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from math import *
import numpy as np
import numpy.fft
import numpy.linalg
#############################
##### LDA XC potentials #####
#############################
# Standard X potential of homogeneous electron gas (C is neglected)
def get_x_potential_lda_slater(s, rho):
return -np.cbrt((3 / pi) * rho)
# XC potential used for the GTH and HGH pseudopotentials.
# See https://arxiv.org/abs/mtrl-th/9512004
def get_xc_potential_lda_teter93(s, rho):
a0 = 0.4581652932831429
a1 = 2.217058676663745
a2 = 0.7405551735357053
a3 = 0.01968227878617998
b1 = 1.0
b2 = 4.504130959426697
b3 = 1.110667363742916
b4 = 0.02359291751427506
rs = np.cbrt(3 / (4 * pi * rho))
# Energy density (epsilon) as a function of rs
eps_num = a0 + rs * (a1 + rs * (a2 + a3 * rs))
eps_den = rs * (b1 + rs * (b2 + rs * (b3 + rs * b4)))
eps = -eps_num / eps_den
# Derivative of energy density with respect to rs
deps_num = a0 * (b1 + rs * (2*b2 + 3*b3*rs + 4*b4*rs**2))
deps_num += rs**3 * (2*a1*b3 + 3*a1*b4*rs - 2*a3*b1 - a3*b2*rs + a3*b4*rs**3)
deps_num += rs**2 * (a1*b2 - a2*b1 + a2*rs**2 * (b3 + 2*b4*rs))
deps = deps_num / eps_den**2
# Chain rule: derivative of rs with respect to rho
chain = - 1 / np.cbrt(36 * sqrt(pi) * rho**4)
return eps + rho * deps * chain
#############################
##### GGA XC potentials #####
#############################
# Because our grid usually has a non-orthogonal rectilinear basis,
# we first need some helper functions to calculate derivatives.
# Concept from tensor calculus, we'll need this later
def inverse_metric_tensor(s):
basis = s.crystal.a / s.Nres
g = np.zeros((3, 3))
for i in range(3):
for j in range(3):
g[i, j] = np.dot(basis[i], basis[j])
# Never fails if basis is linearly independent?
ginv = np.linalg.inv(g)
return ginv
# Compute Laplacian of f on a grid with non-orthogonal basis vectors.
# Shamelessly taken from https://math.stackexchange.com/questions/2632598/
def laplacian(s, f):
g = inverse_metric_tensor(s)
lf = np.zeros_like(f)
for i in range(s.Nres):
for j in range(s.Nres):
for k in range(s.Nres):
# Wrap indices at the boundaries
inext = (i + 1) % s.Nres
jnext = (j + 1) % s.Nres
knext = (k + 1) % s.Nres
# Calculate a lot of finite differences
lf[i, j, k] += -2 * (g[0, 0] + g[1, 1] + g[2, 2]) * f[i, j, k]
lf[i, j, k] += g[0, 0] * (f[i - 1, j, k] + f[inext, j, k])
lf[i, j, k] += g[1, 1] * (f[i, j - 1, k] + f[i, jnext, k])
lf[i, j, k] += g[2, 2] * (f[i, j, k - 1] + f[i, j, knext])
lf[i, j, k] += 0.5 * g[0, 1] * (f[i - 1, j - 1, k] - f[i - 1, jnext, k])
lf[i, j, k] += 0.5 * g[0, 1] * (f[inext, jnext, k] - f[inext, j - 1, k])
lf[i, j, k] += 0.5 * g[0, 2] * (f[i - 1, j, k - 1] - f[i - 1, j, knext])
lf[i, j, k] += 0.5 * g[0, 2] * (f[inext, j, knext] - f[inext, j, k - 1])
lf[i, j, k] += 0.5 * g[1, 2] * (f[i, j - 1, k - 1] - f[i, j - 1, knext])
lf[i, j, k] += 0.5 * g[1, 2] * (f[i, jnext, knext] - f[i, jnext, k - 1])
return lf
# Compute gradient of f on a grid with non-orthogonal basis vectors
def gradient(s, f):
basis = s.crystal.a / s.Nres
g = inverse_metric_tensor(s)
dfdx = np.zeros_like(f)
dfdy = np.zeros_like(f)
dfdz = np.zeros_like(f)
for i in range(s.Nres):
for j in range(s.Nres):
for k in range(s.Nres):
# Technically, we have periodic boundary conditions, but
# in practice we should avoid crossing the boundaries,
# hence the following verbose derivative calculations.
# Derivative along direction indexed by i
if i == 0:
df0 = 1.0 * (f[i + 1, j, k] - f[i, j, k])
elif i == s.Nres - 1:
df0 = 1.0 * (f[i, j, k] - f[i - 1, j, k])
else:
df0 = 0.5 * (f[i + 1, j, k] - f[i - 1, j, k])
# Derivative along direction indexed by j
if j == 0:
df1 = 1.0 * (f[i, j + 1, k] - f[i, j, k])
elif j == s.Nres - 1:
df1 = 1.0 * (f[i, j, k] - f[i, j - 1, k])
else:
df1 = 0.5 * (f[i, j + 1, k] - f[i, j - 1, k])
# Derivative along direction indexed by k
if k == 0:
df2 = 1.0 * (f[i, j, k + 1] - f[i, j, k])
elif k == s.Nres - 1:
df2 = 1.0 * (f[i, j, k] - f[i, j, k - 1])
else:
df2 = 0.5 * (f[i, j, k + 1] - f[i, j, k - 1])
# Get the local gradient at [i, j, k]
df = [df0, df1, df2]
gradf = np.zeros(3, dtype=f.dtype)
for l in range(3):
for m in range(3):
gradf += g[l, m] * df[l] * basis[m]
dfdx[i, j, k] = gradf[0]
dfdy[i, j, k] = gradf[1]
dfdz[i, j, k] = gradf[2]
return dfdx, dfdy, dfdz
# C potential by Lee, Yang and Parr, very popular for DFT.
# See Eqs.(23)&(24) in https://doi.org/10.1103/PhysRevB.37.785
def get_c_potential_gga_lyp(s, rho):
lr = laplacian(s, rho)
drdx, drdy, drdz = gradient(s, rho)
gr2 = drdx**2 + drdy**2 + drdz**2
a = 0.04918
b = 0.132
c = 0.2533
d = 0.349
CF = 0.3 * (3 * pi**2)**(2/3)
# Abbreviations
r = rho
rr = r**(1/3)
er = np.exp(-c / rr)
F1 = 1 / (d/rr + 1)
dF1 = d / (3*rr**2 * (d + rr)**2)
ddF1 = 2*d**2 / (9*rr**8 * (d/rr + 1)**3) - 4*d / (9*rr**7 * (d/rr + 1)**2)
G1 = F1 * er / rr**5
dG1 = (F1 * (c - 5*rr) + 3*dF1*rr**4) * er / (3*r**3)
ddG1 = (F1 * (c - 10*rr) * (c - 4*rr) + 6*dF1*rr**4 * (c - 5*rr) + 9*ddF1*rr**8) * er / (9*rr**13)
V = -a * (dF1*r + F1)
V += -a*b*CF * rr**5 * (dG1*r + (8/3)*G1)
V += -(1/ 4)*a*b * ( ddG1*r*gr2 + dG1 * (3*gr2 + 2*r*lr) + 4*G1*lr)
V += -(1/72)*a*b * (3*ddG1*r*gr2 + dG1 * (5*gr2 + 6*r*lr) + 4*G1*lr)
return V
# X potential by Van Leeuwen and Baerends, inspired by Becke88.
# See Eqs.(50)&(54) in https://doi.org/10.1103/PhysRevA.49.2421
def get_x_potential_gga_lb(s, rho):
drdx, drdy, drdz = gradient(s, rho)
x = np.sqrt(drdx**2 + drdy**2 + drdz**2) / rho**(4/3)
beta = 0.05
fx = - beta * x**2 / (1 + 3 * beta * x * np.arcsinh(x))
V = get_x_potential_lda_slater(s, rho)
return V + fx * rho**(1/3)
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