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```'''
Created on 16 Apr 2013

@author: bakera

'''

import unittest

import numpy as np
from math import pow
from scipy import special
from scipy.linalg import eig

class HartreeFock(object):
'''
simple HF SCF algorithm
'''

def iterate_basis_functions(self,j=0):
for i in np.arange(j,self.basis_function_count,1):
yield i

def iterate_primitives_functions(self):
for i in np.arange(0,self.primitive_count,1):
yield i

def F0(self, t):
'''
error function
'''
if t < 1e-6 :
return 1.0-t/3
else:
return (1.0/2.0)*np.sqrt(np.pi/t)*special.erf(np.sqrt(t))

def __init__(self, R=1.4632):

# counters, 2 is basis functions
self.basis_function_count = 2
# made from each , 3 contacted gaussian functions
self.primitive_count = 3

#
# gaussian a.e^-((x-b)^2)/c), a height, b center, c std
#

# zeta is know as the width of the function, or c?

# 3 primitive gaussians, two fitted values exponent d and contraction coefficient
# a
self.d = np.array([0.444635, 0.535328, 0.1543329])
self.a = np.array([0.109818, 0.405771, 2.22766])

self.zeta_he = 2.0925
self.zeta_h = 1.24

self.a_he = pow(self.zeta_he,2.0)*self.a
self.a_h = pow(self.zeta_h,2.0)*self.a

self.R = R
self.a_prime = np.array([self.a_he, self.a_h]).ravel()
self.r_prime = np.array([0,0,0,self.R,self.R,self.R])

# nuclear charge he, h
self.Z = np.array([2.0,1.0])

self.S = np.zeros((self.basis_function_count,self.basis_function_count), dtype=float)
self.T = np.zeros((self.basis_function_count,self.basis_function_count), dtype=float)
self.V1 = np.zeros((self.basis_function_count,self.basis_function_count), dtype=float)
self.V2 = np.zeros((self.basis_function_count,self.basis_function_count), dtype=float)
self.H = np.zeros((self.basis_function_count,self.basis_function_count), dtype=float)

self.I = np.zeros((self.basis_function_count,self.basis_function_count,self.basis_function_count,self.basis_function_count), dtype=float)

# convergence criteria
self.dP = 1e-4
self.maximum_iterations = 20

def generate(self,i,j,p,q):
a1 = self.a_prime[self.primitive_count*(i)+p]
a2 = self.a_prime[self.primitive_count*(j)+q]
r1 = self.r_prime[self.primitive_count*(i)+p]
r2 = self.r_prime[self.primitive_count*(j)+q]
return a1, a2, r1, r2

def product_two_gaussians(self, a1, a2, r1, r2):
'''
a1 : float

denotes the contraction coefficient of each gaussian or exponent.

r1 : float

denotes the center location of one of the gaussians

r2 : float

denotes the ceter location of second gaussion

return :

as : float

sum of the two centers

ap : float

product of the two centers

disp : float

displacement of two gaussian centers
'''
asum = a1+a2
aproduct = a1*a2
dist = r1-r2
rp = (a1*r1+a2*r2)/(a1+a2)
return asum, aproduct, dist, rp

def orthogonalize(self):
'''

'''
x = np.zeros((2,2), dtype=float)
s_invroot = np.zeros((2,2), dtype=float)

s,U = eig(self.S)
U = np.matrix([[2**-0.5,2**-0.5],[2**-0.5,-1*2**-0.5]])

#for i in self.iterate_basis_functions():
#    s_invroot[i,i] = pow(s[i]+0j,-0.5)

s_invroot[0,0] = pow(s[1]+0j,-0.5)
s_invroot[1,1] = pow(s[0]+0j,-0.5)

#print 's_invroot', s_invroot
x = U*s_invroot

return U,s,x

def scf(self):

U,s,X = self.orthogonalize()
#print 'U:',U
#print 's',s
#print 'X',X

P = np.zeros((2,2), dtype=float)

#
# enter the SCF loop
#

F = np.zeros((2,2), dtype=float)
count = 0
sigma = 1

while sigma > self.dP:

P_prev = P
count += 1

#
# build G matrix
#
G = np.zeros((2,2), dtype=float)
for i in self.iterate_basis_functions():
for j in self.iterate_basis_functions():
for p in self.iterate_basis_functions():
for q in self.iterate_basis_functions():
G[i,j] += P[p,q]*(self.I[i,q,j,p]-0.5*self.I[i,q,p,j])

F = np.matrix(self.H + G)

#print 'Fock Matrix',F

Fp = np.matrix(X.T)*np.matrix(F)*np.matrix(X)

#print 'F prime', Fp

eps, Cp = eig(Fp)
C = np.matrix(X)*np.matrix(Cp)

P = np.zeros((2,2), dtype=float)
for i in self.iterate_basis_functions():
for j in self.iterate_basis_functions():
P[i,j] += 2*C[i,0]*C[j,0]

sigma = 0
for i in self.iterate_basis_functions():
for j in self.iterate_basis_functions():
sigma += (P[i,j]-P_prev[i,j])**2

sigma = ((1.0/4.0)*sigma)**(0.5)

if sigma >= self.maximum_iterations:
sigma = 0

E = 0
for i in self.iterate_basis_functions():
for j in self.iterate_basis_functions():
E += 0.5*P_prev[i,j]*(self.H[j,i]+F[j,i])

Etot = E + max(np.cumprod(self.Z))/self.R
yield count, P, E, Etot, F, Fp, P*np.matrix(self.S)

def compute_integrals(self):

#
#    simple method to evaluate the overlap matrix
#

#
# iterate the basis functions
#

for i in self.iterate_basis_functions():
for j in self.iterate_basis_functions(i):
for p in self.iterate_primitives_functions():
for q in self.iterate_primitives_functions():

a1, a2, r1, r2 = self.generate(i, j, p, q)
asum, aproduct, dist, rp = self.product_two_gaussians(a1, a2, r1, r2)
rat = (aproduct/asum)

#print 'asum, aproduct, dist, rp,rat ', asum, aproduct, dist, rp , rat

#
# overlap matrix S[i,j]
#
s_pq = pow(2,1.5)*pow(rat/asum, 0.75)*pow(np.e, (-1.0*rat*pow(dist,2.0)))
self.S[i,j] += self.d[p]*self.d[q]* s_pq

#
# kinetic energy matrix T[i,j]
#
self.T[i,j] += self.d[p] * self.d[q] * rat * (self.primitive_count-2*rat*pow(dist,2))*s_pq

#
# Nuclear potential of He
#
self.V1[i,j] += self.d[p] * self.d[q] * pow((2.0/np.pi), 3.0/2.0)*pow((aproduct), 3.0/4.0)*-2.0*(np.pi/asum)*self.Z[0]*pow(np.e, -1*(aproduct/asum)*dist**2)*self.F0(asum*rp**2)

#
# Nuclear potentail of H
#
self.V2[i,j] += self.d[p] * self.d[q] * pow((2.0/np.pi), 3.0/2.0)*pow((aproduct), 3.0/4.0)*-2.0*(np.pi/asum)*self.Z[1]*pow(np.e, -1*(aproduct/asum)*dist**2)*self.F0(asum*(rp-self.R)**2)

# reflection in the trace diagonal
self.S[j,i] = self.S[i,j]
self.T[j,i] = self.T[i,j]
self.V1[j,i] = self.V1[i,j]
self.V2[j,i] = self.V2[i,j]

#4 build the Hamiltonian
self.H = self.T + self.V1 + self.V2

#
# calculate the 2 electron integrals
#

for i in self.iterate_basis_functions():
for j in self.iterate_basis_functions(i):
for k in self.iterate_basis_functions(i):
for l in self.iterate_basis_functions(k):

self.I[i,j,k,l] = 0.0

for p in self.iterate_primitives_functions():
for q in self.iterate_primitives_functions():
for r in self.iterate_primitives_functions():
for s in self.iterate_primitives_functions():

a1, a2, r1, r2 = self.generate(i, j, p, q)
a3, a4, r3, r4 = self.generate(k, l, r, s)

asum1 = a1+a3
asum2 = a2+a4
asum = asum1 + asum2
aproduct = a1 * a2 * a3 * a4
rat1 = (a1*a3)/asum1
rat2 = (a2*a4)/asum2
rp = (a1*r1 + a3*r3)/asum1
rq = (a2*r2+a4*r4)/asum2

dist1 = r1-r3
dist2 = r2-r4
dist = rp-rq

#print i,j,k,l, p,q,r,s , a1, a2, r1, r2, a3, a4, r3, r4

self.I[i,j,k,l] +=  self.d[p]*self.d[q]*self.d[s]*self.d[r] * 16.0/(np.sqrt(np.pi))*(pow(aproduct,3.0/4.0)/(asum1*asum2*np.sqrt(asum)))*pow(np.e,-1*rat1*dist1**2-rat2*dist2**2)*self.F0(asum1*asum2/asum*dist**2)

#
# figure out why this set of parameters...todo
#

self.I[i,l,k,j] = self.I[i,j,k,l]
self.I[j,i,l,k] = self.I[i,j,k,l]
self.I[j,k,l,i] = self.I[i,j,k,l]
self.I[k,j,i,l] = self.I[i,j,k,l]
self.I[k,l,i,j] = self.I[i,j,k,l]
self.I[l,i,j,k] = self.I[i,j,k,l]
self.I[l,k,j,i] = self.I[i,j,k,l]

class Test(unittest.TestCase):

def testOverlapMatrix(self):
'''
Longer bond length R=2.5, S(1,2) = S(2,1) gets smaller
less chance of electrons being between atoms
'''
vv = HartreeFock()
vv.compute_integrals()
print 'Overlap matrix S', np.round(vv.S,4)

S = np.array([[1.00000615,0.4507713],[0.4507713, 1.00000615]])
assert(np.round(vv.S,4) == np.round(S,4)).all()

def testKineticEnergyMatrix(self):
vv = HartreeFock()
vv.compute_integrals()
print 'Kinertic Energy matrix T', np.round(vv.T,4)

T = np.array([[2.16434304,0.16701271],[0.16701271, 0.76004365]])
assert(np.round(vv.T,4) == np.round(T,4)).all()

def testNuclearPotential(self):
vv = HartreeFock()
vv.compute_integrals()

print 'Nuclear Energy matrix (He) T', np.round(vv.V1,4)
print 'Nuclear Energy matrix (H) T', np.round(vv.V2,4)
print 'Hamiltonian(core) of single electron in field of the nuclear point charges', np.round(vv.H,4)

print '2 electron integrals ', vv.I

for count, P, E, F, Fp, Etot, PS in vv.scf():
print 'iteration', count, 'Density Matrix', P[0,0], P[0,1], P[1,1], 'Energy',E, Etot, PS

test_results = {}

for r in np.arange(0.1,3.5,0.01):
vv = HartreeFock(R=r)
vv.compute_integrals()
test_results[r] = max([(count, Etot) for count, P, E, Etot, F, Fp, PS in vv.scf()])

radius = [k for k,v in test_results]
energy = [v[1] for k,v in test_results]

#
# visualise data, comment back in.
#

#from pandas import DataFrame, Series
#from pylab import show
#d = {'HeH' : Series(energy, index=radius)}
#df = DataFrame(d)
#df.plot(style='r-+')
#df.to_csv('hehplus.csv')
#show()

def testHelperFunctionErf(self):
print round(HelperFunctions.F0(t=0.01),2) == 0.75

if __name__ == "__main__":
#import sys;sys.argv = ['', 'Test.testName']
unittest.main()
```

#### Diff to Previous Revision

```--- revision 3 2013-04-30 09:08:39
+++ revision 4 2013-04-30 09:09:20
@@ -15,8 +15,7 @@

class HartreeFock(object):
'''
-        simple route to match some numbers before we split and move this into Fortran
-        for vector evaluation across the CPU. Or even parrallel version across picloud.
+       simple HF SCF algorithm
'''

def iterate_basis_functions(self,j=0):
```