Quantum Chemistry technique to calculate various interesting operators for HeH.
Python, 379 lines
Download
Copy to clipboard1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 | '''
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
def testRadiusVary(self):
test_results = {}
for r in np.arange(0.1,3.5,0.01):
print 'radius :',r
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]
print radius, energy
#
# 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()
|
