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
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
def get_hopping_coulomb(locaxis): # Number of orbitals nt2g, nporb, norbs = 6, 6, 24 # On-site Coulomb interaction tensor Ud, JH = edrixs.UJ_to_UdJH(2, 0.3) F0_d, F2_d, F4_d = edrixs.UdJH_to_F0F2F4(Ud, JH) G1_dp, G3_dp = 0.957, 0.569 F0_dp, F2_dp = edrixs.get_F0('dp', G1_dp, G3_dp), 1.107 umat_t2g_i = edrixs.get_umat_slater('t2g', F0_d, F2_d, F4_d) params = [ 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 ] umat_t2gp_n = edrixs.get_umat_slater('t2gp', *params) # static core-hole potential static_v = 2.0 for i in range(0, nt2g): for j in range(nt2g, nt2g + nporb): umat_t2gp_n[i, j, j, i] += static_v umat_i = np.zeros((norbs, norbs, norbs, norbs), dtype=np.complex128) umat_n = np.zeros((norbs, norbs, norbs, norbs), dtype=np.complex128) umat_i[0:6, 0:6, 0:6, 0:6] = umat_t2g_i umat_i[6:12, 6:12, 6:12, 6:12] = umat_t2g_i indx = np.array([[0, 1, 2, 3, 4, 5, 12, 13, 14, 15, 16, 17], [6, 7, 8, 9, 10, 11, 18, 19, 20, 21, 22, 23]]) for m in range(2): for i in range(12): for j in range(12): for k in range(12): for l in range(12): umat_n[indx[m, i], indx[m, j], indx[m, k], indx[m, l]] += umat_t2gp_n[i, j, k, l] emat_i = np.zeros((norbs, norbs), dtype=np.complex128) emat_n = np.zeros((norbs, norbs), dtype=np.complex128) # SOC zeta_d_i, zeta_p_n = 0.35, 1072.6666666666667 soc_d = edrixs.atom_hsoc('t2g', zeta_d_i) soc_p = edrixs.atom_hsoc('p', zeta_p_n) emat_i[0:6, 0:6] += soc_d emat_i[6:12, 6:12] += soc_d emat_n[0:6, 0:6] += soc_d emat_n[6:12, 6:12] += soc_d emat_n[12:18, 12:18] += soc_p emat_n[18:24, 18:24] += soc_p for i in range(2 * nt2g): emat_n[i, i] -= 6 * static_v # Crystal field and hoppings between the two Ir-sites t1, t2, delta = -0.18, 0.036, -0.03 # Uncomment the following line to do calculation without hopping and crystal filed splitting. # t1, t2, delta = 0, 0, -0.03 crys_tmp = np.array( [[0, delta, delta, t1, t2, t1], [delta, 0, delta, t2, t1, t1], [delta, delta, 0, t1, t1, t2], [t1, t2, t1, 0, delta, delta], [t2, t1, t1, delta, 0, delta], [t1, t1, t2, delta, delta, 0]], dtype=np.complex) # transform spin to local axis dmat = np.zeros((2, 2, 2), dtype=np.complex128) ang1, ang2, ang3 = edrixs.rmat_to_euler(locaxis[0]) dmat[0] = edrixs.dmat_spinor(ang1, ang2, ang3) ang1, ang2, ang3 = edrixs.rmat_to_euler(locaxis[1]) dmat[1] = edrixs.dmat_spinor(ang1, ang2, ang3) t_spinor = np.zeros((12, 12), dtype=np.complex128) for i in range(2): off = i * 6 t_spinor[off + 0:off + 2, off + 0:off + 2] = dmat[i] t_spinor[off + 2:off + 4, off + 2:off + 4] = dmat[i] t_spinor[off + 4:off + 6, off + 4:off + 6] = dmat[i] crys_spin = np.zeros((12, 12), dtype=np.complex128) crys_spin[0:12:2, 0:12:2] = crys_tmp crys_spin[1:12:2, 1:12:2] = crys_tmp t_orb = np.zeros((12, 12), dtype=np.complex128) t_orb[0:6, 0:6] = edrixs.tmat_r2c('t2g', True) t_orb[6:12, 6:12] = edrixs.tmat_r2c('t2g', True) crys_spin[:, :] = edrixs.cb_op(crys_spin, np.dot(t_spinor, t_orb)) emat_i[0:12, 0:12] += crys_spin emat_n[0:12, 0:12] += crys_spin # Write to files # ED inputs edrixs.write_emat(emat_i, "ed/hopping_i.in") edrixs.write_umat(umat_i, "ed/coulomb_i.in") # XAS inputs edrixs.write_emat(emat_n, "xas/hopping_n.in") edrixs.write_umat(umat_n, "xas/coulomb_n.in") # RIXS inputs edrixs.write_emat(emat_i, "rixs_pp/hopping_i.in") edrixs.write_umat(umat_i, "rixs_pp/coulomb_i.in") edrixs.write_emat(emat_n, "rixs_pp/hopping_n.in") edrixs.write_umat(umat_n, "rixs_pp/coulomb_n.in") edrixs.write_emat(emat_i, "rixs_ps/hopping_i.in") edrixs.write_umat(umat_i, "rixs_ps/coulomb_i.in") edrixs.write_emat(emat_n, "rixs_ps/hopping_n.in") edrixs.write_umat(umat_n, "rixs_ps/coulomb_n.in")
# t2g system, 6 orbitals and occupied by 2 electrons norb = 6 noccu = 2 # Hubbard U and Hund's coupling J in the form of Kanamori Coulomb interaction U, J = 4, 1 # Hubbard Ud and Hund's coupling JH in the form of Slater Coulomb interaction Ud, JH = edrixs.UJ_to_UdJH(U, J) # Slater integrals F0, F2, F4 = edrixs.UdJH_to_F0F2F4(Ud, JH) # Two fermion terms: spin-orbital coupling (SOC) # The matrix is in the complex spherical Harmonics basis soc_zeta = 0.4 emat_soc = edrixs.atom_hsoc('t2g', soc_zeta) # Four fermion terms: Coulomb interaction # The 4-rank tensor is in the complex spherical Harmonics basis umat = edrixs.get_umat_slater('t2g', F0, F2, F4) # Fock basis in the complex spherical Harmonics basis with the orbital ordering: # |-1,up>, |-1,dn>, |0,up>, |0,dn>, |+1,up>, |+1,dn> # basis: 2d list of integers with 1 or 0, the shape is (15, 6) in this case # where, 15=6*5/2 is the total number of Fock basis and 6 is the total number of # single-particle orbitals basis = edrixs.get_fock_bin_by_N(norb, noccu) # quantum number of orbital angular momentum for t2g: l=1 ll = 1 # Matrices of lx,ly,lz,sx,sy,sz,jx,jy,jz in the single-particle basis # lx: l_orb[0], ly: l_orb[1], lz: l_orb[2] l_orb = edrixs.get_orb_momentum(ll, True) # sx: s_spin[0], sy: s_spin[1], sz: s_spin[2] s_spin = edrixs.get_spin_momentum(ll) # jx: j_so[0], jy: j_so[1], jz: j_so[2]
from example_3_AIM_XAS import (F0_dd, F2_dd, F4_dd, nd, norb_d, norb_bath, v_noccu, imp_mat, CF, 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
# .. math:: # \begin{gather*} # 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} # \end{gather*} # # where :math:`m_s` is the magnetic quantum number for spin # and :math:`m_l` is the quantum number for orbitals. # :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. We can # construct the matrix via umat = edrixs.get_umat_slater('p', F0, F2) ################################################################################ # 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 Slater 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 orbitals
def ed(): # 1-10: Ni-3d valence orbitals, 11-16: Ni-2p core orbitals # Single particle basis: complex shperical Harmonics ndorb, nporb, ntot = 10, 6, 16 emat_i = np.zeros((ntot, ntot), dtype=np.complex) emat_n = np.zeros((ntot, ntot), dtype=np.complex) # 4-index Coulomb interaction tensor, parameterized by # Slater integrals, which are obtained from Cowan's code F2_d, F4_d = 7.9521, 4.9387 # Averaged dd Coulomb interaction is set to be zero F0_d = edrixs.get_F0('d', F2_d, F4_d) G1_dp, G3_dp = 4.0509, 2.3037 # Averaged dp Coulomb interaction is set to be zero F0_dp, F2_dp = edrixs.get_F0('dp', G1_dp, G3_dp), 7.33495 umat_i = edrixs.get_umat_slater( 'dp', F0_d, F2_d, F4_d, # dd 0, 0, 0, 0, # dp 0, 0) # pp umat_n = edrixs.get_umat_slater( 'dp', F0_d, F2_d, F4_d, # dd F0_dp, F2_dp, G1_dp, G3_dp, # dp 0, 0) # pp # Atomic spin-orbit coupling zeta_d, zeta_p = 0.083, 11.24 emat_i[0:ndorb, 0:ndorb] += edrixs.atom_hsoc('d', zeta_d) emat_n[0:ndorb, 0:ndorb] += edrixs.atom_hsoc('d', zeta_d) emat_n[ndorb:ntot, ndorb:ntot] += edrixs.atom_hsoc('p', zeta_p) # Tetragonal crystal field splitting terms, # which are first defined in the real cubic Harmonics basis, # and then transformed to complex shperical Harmonics basis. dt, ds, dq = 0.011428, 0.035714, 0.13 tmp = np.zeros((5, 5), dtype=np.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 tmp[:, :] = edrixs.cb_op(tmp, edrixs.tmat_r2c('d')) emat_i[0:ndorb:2, 0:ndorb:2] += tmp emat_i[1:ndorb:2, 1:ndorb:2] += tmp emat_n[0:ndorb:2, 0:ndorb:2] += tmp emat_n[1:ndorb:2, 1:ndorb:2] += tmp # Build Fock basis in its binary form basis_i = edrixs.get_fock_bin_by_N(ndorb, 8, nporb, nporb) basis_n = edrixs.get_fock_bin_by_N(ndorb, 9, nporb, nporb - 1) ncfg_i, ncfg_n = len(basis_i), len(basis_n) # Build many-body Hamiltonian in Fock basis hmat_i = np.zeros((ncfg_i, ncfg_i), dtype=np.complex) hmat_n = np.zeros((ncfg_n, ncfg_n), dtype=np.complex) hmat_i[:, :] += edrixs.two_fermion(emat_i, basis_i, basis_i) hmat_i[:, :] += edrixs.four_fermion(umat_i, basis_i) hmat_n[:, :] += edrixs.two_fermion(emat_n, basis_n, basis_n) hmat_n[:, :] += edrixs.four_fermion(umat_n, basis_n) # Do exact-diagonalization to get eigenvalues and eigenvectors eval_i, evec_i = np.linalg.eigh(hmat_i) eval_n, evec_n = np.linalg.eigh(hmat_n) # Build dipolar transition operators dipole = np.zeros((3, ntot, ntot), dtype=np.complex) T_abs = np.zeros((3, ncfg_n, ncfg_i), dtype=np.complex) T_emi = np.zeros((3, ncfg_i, ncfg_n), dtype=np.complex) tmp = edrixs.get_trans_oper('dp') for i in range(3): dipole[i, 0:ndorb, ndorb:ntot] = tmp[i] # First, in the Fock basis T_abs[i] = edrixs.two_fermion(dipole[i], basis_n, basis_i) # Then, transfrom to the eigenvector basis T_abs[i] = edrixs.cb_op2(T_abs[i], evec_n, evec_i) T_emi[i] = np.conj(np.transpose(T_abs[i])) return eval_i, eval_n, T_abs, T_emi
G1_dp, G3_dp = 0.957, 0.569 F0_dp, F2_dp = edrixs.get_F0('dp', G1_dp, G3_dp), 1.107 zeta_d_i, zeta_p_n = 0.35, 1072.6666666666667 Ir_loc2g = np.zeros((2, 3, 3), dtype=np.float64) Ir_loc2g[0] = [[+0.70710678, +0.40824829, +0.57735027], [-0.70710678, +0.40824829, +0.57735027], [+0.00000000, -0.81649658, +0.57735027]] Ir_loc2g[1] = [ [-0.70710678, +0.40824829, -0.57735027], [+0.70710678, +0.40824829, -0.57735027], [+0.00000000, -0.81649658, -0.57735027], ] umat_t2g_i = edrixs.get_umat_slater('t2g', F0_d, F2_d, F4_d) umat_t2g_i = edrixs.transform_utensor(umat_t2g_i, edrixs.tmat_c2r('t2g', True)) params = [ 0.0, 0.0, 0.0, # Fk for d F0_dp, F2_dp, # Fk for dp G1_dp, G3_dp, # Gk for dp 0.0, 0.0 # Fk for p ] umat_t2gp_n = edrixs.get_umat_slater('t2gp', *params) tmat = np.zeros((12, 12), dtype=np.complex128)
def get_hopping_coulomb(): norbs = 28 Ud, JH = edrixs.UJ_to_UdJH(6, 0.45) F0_d, F2_d, F4_d = edrixs.UdJH_to_F0F2F4(Ud, JH) scale_dp = 0.9 G1_dp, G3_dp = 0.894 * scale_dp, 0.531 * scale_dp F2_dp = 1.036 * scale_dp Udp_av = 0.0 F0_dp = Udp_av + edrixs.get_F0('dp', G1_dp, G3_dp) params = [F0_d, F2_d, F4_d] umat_i = edrixs.get_umat_slater('t2g', *params) params = [F0_d, F2_d, F4_d, F0_dp, F2_dp, G1_dp, G3_dp, 0.0, 0.0] umat_tmp = edrixs.get_umat_slater('t2gp', *params) tmat = np.zeros((12, 12), dtype=np.complex128) tmat[0:6, 0:6] = edrixs.tmat_c2j(1) tmat[6:12, 6:12] = edrixs.tmat_c2j(1) umat_i[:, :, :, :] = edrixs.transform_utensor(umat_i, edrixs.tmat_c2j(1)) umat_tmp[:, :, :, :] = edrixs.transform_utensor(umat_tmp, tmat) id1 = [0, 1, 2, 3, 4, 5, 8, 9, 10, 11] id2 = [0, 1, 2, 3, 4, 5, 24, 25, 26, 27] umat_n = np.zeros((norbs, norbs, norbs, norbs), dtype=np.complex) for i in range(10): for j in range(10): for k in range(10): for ll in range(10): umat_n[id2[i], id2[j], id2[k], id2[ll]] = umat_tmp[id1[i], id1[j], id1[k], id1[ll]] emat_i = np.zeros((norbs, norbs), dtype=np.complex128) emat_n = np.zeros((norbs, norbs), dtype=np.complex128) # bath energy level & hybridization strength N_site = 3 data = np.loadtxt('bath_sites.in') e1, v1 = data[:, 0], data[:, 1] e2, v2 = data[:, 2], data[:, 3] h = np.zeros((24, 24), dtype=np.complex) h[0, 0] = h[1, 1] = -14.7627972353 h[2, 2] = h[3, 3] = h[4, 4] = h[5, 5] = -15.4689430453 for i in range(N_site): o = 6 + 6 * i # orbital j=1/2 h[o + 0, o + 0] = h[o + 1, o + 1] = e1[i] h[0, o + 0] = h[1, o + 1] = h[o + 0, 0] = h[o + 1, 1] = v1[i] # orbital j=3/2 h[o + 2, o + 2] = h[o + 3, o + 3] = h[o + 4, o + 4] = h[o + 5, o + 5] = e2[i] h[2, o + 2] = h[3, o + 3] = h[4, o + 4] = h[5, o + 5] = v2[i] h[o + 2, 2] = h[o + 3, 3] = h[o + 4, 4] = h[o + 5, 5] = v2[i] emat_i[0:24, 0:24] += h emat_n[0:24, 0:24] += h # static core-hole potential static_V = 2.0 for i in range(6): emat_n[i, i] -= static_V # Write to files # Search GS inputs edrixs.write_emat(emat_i, "search_gs/hopping_i.in") edrixs.write_umat(umat_i, "search_gs/coulomb_i.in") # ED inputs edrixs.write_emat(emat_i, "ed/hopping_i.in") edrixs.write_umat(umat_i, "ed/coulomb_i.in") # XAS inputs edrixs.write_emat(emat_n, "xas/hopping_n.in") edrixs.write_umat(umat_n, "xas/coulomb_n.in") # RIXS inputs edrixs.write_emat(emat_i, "rixs_pp/hopping_i.in") edrixs.write_umat(umat_i, "rixs_pp/coulomb_i.in") edrixs.write_emat(emat_n, "rixs_pp/hopping_n.in") edrixs.write_umat(umat_n, "rixs_pp/coulomb_n.in") edrixs.write_emat(emat_i, "rixs_ps/hopping_i.in") edrixs.write_umat(umat_i, "rixs_ps/coulomb_i.in") edrixs.write_emat(emat_n, "rixs_ps/hopping_n.in") edrixs.write_umat(umat_n, "rixs_ps/coulomb_n.in")
You can change noccu to play with it. """ # number of orbitals norbs = 14 # number of occupancy, change this number and re-run it to see how it will change. noccu = 7 # Slater integrals: FX, x=0, 2, 4, 2*3=6 F2_f, F4_f, F6_f = 9.711 * 0.77, 6.364 * 0.77, 4.677 * 0.77 # set the average Coulomb interaction energy Uf_ave to zero Uf_av = 0.0 F0_f = Uf_av + edrixs.get_F0('f', F2_f, F4_f, F6_f) # get rank-4 Coulomb U-tensor params = [F0_f, F2_f, F4_f, F6_f] umat_i = edrixs.get_umat_slater('f', *params) # SOC strength zeta_f_i = 0.261 * 0.9 hsoc_i = edrixs.atom_hsoc('f', zeta_f_i) # prepare files for ed.x # write control parameters to file edrixs.write_config(ed_solver=2, num_val_orbs=norbs, neval=100, ncv=200, nvector=1, idump=True) # write nonzeros terms of two-fermion terms hsoc_i to file 'hopping_i.in', read by ed.x edrixs.write_emat(hsoc_i, 'hopping_i.in', 1E-10) # write nonzeros terms of four-fermion terms umat to file 'coulomb_i.in', read by ed.x
def get_hopping_coulomb(locaxis): # Number of orbitals for each site ndorb, nporb = 6, 4 # Number of sites nsite = 2 # Total number of orbitals ntot = nsite * (ndorb + nporb) # orbital orders: # 0-5: 1st-site-t2g # 6-11: 2nd-site-t2g # 12-15: 1st-site-2p # 16-19: 2nd-site-2p # On-site Coulomb interaction tensor U, J = 2.0, 0.3 Ud, JH = edrixs.UJ_to_UdJH(U, J) F0_dd, F2_dd, F4_dd = edrixs.UdJH_to_F0F2F4(Ud, JH) # k=0, 2, 2*l G1_dp, G3_dp = 0.957, 0.569 # k=|2-1|, |2+1| F0_dp, F2_dp = edrixs.get_F0('dp', G1_dp, G3_dp), 1.107 # k=0, min(2*2, 2*1) # just one site t2g-subspace umat_tmp_i = edrixs.get_umat_slater('t2g', F0_dd, F2_dd, F4_dd) params = [ F0_dd, F2_dd, F4_dd, # FX for dd F0_dp, F2_dp, # FX for dp G1_dp, G3_dp, # GX for dp 0, 0 # FX for pp ] # just one site umat_tmp_n = edrixs.get_umat_slater('t2gp32', *params) # 2p_3/2 -> t2g # static core-hole potential static_v = 2.0 for i in range(0, ndorb): for j in range(ndorb, ndorb + nporb): umat_tmp_n[i, j, j, i] += static_v # two sites as a whole umat_i = np.zeros((ntot, ntot, ntot, ntot), dtype=np.complex) umat_n = np.zeros((ntot, ntot, ntot, ntot), dtype=np.complex) umat_i[0:ndorb, 0:ndorb, 0:ndorb, 0:ndorb] = umat_tmp_i # 1st site 5d-valence umat_i[ndorb:2 * ndorb, ndorb:2 * ndorb, ndorb:2 * ndorb, ndorb:2 * ndorb] = umat_tmp_i # 2nd site 5d-valence indx = np.array([ [ 0, 1, 2, 3, 4, 5, # orbital indices for 1st site 5d-t2g 12, 13, 14, 15 ], # orbital indices for 1st site 2p-core [ 6, 7, 8, 9, 10, 11, # orbital indices for 2nd site 5d-t2g 16, 17, 18, 19 ] # orbital indices for 2nd site 2p-core ]) # copy umat_tmp_n (one site) to umat_n (two sites) ndp = ndorb + nporb for m in range(nsite): for i in range(ndp): for j in range(ndp): for k in range(ndp): for l in range(ndp): umat_n[indx[m, i], indx[m, j], indx[m, k], indx[m, l]] += umat_tmp_n[i, j, k, l] # two fermion terms, SOC, crystal field, and hopping between the two sites emat_i = np.zeros((ntot, ntot), dtype=np.complex) emat_n = np.zeros((ntot, ntot), dtype=np.complex) # SOC zeta_d_i = 0.35 soc_d = edrixs.atom_hsoc('t2g', zeta_d_i) emat_i[0:ndorb, 0:ndorb] += soc_d emat_i[ndorb:2 * ndorb, ndorb:2 * ndorb] += soc_d emat_n[0:ndorb, 0:ndorb] += soc_d emat_n[ndorb:2 * ndorb, ndorb:2 * ndorb] += soc_d # Terms from static core-hole potential for i in range(2 * ndorb): emat_n[i, i] -= nporb * static_v # Crystal field and hoppings between the two Ir-sites d = -0.03 # trgional splitting in t2g-subspace # Uncomment the following line to do calculation without hopping and crystal filed splitting. t1, t2 = -0.18, 0.036 # hopping between the two-sites in t2g-subspace cf_tmp = np.array([ # dzx_1, dzy_1, dxy_1, dzx_2, dzy_2, dxy_2 [0, d, d, t1, t2, t1], # dzx_1 [d, 0, d, t2, t1, t1], # dzy_1 [d, d, 0, t1, t1, t2], # dxy_1 [t1, t2, t1, 0, d, d], # dzx_2 [t2, t1, t1, d, 0, d], # dzy_2 [t1, t1, t2, d, d, 0], # dxy_2 ]) # Including spin degree of freedom, in global axis cf_spin = np.zeros((2 * ndorb, 2 * ndorb), dtype=np.complex) cf_spin[0:2 * ndorb:2, 0:2 * ndorb:2] = cf_tmp cf_spin[1:2 * ndorb:2, 1:2 * ndorb:2] = cf_tmp # Transform spin basis to local axis # 1/2-spinor matrix t_spinor = np.zeros((2 * ndorb, 2 * ndorb), dtype=np.complex128) for i in range(nsite): alpha, beta, gamma = edrixs.rmat_to_euler(locaxis[i]) dmat = edrixs.dmat_spinor(alpha, beta, gamma) for j in range(ndorb // 2): off = i * ndorb + j * 2 t_spinor[off:off + 2, off:off + 2] = dmat # Transform orbital basis from real cubic to complex harmonics t_orb = np.zeros((2 * ndorb, 2 * ndorb), dtype=np.complex128) t_orb[0:ndorb, 0:ndorb] = edrixs.tmat_r2c('t2g', True) t_orb[ndorb:2 * ndorb, ndorb:2 * ndorb] = edrixs.tmat_r2c('t2g', True) # Do the tranformation cf_spin[:, :] = edrixs.cb_op(cf_spin, np.dot(t_spinor, t_orb)) emat_i[0:2 * ndorb, 0:2 * ndorb] += cf_spin emat_n[0:2 * ndorb, 0:2 * ndorb] += cf_spin # Write emat and umat to files # ED inputs edrixs.write_emat(emat_i, "ed/hopping_i.in") edrixs.write_umat(umat_i, "ed/coulomb_i.in") # XAS inputs edrixs.write_emat(emat_n, "xas/hopping_n.in") edrixs.write_umat(umat_n, "xas/coulomb_n.in") # RIXS inputs edrixs.write_emat(emat_i, "rixs_pp/hopping_i.in") edrixs.write_umat(umat_i, "rixs_pp/coulomb_i.in") edrixs.write_emat(emat_n, "rixs_pp/hopping_n.in") edrixs.write_umat(umat_n, "rixs_pp/coulomb_n.in") edrixs.write_emat(emat_i, "rixs_ps/hopping_i.in") edrixs.write_umat(umat_i, "rixs_ps/coulomb_i.in") edrixs.write_emat(emat_n, "rixs_ps/hopping_n.in") edrixs.write_umat(umat_n, "rixs_ps/coulomb_n.in")
# # 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.