def norm_slater_cartesian(a, b, c, n, exp):
"""Normaliation of STos with cartesian harmonics. \n
* Monte Carlo Methods in Ab Initio Quantum Chemistry page 279
Args:
a (torch.tensor): exponent of x
b (torch.tensor): exponent of y
c (torch.tensor): exponent of z
n (torch.tensor): exponent of r
exp (torch.tensor): Sater exponent
Returns:
torch.tensor: normalization factor
"""
from scipy.special import factorial2 as f2
lvals = a + b + c + n + 1.
lfact = torch.tensor([np.math.factorial(2 * i)
for i in lvals]).type(torch.get_default_dtype())
prefact = 4 * np.pi * lfact / ((2 * exp)**(2 * lvals + 1))
num = torch.tensor(
f2(2 * a.astype('int') - 1) * f2(2 * b.astype('int') - 1) *
f2(2 * c.astype('int') - 1)).type(torch.get_default_dtype())
denom = torch.tensor(f2((2 * a + 2 * b + 2 * c + 1).astype('int'))).type(
torch.get_default_dtype())
return torch.sqrt(1. / (prefact * num / denom))
def norm_gaussian_cartesian(a, b, c, exp):
"""Normaliation of GTOs with cartesian harmonics. \n
* Monte Carlo Methods in Ab Initio Quantum Chemistry page 279
Args:
a (torch.tensor): exponent of x
b (torch.tensor): exponent of y
c (torch.tensor): exponent of z
exp (torch.tensor): Sater exponent
Returns:
torch.tensor: normalization factor
"""
from scipy.special import factorial2 as f2
pref = torch.tensor((2 * exp / np.pi)**(0.75))
am1 = (2 * a - 1).astype('int')
x = (4 * exp)**(a / 2) / torch.sqrt(torch.tensor(f2(am1)))
bm1 = (2 * b - 1).astype('int')
y = (4 * exp)**(b / 2) / torch.sqrt(torch.tensor(f2(bm1)))
cm1 = (2 * c - 1).astype('int')
z = (4 * exp)**(c / 2) / torch.sqrt(torch.tensor(f2(cm1)))
return (pref * x * y * z).type(torch.get_default_dtype())
def _norm_gaussian_cart(self):
'''Normlization of cartesian gaussian functions
taken from http://www.chem.unifr.ch/cd/lectures/files/module5.pdf
'''
from scipy.special import factorial2 as f2
L = self.lmn_cart.sum(1)
num = 2**L * self.bas_coeffs**((2 * L + 3) / 4)
denom = torch.sqrt(f2(2 * self.lmn - 1).prod(1))
return (2. / np.pi)**(3. / 4.) * num / denom
def norm_gaussian_cartesian(a, b, c, exp):
"""normaliation of cartesian gaussian
Monte Carlo Methods in Ab Initio Quantum Chemistry page 280
Arguments:
a {[type]} -- exponent of x
b {[type]} -- exponent of y
c {[type]} -- exponent of z
exp {[type]} -- coefficient of the expo
"""
from scipy.special import factorial2 as f2
pref = torch.tensor((2 * exp / np.pi)**(0.75))
x = (4 * exp)**(a / 2) / torch.sqrt(
torch.tensor(f2(
(2 * a - 1).astype('int')))).type(torch.get_default_dtype())
y = (4 * exp)**(b / 2) / torch.sqrt(
torch.tensor(f2(
(2 * b - 1).astype('int')))).type(torch.get_default_dtype())
z = (4 * exp)**(c / 2) / torch.sqrt(
torch.tensor(f2(
(2 * c - 1).astype('int')))).type(torch.get_default_dtype())
return pref * x * y * z
def norm_slater_cartesian(a, b, c, n, exp):
"""normaliation of cartesian slater
Monte Carlo Methods in Ab Initio Quantum Chemistry page 279
Arguments:
a {[type]} -- exponent of x
b {[type]} -- exponent of y
c {[type]} -- exponent of z
n {[type]} -- exponent of r
exp {[type]} -- coefficient of the expo
"""
from scipy.special import factorial2 as f2
l = a + b + c + n + 1.
lfact = torch.tensor([np.math.factorial(2 * i)
for i in l]).type(torch.get_default_dtype())
prefact = 4 * np.pi * lfact / ((2 * exp)**(2 * l + 1))
num = torch.tensor(
f2(2 * a.astype('int') - 1) * f2(2 * b.astype('int') - 1) *
f2(2 * c.astype('int') - 1)).type(torch.get_default_dtype())
denom = torch.tensor(f2((2 * a + 2 * b + 2 * c + 1).astype('int'))).type(
torch.get_default_dtype())
return torch.sqrt(1. / (prefact * num / denom))
def norm_gaussian_spherical(bas_n, bas_exp):
""" Normlization of GTOs.
[1] Computational Quantum Chemistry: An interactive Intrduction to basis set theory
eq : 1.14 page 23.'''
Returns:
torch.tensor -- normalization factor
"""
from scipy.special import factorial2 as f2
bas_n = bas_n + 1.
exp1 = 0.25 * (2. * bas_n + 1.)
A = bas_exp**exp1
B = 2**(2. * bas_n + 3. / 2)
C = torch.tensor(f2(2 * bas_n.int() - 1) * np.pi**0.5).type(
torch.get_default_dtype())
return torch.sqrt(B / C) * A
def norm_gaussian_spherical(bas_n, bas_exp):
"""Normlization of GTOs with spherical harmonics. \n
* Computational Quantum Chemistry: An interactive Intrduction to basis set theory \n
eq : 1.14 page 23.
Args:
bas_n (torch.tensor): prinicpal quantum number
bas_exp (torch.tensor): slater exponents
Returns:
torch.tensor: normalization factor
"""
from scipy.special import factorial2 as f2
bas_n = bas_n + 1.
exp1 = 0.25 * (2. * bas_n + 1.)
A = bas_exp**exp1
B = 2**(2. * bas_n + 3. / 2)
C = torch.tensor(f2(2 * bas_n.int() - 1) * np.pi**0.5).type(
torch.get_default_dtype())
return torch.sqrt(B / C) * A