import numpy as np
[docs]def S(i,j,A,B,a,b):
''' Recursive definition of Hermite Gaussian coefficients.
Returns a float.
Parameters
----------
a :
orbital exponent on Gaussian 'a' (e.g. alpha in the text)
b :
orbital exponent on Gaussian 'b' (e.g. beta in the text)
i,j:
orbital angular momentum number on Gaussian 'a' and 'b'
t:
number nodes in Hermite (depends on type of integral,
e.g. always zero for overlap integrals)
Qx: distance between origins of Gaussian 'a' and 'b'
'''
p = a + b
q = (a*b)/p
P = (1./p)*(a*A + b*B)
PA = P-A
PB = P-B
AB = A-B
if i == j == 0:
# base case
return np.exp(-q*AB*AB) # K_AB
elif i<0 or j<0 :
return 0.
elif i == 0 and j>0:
# decrement index j
return PB*S(i,j-1,A,B,a,b) +\
1./(2*p)*i*S(i-1,j-1,A,B,a,b) +\
1./(2*p)*(j-1)*S(i,j-2,A,B,a,b)
else:
# decrement index i
return PA*S(i-1,j,A,B,a,b) +\
1./(2*p)*(i-1)*S(i-2,j,A,B,a,b) +\
1./(2*p)*j*S(i-1,j-1,A,B,a,b)
[docs]def Sxyz(a,lmn1,A,b,lmn2,B):
''' Evaluates overlap integral between two Gaussians
Returns a float.
Parameters
----------
a: orbital exponent on Gaussian 'a' (e.g. alpha in the text)
b: orbital exponent on Gaussian 'b' (e.g. beta in the text)
lmn1: int tuple containing orbital angular momentum (e.g. (1,0,0))
for Gaussian 'a'
lmn2: int tuple containing orbital angular momentum for Gaussian 'b'
A: list containing origin of Gaussian 'a', e.g. [1.0, 2.0, 0.0]
B: list containing origin of Gaussian 'b'
'''
i,k,m = lmn1 # shell angular momentum on Gaussian 'a'
j,l,n = lmn2 # shell angular momentum on Gaussian 'b'
Sx = S(i,j,A[0],B[0],a,b) # X
Sy = S(k,l,A[1],B[1],a,b) # Y
Sz = S(m,n,A[2],B[2],a,b) # Z
return Sx*Sy*Sz*np.power(np.pi/(a+b),1.5)
[docs]def overlap(a,b):
'''Evaluates overlap between two contracted Gaussians
Returns float.
Arguments:
a: contracted Gaussian 'a', BasisFunction object
b: contracted Gaussian 'b', BasisFunction object
'''
s = 0.0
for ia, ca in enumerate(a.coefs):
for ib, cb in enumerate(b.coefs):
s += a.norm[ia]*b.norm[ib]*ca*cb*\
Sxyz(a.exps[ia],a.shell,a.origin,
b.exps[ib],b.shell,b.origin)
return s