''' 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()