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
get the matrix or 4-rank tensor in the single-particle basis, and then call build_opers() to get the many-body operators. """ # 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)
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")
_ = ax.set_xticks([]) ################################################################################ # 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=1` 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):
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
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[m, 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((2, norbs, norbs), dtype=np.complex128) soc_d = edrixs.cb_op(edrixs.atom_hsoc('t2g', zeta_d_i), edrixs.tmat_c2r('t2g', True)) soc_p = edrixs.cb_op(edrixs.atom_hsoc('p', zeta_p_n), edrixs.tmat_c2r('p', True)) emat_i[0:6, 0:6] += soc_d emat_i[6:12, 6:12] += soc_d emat_n[0, 0:6, 0:6] += soc_d emat_n[0, 6:12, 6:12] += soc_d emat_n[0, 12:18, 12:18] += soc_p emat_n[1, 0:6, 0:6] += soc_d emat_n[1, 6:12, 6:12] += soc_d emat_n[1, 18:24, 18:24] += soc_p # crystal field and hopping between the two Ir atom a, b, c = -0.18, 0.036, -0.03
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 edrixs.write_umat(umat_i, 'coulomb_i.in', 1E-10) # write fock basis of decimal format to file 'fock_i.in', read by ed.x edrixs.write_fock_dec_by_N(norbs, noccu, "fock_i.in")
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")