#!@TRIQS_PYTHON_EXECUTABLE@
import numpy as np
import collections
from itertools import product
'''
python functions for reading Uijkl from a VASP cRPA run and the evaluating the matrix
elements for different basis sets.
Copyright (C) 2020, A. Hampel and M. Merkel from Materials Theory Group
at ETH Zurich
'''
[docs]
def read_uijkl(path_to_uijkl, n_sites, n_orb):
'''
reads the VASP UIJKL files or the vijkl file if wanted
Parameters
----------
path_to_uijkl : string
path to Uijkl like file
n_sites: int
number of different atoms (Wannier centers)
n_orb : int
number of orbitals per atom
Returns
-------
uijkl : numpy array
uijkl Coulomb tensor
'''
dim = n_sites*n_orb
uijkl = np.zeros((dim, dim, dim, dim), dtype=complex)
data = np.loadtxt(path_to_uijkl)
for line in range(0, len(data[:, 0])):
i = int(data[line, 0])-1
j = int(data[line, 1])-1
k = int(data[line, 2])-1
l = int(data[line, 3])-1
uijkl[i, j, k, l] = data[line, 4]+1j*data[line, 5]
return uijkl
[docs]
def construct_U_kan(n_orb, U, J, Up=None, Jc=None):
'''
construct Kanamori Uijkl tensor for given U, J, Up, and Jc
Parameters
----------
n_orb : int
number of orbitals
U : float
U value for elements Uiiii
J : float
Hunds coupling J for tensor elements Uijji
Up : float, optional, default=U-2J
inter orbital exchange term Uijij
Jc : float, optional, default=J
Uiijj term, is the same as J for real valued wave functions
Returns
-------
uijkl : numpy array
uijkl Coulomb tensor
'''
orb_range = range(0, n_orb)
U_kan = np.zeros((n_orb, n_orb, n_orb, n_orb))
if not Up:
Up = U-2*J
if not Jc:
Jc = J
for i, j, k, l in product(orb_range, orb_range, orb_range, orb_range):
if i == j == k == l: # Uiiii
U_kan[i, j, k, l] = U
elif i == k and j == l: # Uijij
U_kan[i, j, k, l] = Up
elif i == l and j == k: # Uijji
U_kan[i, j, k, l] = J
elif i == j and k == l: # Uiijj
U_kan[i, j, k, l] = Jc
return U_kan
[docs]
def red_to_2ind(uijkl, n_sites, n_orb, out=False):
'''
reduces the 4index coulomb matrix to a 2index matrix and
follows the procedure given in PRB96 seth,peil,georges:
U_antipar = U_mm'^oo' = U_mm'mm' (Coulomb Int)
U_par = U_mm'^oo = U_mm'mm' - U_mm'm'm (for intersite interaction)
U_ijij (Hunds coupling)
the indices in VASP are switched: U_ijkl ---VASP--> U_ikjl
Parameters
----------
uijkl : numpy array
4d numpy array of Coulomb tensor
n_sites: int
number of different atoms (Wannier centers)
n_orb : int
number of orbitals per atom
out : bool
verbose mode
Returns
-------
Uij_anti : numpy array
red 2 index matrix U_mm'mm'
Uiijj : numpy array
red 2 index matrix U_iijj
Uijji : numpy array
red 2 index matrix Uijji
Uij_par : numpy array
red 2 index matrix U_mm\'mm\' - U_mm\'m\'m
'''
dim = n_sites*n_orb
# create 2 index matrix
Uij_anti = np.zeros((dim, dim))
Uij_par = np.zeros((dim, dim))
Uiijj = np.zeros((dim, dim))
Uijji = np.zeros((dim, dim))
for i in range(0, dim):
for j in range(0, dim):
# the indices in VASP are switched: U_ijkl ---VASP--> U_ikjl
Uij_anti[i, j] = uijkl[i, i, j, j]
Uij_par[i, j] = uijkl[i, i, j, j]-uijkl[i, j, j, i]
Uiijj[i, j] = uijkl[i, j, i, j]
Uijji[i, j] = uijkl[i, j, j, i]
np.set_printoptions(precision=3, suppress=True)
if out:
print('reduced U anti-parallel = U_mm\'\^oo\' = U_mm\'mm\' matrix : \n', Uij_anti)
print('reduced U parallel = U_mm\'\^oo = U_mm\'mm\' - U_mm\'m\'m matrix : \n', Uij_par)
print('reduced Uijji : \n', Uijji)
print('reduced Uiijj : \n', Uiijj)
return Uij_anti, Uiijj, Uijji, Uij_par
[docs]
def calc_kan_params(uijkl, n_sites, n_orb, out=False):
'''
calculates the kanamori interaction parameters from a
given Uijkl matrix. Follows the procedure given in
PHYSICAL REVIEW B 86, 165105 (2012) Vaugier,Biermann
formula 30,31,32
Parameters
----------
uijkl : numpy array
4d numpy array of Coulomb tensor
n_sites: int
number of different atoms (Wannier centers)
n_orb : int
number of orbitals per atom
out : bool
verbose mode
Returns
-------
int_params : direct
kanamori parameters
'''
int_params = collections.OrderedDict()
# calculate intra-orbital U
U = 0.0
for i in range(0, n_orb):
U += uijkl[i, i, i, i]
U = U/(n_orb)
int_params['U'] = U
# calculate the U'
Uprime = 0.0
for i in range(0, n_orb):
for j in range(0, n_orb):
if i != j:
Uprime += uijkl[i, i, j, j]
Uprime = Uprime / (n_orb*(n_orb-1))
int_params['Uprime'] = Uprime
# calculate J
J = 0.0
for i in range(0, n_orb):
for j in range(0, n_orb):
if i != j:
J += uijkl[i, j, i, j]
J = J / (n_orb*(n_orb-1))
int_params['J'] = J
if out:
print('U= ', "{:.4f}".format(U))
print('U\'= ', "{:.4f}".format(Uprime))
print('J= ', "{:.4f}".format(J))
return int_params
[docs]
def fit_kanamori(uijkl, n_orb, switch_jk=False, fit_2=True, fit_3=False, fit_4=True):
'''
Fit Kanamori Hamiltonian with scipy to 2,3, and / or 4 parameters
Parameters
-----------
uijkl: np.array (n_orb x n_orb x n_orb x n_orb)
input four index tensor
n_orb: int
number of orbitals
switch_jk: bool, default=False
flip two inner indices in input U tensor (for Vasp)
fit_2: bool, default=True
fit two parameter form
fit_3: bool, default=False
fit three parameter form (U,Up,J=Jc)
fit_4: bool, default=True
fit four parameter form
Returns
-------
Uijkl_fit: np.array (n_orb x n_orb x n_orb x n_orb)
fitted Uijkl tensor
'''
from scipy.optimize import minimize
def minimizer_2params(parameters):
U, J = parameters
Uijkl_fit = construct_U_kan(n_orb, U, J)
return np.sum((uijkl - Uijkl_fit)**2)
def minimizer_3params(parameters):
U, J, Up = parameters
Uijkl_fit = construct_U_kan(n_orb, U, J, Up)
return np.sum((uijkl - Uijkl_fit)**2)
def minimizer_4params(parameters):
U, J, Up, Jc = parameters
Uijkl_fit = construct_U_kan(n_orb, U, J, Up, Jc)
return np.sum((uijkl - Uijkl_fit)**2)
# check if J = JC (Hunds exchange and pair hopping have same amplitude)
# true for real values wave functions
if np.max(uijkl.imag) > 0.0:
print(f"Largest imaginary part of Uijkl: {np.max(uijkl.imag)}. Kanamori Hint assumed to be real valued. Neglecting imag part")
uijkl = uijkl.real
if switch_jk:
uijkl = np.moveaxis(uijkl, 1, 2)
# fit U, J
if fit_2:
initial_guess = (4, 1)
result = minimize(minimizer_2params, initial_guess)
U, J = result.x
Uijkl_fit = construct_U_kan(n_orb, U, J)
print('Result 2 parameter fit: \nU = {:.4f} eV, J = {:.4f} eV'.format(U, J))
print(f'optimize error {result.fun:.3e}')
max_ind = np.unravel_index(np.argmax(np.abs(Uijkl_fit-uijkl), axis=None), Uijkl_fit.shape)
print(f'U max diff: U{max_ind}= {np.abs(Uijkl_fit-uijkl)[max_ind]:.4e}')
print('\n-------------------------\n')
# fit U, J, Up
if fit_3:
initial_guess = (4, 1, 2)
result = minimize(minimizer_3params, initial_guess)
U, J, Up = result.x
Uijkl_fit = construct_U_kan(n_orb, U, J, Up)
print('Result 3 parameter fit: \nU = {:.4f} eV, U\' = {:.4f} eV J = {:.4f} eV'.format(U, Up, J))
print(f'optimize error {result.fun:.3e}')
max_ind = np.unravel_index(np.argmax(np.abs(Uijkl_fit-uijkl), axis=None), Uijkl_fit.shape)
print(f'U max diff: U{max_ind}= {np.abs(Uijkl_fit-uijkl)[max_ind]:.4e}')
print(f'U=U\'-2J deviation: {U-Up-2*J:.4f}')
print('\n-------------------------\n')
if fit_4:
# fit U, J, Up, Jc
initial_guess = (4, 1, 2, 1)
result = minimize(minimizer_4params, initial_guess)
U, J, Up, Jc = result.x
Uijkl_fit = construct_U_kan(n_orb, U, Up, J, Jc)
print('Result 4 parameter fit: \nU = {:.4f} eV, U\' = {:.4f} eV, J = {:.4f} eV, Jc = {:.4f} eV'.format(U, Up, J, Jc))
print(f'optimize error {result.fun:.3e}')
print(f'U max diff: U{max_ind}= {np.abs(Uijkl_fit-uijkl)[max_ind]:.4e}')
max_ind = np.unravel_index(np.argmax(np.abs(Uijkl_fit-uijkl), axis=None), Uijkl_fit.shape)
print(f'U=U\'-2J deviation: {U-Up-2*J:.4f}')
print(f'J=Jc deviation: {J-Jc:.4f}')
return Uijkl_fit
[docs]
def calc_u_avg_fulld(uijkl, n_sites, n_orb, out=False):
'''
calculates the coulomb integrals from a
given Uijkl matrix for full d shells. Follows the procedure given
in Pavarini - 2014 - arXiv - 1411 6906 - julich school U matrix
page 8 or as done in
PHYSICAL REVIEW B 86, 165105 (2012) Vaugier,Biermann
formula 23, 25
works atm only for full d shell (l=2)
Returns F0=U, and J=(F2+F4)/14
Parameters
----------
uijkl : numpy array
4d numpy array of Coulomb tensor
n_sites: int
number of different atoms (Wannier centers)
n_orb : int
number of orbitals per atom
out : bool
verbose mode
Returns
-------
int_params : direct
Slater parameters
'''
int_params = collections.OrderedDict()
Uij_anti, Uiijj, Uijji, Uij_par = red_to_2ind(uijkl, n_sites, n_orb, out=out)
# U_antipar = U_mm'^oo' = U_mm'mm' (Coulomb Int)
# U_par = U_mm'^oo = U_mm'mm' - U_mm'm'm (for intersite interaction)
# here we assume cubic harmonics (real harmonics) as basis functions in the order
# dz2 dxz dyz dx2-y2 dxy
# triqs basis: basis ordered as (xy,yz,z^2,xz,x^2-y^2)
# calculate J
J_cubic = 0.0
for i in range(0, n_orb):
for j in range(0, n_orb):
if i != j:
J_cubic += Uijji[i, j]
J_cubic = J_cubic/(20.0)
# 20 for 2l(2l+1)
int_params['J_cubic'] = J_cubic
# conversion from cubic to spherical:
J = 7.0 * J_cubic / 5.0
int_params['J'] = J
# calculate intra-orbital U
U_0 = 0.0
for i in range(0, n_orb):
U_0 += Uij_anti[i, i]
U_0 = U_0 / float(n_orb)
int_params['U_0'] = U_0
# now conversion from cubic to spherical
U = U_0 - (8.0*J_cubic/5.0)
int_params['U'] = U
if out:
print('cubic U_0= ', "{:.4f}".format(U_0))
print('cubic J_cubic= ', "{:.4f}".format(J_cubic))
print('spherical F0=U= ', "{:.4f}".format(U))
print('spherical J=(F2+f4)/14 = ', "{:.4f}".format(J))
return int_params
[docs]
def calculate_interaction_from_averaging(uijkl, n_sites, n_orb, out=False):
'''
calculates U,J by averaging directly the Uijkl matrix
ignoring if tensor is given in spherical or cubic basis.
The assumption here is that the averaging gives indepentendly
of the choosen basis (cubic or spherical harmonics) the same results
if Uijkl is a true Slater matrix.
Returns F0=U, and J=(F2+F4)/14
Parameters
----------
uijkl : numpy array
4d numpy array of Coulomb tensor
n_sites: int
number of different atoms (Wannier centers)
n_orb : int
number of orbitals per atom
out : bool
verbose mode
Returns
-------
U, J: tuple
Slater parameters
'''
l = 2
Uij_anti, Uiijj, Uijji, Uij_par = red_to_2ind(uijkl, n_sites, n_orb, out=out)
# Calculates Slater-averaged parameters directly
U = [None] * n_sites
J = [None] * n_sites
for impurity in range(n_sites):
u_ijij_imp = Uij_anti[impurity*n_orb:(impurity+1)*n_orb, impurity*n_orb:(impurity+1)*n_orb]
U[impurity] = np.mean(u_ijij_imp)
u_iijj_imp = Uiijj[impurity*n_orb:(impurity+1)*n_orb, impurity*n_orb:(impurity+1)*n_orb]
J[impurity] = np.sum(u_iijj_imp) / (2*l*(2*l+1)) - U[impurity] / (2*l)
U = np.mean(U)
J = np.mean(J)
if out:
print('spherical F0=U= ', "{:.4f}".format(U))
print('spherical J=(F2+f4)/14 = ', "{:.4f}".format(J))
return U, J
[docs]
def fit_slater_fulld(uijkl, n_sites, U_init, J_init, fixed_F4_F2=True):
'''
finds best Slater parameters U, J for given Uijkl tensor
using the triqs U_matrix operator routine
assumes F4/F2=0.625
'''
from triqs.operators.util.U_matrix import U_matrix_slater, reduce_4index_to_2index
from scipy.optimize import minimize
# transform U matrix orbital basis ijkl to nmop, note the last two indices need to be switched in the T matrices
def transformU(U_matrix, T):
return np.einsum("im,jn,ijkl,lo,kp->mnpo", np.conj(T), np.conj(T), U_matrix, T, T)
def minimizer(parameters):
U_int, J_hund = parameters
Umat_full = U_matrix_slater(l=2, U_int=U_int, J_hund=J_hund, basis='cubic')
Umat_full = transformU(Umat_full, rot_def_to_w90)
Umat, Upmat = reduce_4index_to_2index(Umat_full)
u_iijj_crpa = Uiijj[:5, :5]
u_iijj_slater = Upmat - Umat
u_ijij_crpa = Uij_anti[:5, :5]
u_ijij_slater = Upmat
return np.sum((u_iijj_crpa - u_iijj_slater)**2 + (u_ijij_crpa - u_ijij_slater)**2)
def minimizer_radial(parameters):
F0, F2, F4 = parameters
Umat_full = U_matrix_slater(l=2, radial_integrals=[F0, F2, F4], basis='cubic')
Umat_full = transformU(Umat_full, rot_def_to_w90)
Umat, Upmat = reduce_4index_to_2index(Umat_full)
u_iijj_crpa = Uiijj[:5, :5]
u_iijj_slater = Upmat - Umat
u_ijij_crpa = Uij_anti[:5, :5]
u_ijij_slater = Upmat
return np.sum((u_iijj_crpa - u_iijj_slater)**2 + (u_ijij_crpa - u_ijij_slater)**2)
# rot triqs d basis to w90 default basis!
# check your order of orbitals assuming:
# dz2, dxz, dyz, dx2-y2, dxy
rot_def_to_w90 = np.array([[0, 0, 0, 0, 1],
[0, 0, 1, 0, 0],
[1, 0, 0, 0, 0],
[0, 1, 0, 0, 0],
[0, 0, 0, 1, 0]])
Uij_anti, Uiijj, Uijji, Uij_par = red_to_2ind(uijkl, n_sites, n_orb=5, out=False)
if fixed_F4_F2:
result = minimize(minimizer, (U_init, J_init))
U_int, J_hund = result.x
print('Final results from fit: U = {:.3f} eV, J = {:.3f} eV'.format(U_int, J_hund))
print('optimize error', result.fun)
else:
# start with 0.63 as guess
F0 = U_init
F2 = J_init * 14.0 / (1.0 + 0.63)
F4 = 0.630 * F2
initial_guess = (F0, F2, F4)
print('Initial guess: F0 = {0[0]:.3f} eV, F2 = {0[1]:.3f} eV, F4 = {0[2]:.3f} eV'.format(initial_guess))
result = minimize(minimizer_radial, initial_guess)
F0, F2, F4 = result.x
print('Final results from fit: F0 = {:.3f} eV, F2 = {:.3f} eV, F4 = {:.3f} eV'.format(F0, F2, F4))
print('(F2+F4)/14 = {:.3f} eV'.format((F2+F4)/14))
print('F4/F2 = {:.3f} eV'.format(F4/F2))
print('optimize error', result.fun)
U_int = F0
J_hund = (F2+F4)/14
return U_int, J_hund
# example for a five orbital model
# uijkl=read_uijkl('UIJKL',1,5)
# calc_u_avg_fulld(uijkl,n_sites=1,n_orb=5,out=True)
# print('now via fitting with fixed F4/F2=0.63 ratio')
# fit_slater_fulld(uijkl,1,3,1, fixed_F4_F2 = True)
# print('now directly fitting F0,F2,F4')
# fit_slater_fulld(uijkl,1,3,1, fixed_F4_F2 = False)
# calculate_interaction_from_averaging(uijkl, 1, 5, out=True)
# example for 3 orbital kanamori
# uijkl=read_uijkl('UIJKL',1,3)
# calc_kan_params(uijkl,1,3,out=True)