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.
