li, Jonathan Pelliciari, Valentina Bisogni, Mark P. M. Dean, Stefanos Kourtis, `arXiv:2301.02677 (2023) `_ - -* Momentum-independent magnetic excitation continuum in the honeycomb iridate H\ :sub:`3`\ LiIr\ :sub:`2`\ O\ :sub:`6`\, A. de la Torre, B. Zager, F. Bahrami, M. H. Upton, J. Kim, G. Fabbris, G. -H. Lee, W. Yang, D. Haskel, F. Tafti, K. W. Plumb, `Nature Comm. 14, 5018 (2023) `_ - -* Low-energy electronic interactions in ferrimagnetic Sr\ :sub:`2`\ CrReO\ :sub:`6`\ thin films, Guillaume Marcaud, Alex Taekyung Lee, Adam J. Hauser, F. Y. Yang, Sangjae Lee, Diego Casa, Mary Upton, Thomas Gog, Kayahan Saritas, Yilin Wang, Mark P. M. Dean, Hua Zhou, Zhan Zhang, F. J. Walker, Ignace Jarrige, Sohrab Ismail-Beigi, and Charles Ahn, `Phys. Rev. B 108, 075132 (2023) `_ - -* Softening of dd-excitation in the resonant inelastic x-ray scattering spectra as a signature of Hund's coupling in nickelates, U Kumar, C Melnick, G Kotliar, `arXiv:2310.00983 (2023) `_ - -* Interplay of broken symmetry and delocalized excitations in the insulating state of 1T-TaS\ :sub:`2`\, Xun Jia, Anubhab Haldar, Jungho Kim, Yilin Wang, Gilberto Fabbris, Karl Ludwig, Stefanos Kourtis, Mary Upton, Yu Liu, Wenjian Lu, Xuan Luo, Yu-Ping Sun, Diego Casa, Sahar Sharifzadeh, Pierre T. Darancet, Yue Cao, `Phys. Rev. B 108, 205105 (2023) `_ - -* Local site behavior of the 5d and 4f ions in the frustrated pyrochlore Ho\ :sub:`2`\Os\ :sub:`2`\O\ :sub:`7`\, S. Calder, Z. Y. Zhao, M. H. Upton, and J.-Q. Yan, `Phys. Rev. B 109, 054408 (2024) `_ - -* Elucidating the Role of Dimensionality on the Electronic Structure of the Van der Waals Antiferromagnet NiPS\ :sub:`3`\ , M. F. DiScala, D. Staros, A. de la Torre, A. Lopez, D. Wong, C. Schulz, M. Bartkowiak, B. Rubenstein, K. W. Plumb, `Advanced Physics Research 3, 2300096 (2024) `_ - -* Experimentally Assessing the Electronic Structure and Spin-State Energetics in MnFe Dimers Using 1s3p Resonant Inelastic X-ray Scattering, Rebeca G. Castillo, Benjamin E. Van Kuiken, Thomas Weyhermüller, and Serena DeBeer, `Inorganic Chemistry (2024) `_ \ No newline at end of file diff --git a/edrixs/_sources/user/pythontips.rst.txt b/edrixs/_sources/user/pythontips.rst.txt deleted file mode 100644 index d9d494c2fa..0000000000 --- a/edrixs/_sources/user/pythontips.rst.txt +++ /dev/null @@ -1,66 +0,0 @@ -.. _pythontips: - -************************** -Tips for python and edrixs -************************** - -In the design of edrixs, we made a deliberate choice to use -`python `_ for the application programming interface. This is -because of its readability, easy of use and flexibility to run and combine it -in many different ways. - -The standard way to run a python script ``myscript.py`` is:: - - python myscript.py - -This will generate the outputs of the script such as plots you choose to save -and print statements will be returned into the terminal. You can save the print -output by simply redirecting the output to a file:: - - python myscript.py > myoutput.txt - -Python also includes an excellent plotting package -`matplotlib `_, which one can use to make publication -quality plots. - -While our aim is that the huge majority of tasks can be done without modifying -the underlying code all the python layers of code are easy to modify if you -would like to. If you want to modify, say, solvers.py. we would suggest copying -solvers.py to your working directory under a different name e.g. -:code:`my_solvers.py`. Executing :code:`%run my_solvers.py` -from within your script will then load the functions from the file into your -namespace. Just be sure to tell the script to load functions from edrixs -(and not via a relative file import) i.e. :code:`from .soc import atom_hsoc` -should be :code:`from edrixs.soc import atom_hsoc`. - -For more exploratory usage, `IPython `_ or -`Jupyter `_ -can be very useful. (See :ref:`edrixsanddocker` for invoking jupyter over docker.) -After starting IPython by typing ``ipython`` at the command line -you might find it useful to execute your script via:: - - %run -i myscript.py - -The *interactive* flag ``-i`` means that all the variables and functions you loaded -will be available to you at the command line. These can be straightforwardly printed -or inspected via the ``?`` or ``??`` flags, which show you the object documentation -and the object code respectively.:: - - from edrixs import cd_cubic_d - cd_cubic_d?? - -Including ``%matplotlib widget`` in your script -will facilitate interactive plots. All these interactive options are also available -in the rich outputs possible within the -`juptyer lab `_ interface. - -A brute force option to look at a variable deep within the code is to use a debugger:: - - python3 -m pdb myscript.py - -Use ``import pdb; pdb.set_trace()`` to set the place where you want to enter the -debugger. See `here `_ for more details. - -If you are feeling even braver, you can browse the Fortran code which does the -heavyweight computation. Either in the source edrixs directory or via the online -`edrixs repo `_. diff --git a/edrixs/_sources/user/quickstart.rst.txt b/edrixs/_sources/user/quickstart.rst.txt deleted file mode 100644 index 46f29db1df..0000000000 --- a/edrixs/_sources/user/quickstart.rst.txt +++ /dev/null @@ -1,46 +0,0 @@ -=================== -Quickstart tutorial -=================== - -If you are already familiar with multiplet calculations this page and our -:ref:`examples page ` are a good part to start. Otherwise, checkout -our Pedagogical examples to learn the concepts -behind how edrixs works. The first example explains :ref:`Exact diagonalization -`. - -Use edrixs as an ED calculator ------------------------------- -edrixs can be used as a simple ED calculator to get eigenvalues (eigenvectors) of a many-body Hamiltonian with small size dimension (:math:`< 1,000`). -We will give an example to get eigenvalues for a :math:`t_{2g}`-orbital system (:math:`l_{eff}=1`). There are 6 orbitals including spin. - -Launch your favorite python terminal:: - - >>> import edrixs - >>> import scipy - >>> norb = 6 - >>> noccu = 2 - >>> Ud, JH = edrixs.UJ_to_UdJH(4, 1) - >>> F0, F2, F4 = edrixs.UdJH_to_F0F2F4(Ud, JH) - >>> umat = edrixs.get_umat_slater('t2g', F0, F2, F4) - >>> emat = edrixs.atom_hsoc('t2g', 0.2) - >>> basis = edrixs.get_fock_bin_by_N(norb, noccu) - >>> H = edrixs.build_opers(4, umat, basis) - >>> e1, v1 = scipy.linalg.eigh(H) - >>> H += edrixs.build_opers(2, emat, basis) - >>> e2, v2 = scipy.linalg.eigh(H) - >>> print(e1) - [1. 1. 1. 1. 1. 1. 1. 1. 1. 3. 3. 3. 3. 3. 6.] - >>> print(e2) - [0.890519 0.890519 0.890519 0.890519 0.890519 1.1 1.1 1.1 - 1.183391 3.009481 3.009481 3.009481 3.009481 3.009481 6.016609] - - -Hello RIXS! ------------ - -This is a "Hello World!" example for RIXS simulations at Ni (:math:`3d^8`) :math:`L_{2/3}` edges. -:math:`L_3` means transition from Ni-:math:`2p_{3/2}` to Ni-:math:`3d`, and -:math:`L_2` means transition from Ni-:math:`2p_{1/2}` to Ni-:math:`3d`. - -.. plot:: pyplots/helloworld.py - :include-source: diff --git a/edrixs/_sources/user/usedocker.rst.txt b/edrixs/_sources/user/usedocker.rst.txt deleted file mode 100644 index 9e7ae0568b..0000000000 --- a/edrixs/_sources/user/usedocker.rst.txt +++ /dev/null @@ -1,53 +0,0 @@ -.. _edrixsanddocker: - -********************* -edrixs and docker -********************* - -Run edrixs in a docker container --------------------------------- - -To make life easier, we have built a docker image based on Ubuntu Linux (22.04) for edrixs, so you don't need to struggle with the installation anymore. -The docker image can be used on any OS as long as the `docker `_ application is available. -Follow these steps to use the docker image: - -* Install the `docker `_ application on your system. - -* Once Docker is running, create a directory to store data and create a file called ``docker-compose.yml`` with contents :: - - version: '3' - services: - edrixs-jupyter: - image: edrixs/edrixs - volumes: - - ./:/home/rixs - working_dir: /home/rixs - ports: - - 8888:8888 - command: "jupyter lab --ip=0.0.0.0 --port=8888 --no-browser --allow-root" - edrixs-ipython: - image: edrixs/edrixs - volumes: - - ./:/home/rixs - working_dir: /home/rixs - - and execute :: - - docker compose up - - This will return a url, which you can open to connect to the jupyter session. - -* If you would like to access a terminal rather than jupyter run :: - - docker compose run --rm edrixs-ipython - - -Sharing your code ------------------ - -Using Docker is a nice way to straightforwardly share your code with others. The standard way to specify which docker image is needed to run your code is to include a file named ``Dockerfile`` with the following contents :: - - FROM edrixs/edrixs - -You might like to checkout the `jupyter-repo2docker -`_ project, which helps automate the process of building and connecting to docker images. The `mybinder `_ project might also be helpful as this will open a github respository of notebooks in an executable environment, making your code immediately reproducible by anyone, anywhere. diff --git a/edrixs/_sources/user/whatisedrixs.rst.txt b/edrixs/_sources/user/whatisedrixs.rst.txt deleted file mode 100644 index c62d381f35..0000000000 --- a/edrixs/_sources/user/whatisedrixs.rst.txt +++ /dev/null @@ -1,9 +0,0 @@ -**************** -What is edrixs? -**************** - -EDRIXS is an open source toolkit for simulating XAS and RIXS spectra based on exact diagonalization of model Hamiltonians. -It was started as part of `COMSCOPE project `_ in the -Center for Computational Material Spectroscopy and Design, Brookhaven National Laboratory and is now maintained and -developed in collaboration between the `Condensed Matter Physics and Materials Science Division `_ -and the `National Syncrotron Light Source II `_. \ No newline at end of file diff --git a/edrixs/_static/_sphinx_javascript_frameworks_compat.js b/edrixs/_static/_sphinx_javascript_frameworks_compat.js deleted file mode 100644 index 8549469dc2..0000000000 --- a/edrixs/_static/_sphinx_javascript_frameworks_compat.js +++ /dev/null @@ -1,134 +0,0 @@ -/* - * _sphinx_javascript_frameworks_compat.js - * ~~~~~~~~~~ - * - * Compatability shim for jQuery and underscores.js. - * - * WILL BE REMOVED IN Sphinx 6.0 - * xref RemovedInSphinx60Warning - * - */ - -/** - * select a different prefix for underscore - */ -$u = _.noConflict(); - - -/** - * small helper function to urldecode strings - * - * See https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Global_Objects/decodeURIComponent#Decoding_query_parameters_from_a_URL - */ -jQuery.urldecode = function(x) { - if (!x) { - return x - } - return decodeURIComponent(x.replace(/\+/g, ' ')); -}; - -/** - * small helper function to urlencode strings - */ -jQuery.urlencode = encodeURIComponent; - -/** - * This function returns the parsed url parameters of the - * current request. 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exports&&"object"==typeof module?module.exports=e():"function"==typeof define&&define.amd?define([],e):"object"==typeof exports?exports.ClipboardJS=e():t.ClipboardJS=e()}(this,function(){return n={686:function(t,e,n){"use strict";n.d(e,{default:function(){return o}});var e=n(279),i=n.n(e),e=n(370),u=n.n(e),e=n(817),c=n.n(e);function a(t){try{return document.execCommand(t)}catch(t){return}}var f=function(t){t=c()(t);return a("cut"),t};var l=function(t){var e,n,o,r=1 - - - - diff --git a/edrixs/_static/copybutton.css b/edrixs/_static/copybutton.css deleted file mode 100644 index f1916ec7d1..0000000000 --- a/edrixs/_static/copybutton.css +++ /dev/null @@ -1,94 +0,0 @@ -/* Copy buttons */ -button.copybtn { - position: absolute; - display: flex; - top: .3em; - right: .3em; - width: 1.7em; - height: 1.7em; - opacity: 0; - transition: opacity 0.3s, border .3s, background-color .3s; - user-select: none; - padding: 0; - border: none; - outline: none; - border-radius: 0.4em; - /* The colors that GitHub uses */ - border: #1b1f2426 1px solid; - background-color: #f6f8fa; - color: #57606a; -} - -button.copybtn.success { - border-color: #22863a; - color: #22863a; -} - -button.copybtn svg { - stroke: currentColor; - width: 1.5em; - height: 1.5em; - padding: 0.1em; -} - -div.highlight { - position: relative; -} - -/* Show the copybutton */ -.highlight:hover button.copybtn, button.copybtn.success { - opacity: 1; -} - -.highlight button.copybtn:hover { - background-color: rgb(235, 235, 235); -} - -.highlight button.copybtn:active { - background-color: rgb(187, 187, 187); -} - -/** - * A minimal CSS-only tooltip copied from: - * https://codepen.io/mildrenben/pen/rVBrpK - * - * To use, write HTML like the following: - * - *

Short

- */ - .o-tooltip--left { - position: relative; - } - - .o-tooltip--left:after { - opacity: 0; - visibility: hidden; - position: absolute; - content: attr(data-tooltip); - padding: .2em; - font-size: .8em; - left: -.2em; - background: grey; - color: white; - white-space: nowrap; - z-index: 2; - border-radius: 2px; - transform: translateX(-102%) translateY(0); - transition: opacity 0.2s cubic-bezier(0.64, 0.09, 0.08, 1), transform 0.2s cubic-bezier(0.64, 0.09, 0.08, 1); -} - -.o-tooltip--left:hover:after { - display: block; - opacity: 1; - visibility: visible; - transform: translateX(-100%) translateY(0); - transition: opacity 0.2s cubic-bezier(0.64, 0.09, 0.08, 1), transform 0.2s cubic-bezier(0.64, 0.09, 0.08, 1); - transition-delay: .5s; -} - -/* By default the copy button shouldn't show up when printing a page */ -@media print { - button.copybtn { - display: none; - } -} diff --git a/edrixs/_static/copybutton.js b/edrixs/_static/copybutton.js deleted file mode 100644 index 2ea7ff3e21..0000000000 --- a/edrixs/_static/copybutton.js +++ /dev/null @@ -1,248 +0,0 @@ -// Localization support -const messages = { - 'en': { - 'copy': 'Copy', - 'copy_to_clipboard': 'Copy to clipboard', - 'copy_success': 'Copied!', - 'copy_failure': 'Failed to copy', - }, - 'es' : { - 'copy': 'Copiar', - 'copy_to_clipboard': 'Copiar al portapapeles', - 'copy_success': '¡Copiado!', - 'copy_failure': 'Error al copiar', - }, - 'de' : { - 'copy': 'Kopieren', - 'copy_to_clipboard': 'In die Zwischenablage kopieren', - 'copy_success': 'Kopiert!', - 'copy_failure': 'Fehler beim Kopieren', - }, - 'fr' : { - 'copy': 'Copier', - 'copy_to_clipboard': 'Copier dans le presse-papier', - 'copy_success': 'Copié !', - 'copy_failure': 'Échec de la copie', - }, - 'ru': { - 'copy': 'Скопировать', - 'copy_to_clipboard': 'Скопировать в буфер', - 'copy_success': 'Скопировано!', - 'copy_failure': 'Не удалось скопировать', - }, - 'zh-CN': { - 'copy': '复制', - 'copy_to_clipboard': '复制到剪贴板', - 'copy_success': '复制成功!', - 'copy_failure': '复制失败', - }, - 'it' : { - 'copy': 'Copiare', - 'copy_to_clipboard': 'Copiato negli appunti', - 'copy_success': 'Copiato!', - 'copy_failure': 'Errore durante la copia', - } -} - -let locale = 'en' -if( document.documentElement.lang !== undefined - && messages[document.documentElement.lang] !== undefined ) { - locale = document.documentElement.lang -} - -let doc_url_root = DOCUMENTATION_OPTIONS.URL_ROOT; -if (doc_url_root == '#') { - doc_url_root = ''; -} - -/** - * SVG files for our copy buttons - */ -let iconCheck = ` - ${messages[locale]['copy_success']} - - -` - -// If the user specified their own SVG use that, otherwise use the default -let iconCopy = ``; -if (!iconCopy) { - iconCopy = ` - ${messages[locale]['copy_to_clipboard']} - - - -` -} - -/** - * Set up copy/paste for code blocks - */ - -const runWhenDOMLoaded = cb => { - if (document.readyState != 'loading') { - cb() - } else if (document.addEventListener) { - document.addEventListener('DOMContentLoaded', cb) - } else { - document.attachEvent('onreadystatechange', function() { - if (document.readyState == 'complete') cb() - }) - } -} - -const codeCellId = index => `codecell${index}` - -// Clears selected text since ClipboardJS will select the text when copying -const clearSelection = () => { - if (window.getSelection) { - window.getSelection().removeAllRanges() - } else if (document.selection) { - document.selection.empty() - } -} - -// Changes tooltip text for a moment, then changes it back -// We want the timeout of our `success` class to be a bit shorter than the -// tooltip and icon change, so that we can hide the icon before changing back. -var timeoutIcon = 2000; -var timeoutSuccessClass = 1500; - -const temporarilyChangeTooltip = (el, oldText, newText) => { - el.setAttribute('data-tooltip', newText) - el.classList.add('success') - // Remove success a little bit sooner than we change the tooltip - // So that we can use CSS to hide the copybutton first - setTimeout(() => el.classList.remove('success'), timeoutSuccessClass) - setTimeout(() => el.setAttribute('data-tooltip', oldText), timeoutIcon) -} - -// Changes the copy button icon for two seconds, then changes it back -const temporarilyChangeIcon = (el) => { - el.innerHTML = iconCheck; - setTimeout(() => {el.innerHTML = iconCopy}, timeoutIcon) -} - -const addCopyButtonToCodeCells = () => { - // If ClipboardJS hasn't loaded, wait a bit and try again. This - // happens because we load ClipboardJS asynchronously. - if (window.ClipboardJS === undefined) { - setTimeout(addCopyButtonToCodeCells, 250) - return - } - - // Add copybuttons to all of our code cells - const COPYBUTTON_SELECTOR = 'div.highlight pre'; - const codeCells = document.querySelectorAll(COPYBUTTON_SELECTOR) - codeCells.forEach((codeCell, index) => { - const id = codeCellId(index) - codeCell.setAttribute('id', id) - - const clipboardButton = id => - `` - codeCell.insertAdjacentHTML('afterend', clipboardButton(id)) - }) - -function escapeRegExp(string) { - return string.replace(/[.*+?^${}()|[\]\\]/g, '\\$&'); // $& means the whole matched string -} - -/** - * Removes excluded text from a Node. - * - * @param {Node} target Node to filter. - * @param {string} exclude CSS selector of nodes to exclude. - * @returns {DOMString} Text from `target` with text removed. - */ -function filterText(target, exclude) { - const clone = target.cloneNode(true); // clone as to not modify the live DOM - if (exclude) { - // remove excluded nodes - clone.querySelectorAll(exclude).forEach(node => node.remove()); - } - return clone.innerText; -} - -// Callback when a copy button is clicked. Will be passed the node that was clicked -// should then grab the text and replace pieces of text that shouldn't be used in output -function formatCopyText(textContent, copybuttonPromptText, isRegexp = false, onlyCopyPromptLines = true, removePrompts = true, copyEmptyLines = true, lineContinuationChar = "", hereDocDelim = "") { - var regexp; - var match; - - // Do we check for line continuation characters and "HERE-documents"? - var useLineCont = !!lineContinuationChar - var useHereDoc = !!hereDocDelim - - // create regexp to capture prompt and remaining line - if (isRegexp) { - regexp = new RegExp('^(' + copybuttonPromptText + ')(.*)') - } else { - regexp = new RegExp('^(' + escapeRegExp(copybuttonPromptText) + ')(.*)') - } - - const outputLines = []; - var promptFound = false; - var gotLineCont = false; - var gotHereDoc = false; - const lineGotPrompt = []; - for (const line of textContent.split('\n')) { - match = line.match(regexp) - if (match || gotLineCont || gotHereDoc) { - promptFound = regexp.test(line) - lineGotPrompt.push(promptFound) - if (removePrompts && promptFound) { - outputLines.push(match[2]) - } else { - outputLines.push(line) - } - gotLineCont = line.endsWith(lineContinuationChar) & useLineCont - if (line.includes(hereDocDelim) & useHereDoc) - gotHereDoc = !gotHereDoc - } else if (!onlyCopyPromptLines) { - outputLines.push(line) - } else if (copyEmptyLines && line.trim() === '') { - outputLines.push(line) - } - } - - // If no lines with the prompt were found then just use original lines - if (lineGotPrompt.some(v => v === true)) { - textContent = outputLines.join('\n'); - } - - // Remove a trailing newline to avoid auto-running when pasting - if (textContent.endsWith("\n")) { - textContent = textContent.slice(0, -1) - } - return textContent -} - - -var copyTargetText = (trigger) => { - var target = document.querySelector(trigger.attributes['data-clipboard-target'].value); - - // get filtered text - let exclude = '.linenos'; - - let text = filterText(target, exclude); - return formatCopyText(text, '', false, true, true, true, '', '') -} - - // Initialize with a callback so we can modify the text before copy - const clipboard = new ClipboardJS('.copybtn', {text: copyTargetText}) - - // Update UI with error/success messages - clipboard.on('success', event => { - clearSelection() - temporarilyChangeTooltip(event.trigger, messages[locale]['copy'], messages[locale]['copy_success']) - temporarilyChangeIcon(event.trigger) - }) - - clipboard.on('error', event => { - temporarilyChangeTooltip(event.trigger, messages[locale]['copy'], messages[locale]['copy_failure']) - }) -} - -runWhenDOMLoaded(addCopyButtonToCodeCells) \ No newline at end of file diff --git a/edrixs/_static/copybutton_funcs.js b/edrixs/_static/copybutton_funcs.js deleted file mode 100644 index dbe1aaad79..0000000000 --- a/edrixs/_static/copybutton_funcs.js +++ /dev/null @@ -1,73 +0,0 @@ -function escapeRegExp(string) { - return string.replace(/[.*+?^${}()|[\]\\]/g, '\\$&'); // $& means the whole matched string -} - -/** - * Removes excluded text from a Node. - * - * @param {Node} target Node to filter. - * @param {string} exclude CSS selector of nodes to exclude. - * @returns {DOMString} Text from `target` with text removed. - */ -export function filterText(target, exclude) { - const clone = target.cloneNode(true); // clone as to not modify the live DOM - if (exclude) { - // remove excluded nodes - clone.querySelectorAll(exclude).forEach(node => node.remove()); - } - return clone.innerText; -} - -// Callback when a copy button is clicked. Will be passed the node that was clicked -// should then grab the text and replace pieces of text that shouldn't be used in output -export function formatCopyText(textContent, copybuttonPromptText, isRegexp = false, onlyCopyPromptLines = true, removePrompts = true, copyEmptyLines = true, lineContinuationChar = "", hereDocDelim = "") { - var regexp; - var match; - - // Do we check for line continuation characters and "HERE-documents"? - var useLineCont = !!lineContinuationChar - var useHereDoc = !!hereDocDelim - - // create regexp to capture prompt and remaining line - if (isRegexp) { - regexp = new RegExp('^(' + copybuttonPromptText + ')(.*)') - } else { - regexp = new RegExp('^(' + escapeRegExp(copybuttonPromptText) + ')(.*)') - } - - const outputLines = []; - var promptFound = false; - var gotLineCont = false; - var gotHereDoc = false; - const lineGotPrompt = []; - for (const line of textContent.split('\n')) { - match = line.match(regexp) - if (match || gotLineCont || gotHereDoc) { - promptFound = regexp.test(line) - lineGotPrompt.push(promptFound) - if (removePrompts && promptFound) { - outputLines.push(match[2]) - } else { - outputLines.push(line) - } - gotLineCont = line.endsWith(lineContinuationChar) & useLineCont - if (line.includes(hereDocDelim) & useHereDoc) - gotHereDoc = !gotHereDoc - } else if (!onlyCopyPromptLines) { - outputLines.push(line) - } else if (copyEmptyLines && line.trim() === '') { - outputLines.push(line) - } - } - - // If no lines with the prompt were found then just use original lines - if (lineGotPrompt.some(v => v === true)) { - textContent = outputLines.join('\n'); - } - - // Remove a trailing newline to avoid auto-running when pasting - if (textContent.endsWith("\n")) { - textContent = textContent.slice(0, -1) - } - return textContent -} diff --git a/edrixs/_static/css/badge_only.css b/edrixs/_static/css/badge_only.css deleted file mode 100644 index 88ba55b965..0000000000 --- a/edrixs/_static/css/badge_only.css +++ /dev/null @@ -1 +0,0 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