def nuclear_attraction(alpha1,lmn1,A,alpha2,lmn2,B,C):
"""
Full form of the nuclear attraction integral
>>> isnear(nuclear_attraction(1,(0,0,0),array((0,0,0),'d'),1,(0,0,0),array((0,0,0),'d'),array((0,0,0),'d')),-3.141593)
True
"""
l1,m1,n1 = lmn1
l2,m2,n2 = lmn2
gamma = alpha1+alpha2
P = gaussian_product_center(alpha1,A,alpha2,B)
rab2 = norm2(A-B)
rcp2 = norm2(C-P)
dPA = P-A
dPB = P-B
dPC = P-C
Ax = A_array(l1,l2,dPA[0],dPB[0],dPC[0],gamma)
Ay = A_array(m1,m2,dPA[1],dPB[1],dPC[1],gamma)
Az = A_array(n1,n2,dPA[2],dPB[2],dPC[2],gamma)
total = 0.
for I in range(l1+l2+1):
for J in range(m1+m2+1):
for K in range(n1+n2+1):
total += Ax[I]*Ay[J]*Az[K]*Fgamma(I+J+K,rcp2*gamma)
return -2*pi/gamma*exp(-alpha1*alpha2*rab2/gamma)*total
def nuclear_attraction(alpha1, lmn1, A, alpha2, lmn2, B, C):
"""
Full form of the nuclear attraction integral
>>> isnear(nuclear_attraction(1,(0,0,0),array((0,0,0),'d'),1,(0,0,0),array((0,0,0),'d'),array((0,0,0),'d')),-3.141593)
True
"""
l1, m1, n1 = lmn1
l2, m2, n2 = lmn2
gamma = alpha1 + alpha2
P = gaussian_product_center(alpha1, A, alpha2, B)
rab2 = norm2(A - B)
rcp2 = norm2(C - P)
dPA = P - A
dPB = P - B
dPC = P - C
Ax = A_array(l1, l2, dPA[0], dPB[0], dPC[0], gamma)
Ay = A_array(m1, m2, dPA[1], dPB[1], dPC[1], gamma)
Az = A_array(n1, n2, dPA[2], dPB[2], dPC[2], gamma)
total = 0.
for I in range(l1 + l2 + 1):
for J in range(m1 + m2 + 1):
for K in range(n1 + n2 + 1):
total += Ax[I] * Ay[J] * Az[K] * Fgamma(
I + J + K, rcp2 * gamma)
return -2 * pi / gamma * exp(-alpha1 * alpha2 * rab2 / gamma) * total
def coulomb_repulsion(xyza,norma,lmna,alphaa,
xyzb,normb,lmnb,alphab,
xyzc,normc,lmnc,alphac,
xyzd,normd,lmnd,alphad):
"""
Return the coulomb repulsion between four primitive gaussians a,b,c,d with the given origin
, x,y,z, normalization constants norm, angular momena l,m,n, and exponent alpha.
>>> p1 = array((0.,0.,0.),'d')
>>> p2 = array((0.,0.,1.),'d')
>>> lmn = (0,0,0)
>>> isclose(coulomb_repulsion(p1,1.,lmn,1.,p1,1.,lmn,1.,p1,1.,lmn,1.,p1,1.,lmn,1.),4.373355)
True
>>> isclose(coulomb_repulsion(p1,1.,lmn,1.,p1,1.,lmn,1.,p2,1.,lmn,1.,p2,1.,lmn,1.),3.266127)
True
>>> isclose(coulomb_repulsion(p1,1.,lmn,1.,p2,1.,lmn,1.,p1,1.,lmn,1.,p2,1.,lmn,1.),1.6088672)
True
"""
la,ma,na = lmna
lb,mb,nb = lmnb
lc,mc,nc = lmnc
ld,md,nd = lmnd
xa,ya,za = xyza
xb,yb,zb = xyzb
xc,yc,zc = xyzc
xd,yd,zd = xyzd
rab2 = norm2(xyza-xyzb)
rcd2 = norm2(xyzc-xyzd)
xyzp = gaussian_product_center(alphaa,xyza,alphab,xyzb)
xp,yp,zp = xyzp
xyzq = gaussian_product_center(alphac,xyzc,alphad,xyzd)
xq,yq,zq = xyzq
rpq2 = norm2(xyzp-xyzq)
gamma1 = alphaa+alphab
gamma2 = alphac+alphad
delta = 0.25*(1/gamma1+1/gamma2)
Bx = B_array(la,lb,lc,ld,xp,xa,xb,xq,xc,xd,gamma1,gamma2,delta)
By = B_array(ma,mb,mc,md,yp,ya,yb,yq,yc,yd,gamma1,gamma2,delta)
Bz = B_array(na,nb,nc,nd,zp,za,zb,zq,zc,zd,gamma1,gamma2,delta)
sum = 0.
for I in range(la+lb+lc+ld+1):
for J in range(ma+mb+mc+md+1):
for K in range(na+nb+nc+nd+1):
sum = sum + Bx[I]*By[J]*Bz[K]*Fgamma(I+J+K,0.25*rpq2/delta)
return 2*pow(pi,2.5)/(gamma1*gamma2*sqrt(gamma1+gamma2)) \
*exp(-alphaa*alphab*rab2/gamma1) \
*exp(-alphac*alphad*rcd2/gamma2)*sum*norma*normb*normc*normd
def coulomb_repulsion(xyza, norma, lmna, alphaa, xyzb, normb, lmnb, alphab,
xyzc, normc, lmnc, alphac, xyzd, normd, lmnd, alphad):
"""
Return the coulomb repulsion between four primitive gaussians a,b,c,d with the given origin
, x,y,z, normalization constants norm, angular momena l,m,n, and exponent alpha.
>>> p1 = array((0.,0.,0.),'d')
>>> p2 = array((0.,0.,1.),'d')
>>> lmn = (0,0,0)
>>> isclose(coulomb_repulsion(p1,1.,lmn,1.,p1,1.,lmn,1.,p1,1.,lmn,1.,p1,1.,lmn,1.),4.373355)
True
>>> isclose(coulomb_repulsion(p1,1.,lmn,1.,p1,1.,lmn,1.,p2,1.,lmn,1.,p2,1.,lmn,1.),3.266127)
True
>>> isclose(coulomb_repulsion(p1,1.,lmn,1.,p2,1.,lmn,1.,p1,1.,lmn,1.,p2,1.,lmn,1.),1.6088672)
True
"""
la, ma, na = lmna
lb, mb, nb = lmnb
lc, mc, nc = lmnc
ld, md, nd = lmnd
xa, ya, za = xyza
xb, yb, zb = xyzb
xc, yc, zc = xyzc
xd, yd, zd = xyzd
rab2 = norm2(xyza - xyzb)
rcd2 = norm2(xyzc - xyzd)
xyzp = gaussian_product_center(alphaa, xyza, alphab, xyzb)
xp, yp, zp = xyzp
xyzq = gaussian_product_center(alphac, xyzc, alphad, xyzd)
xq, yq, zq = xyzq
rpq2 = norm2(xyzp - xyzq)
gamma1 = alphaa + alphab
gamma2 = alphac + alphad
delta = 0.25 * (1 / gamma1 + 1 / gamma2)
Bx = B_array(la, lb, lc, ld, xp, xa, xb, xq, xc, xd, gamma1, gamma2, delta)
By = B_array(ma, mb, mc, md, yp, ya, yb, yq, yc, yd, gamma1, gamma2, delta)
Bz = B_array(na, nb, nc, nd, zp, za, zb, zq, zc, zd, gamma1, gamma2, delta)
sum = 0.
for I in range(la + lb + lc + ld + 1):
for J in range(ma + mb + mc + md + 1):
for K in range(na + nb + nc + nd + 1):
sum = sum + Bx[I] * By[J] * Bz[K] * Fgamma(
I + J + K, 0.25 * rpq2 / delta)
return 2*pow(pi,2.5)/(gamma1*gamma2*sqrt(gamma1+gamma2)) \
*exp(-alphaa*alphab*rab2/gamma1) \
*exp(-alphac*alphad*rcd2/gamma2)*sum*norma*normb*normc*normd
def term0(aexpn,xyza,bexpn,xyzb,cexpn,xyzc,dexpn,xyzd,mtot):
zeta,eta = float(aexpn+bexpn),float(cexpn+dexpn)
zpe = zeta+eta
xyzp = gaussian_product_center(aexpn,xyza,bexpn,xyzb)
xyzq = gaussian_product_center(cexpn,xyzc,dexpn,xyzd)
rab2 = norm2(xyza-xyzb)
Kab = sqrt(2)*pow(pi,1.25)/(aexpn+bexpn)\
*exp(-aexpn*bexpn/(aexpn+bexpn)*rab2)
rcd2 = norm2(xyzc-xyzd)
Kcd = sqrt(2)*pow(pi,1.25)/(cexpn+dexpn)\
*exp(-cexpn*dexpn/(cexpn+dexpn)*rcd2)
rpq2 = norm2(xyzp-xyzq)
T = zeta*eta/zpe*rpq2
return [Fgamma(m,T)*Kab*Kcd/sqrt(zpe) for m in xrange(mtot+1)]
def term0(aexpn, xyza, bexpn, xyzb, cexpn, xyzc, dexpn, xyzd, mtot):
zeta, eta = float(aexpn + bexpn), float(cexpn + dexpn)
zpe = zeta + eta
xyzp = gaussian_product_center(aexpn, xyza, bexpn, xyzb)
xyzq = gaussian_product_center(cexpn, xyzc, dexpn, xyzd)
rab2 = norm2(xyza - xyzb)
Kab = sqrt(2)*pow(pi,1.25)/(aexpn+bexpn)\
*exp(-aexpn*bexpn/(aexpn+bexpn)*rab2)
rcd2 = norm2(xyzc - xyzd)
Kcd = sqrt(2)*pow(pi,1.25)/(cexpn+dexpn)\
*exp(-cexpn*dexpn/(cexpn+dexpn)*rcd2)
rpq2 = norm2(xyzp - xyzq)
T = zeta * eta / zpe * rpq2
return [Fgamma(m, T) * Kab * Kcd / sqrt(zpe) for m in xrange(mtot + 1)]
def overlap(alpha1,lmn1,A,alpha2,lmn2,B):
"""
Full form of the overlap integral. Taken from THO eq. 2.12
>>> isnear(overlap(1,(0,0,0),array((0,0,0),'d'),1,(0,0,0),array((0,0,0),'d')),1.968701)
True
"""
l1,m1,n1 = lmn1
l2,m2,n2 = lmn2
rab2 = norm2(A-B)
gamma = alpha1+alpha2
P = gaussian_product_center(alpha1,A,alpha2,B)
pre = pow(pi/gamma,1.5)*exp(-alpha1*alpha2*rab2/gamma)
wx = overlap1d(l1,l2,P[0]-A[0],P[0]-B[0],gamma)
wy = overlap1d(m1,m2,P[1]-A[1],P[1]-B[1],gamma)
wz = overlap1d(n1,n2,P[2]-A[2],P[2]-B[2],gamma)
return pre*wx*wy*wz
def overlap(alpha1, lmn1, A, alpha2, lmn2, B):
"""
Full form of the overlap integral. Taken from THO eq. 2.12
>>> isnear(overlap(1,(0,0,0),array((0,0,0),'d'),1,(0,0,0),array((0,0,0),'d')),1.968701)
True
"""
l1, m1, n1 = lmn1
l2, m2, n2 = lmn2
rab2 = norm2(A - B)
gamma = alpha1 + alpha2
P = gaussian_product_center(alpha1, A, alpha2, B)
pre = pow(pi / gamma, 1.5) * exp(-alpha1 * alpha2 * rab2 / gamma)
wx = overlap1d(l1, l2, P[0] - A[0], P[0] - B[0], gamma)
wy = overlap1d(m1, m2, P[1] - A[1], P[1] - B[1], gamma)
wz = overlap1d(n1, n2, P[2] - A[2], P[2] - B[2], gamma)
return pre * wx * wy * wz
def vrr_array(xyza,lmna,alphaa,xyzb,alphab,
xyzc,lmnc,alphac,xyzd,alphad):
la,ma,na = lmna
lc,mc,nc = lmnc
xa,ya,za = xyza
xb,yb,zb = xyzb
xc,yc,zc = xyzc
xd,yd,zd = xyzd
px,py,pz = xyzp = gaussian_product_center(alphaa,xyza,alphab,xyzb)
qx,qy,qz = xyzq = gaussian_product_center(alphac,xyzc,alphad,xyzd)
zeta,eta = float(alphaa+alphab),float(alphac+alphad)
wx,wy,wz = xyzw = gaussian_product_center(zeta,xyzp,eta,xyzq)
zpe = zeta+eta
rab2 = norm2(xyza-xyzb)
Kab = sqrt(2)*pow(pi,1.25)/(alphaa+alphab)\
*exp(-alphaa*alphab/(alphaa+alphab)*rab2)
rcd2 = norm2(xyzc-xyzd)
Kcd = sqrt(2)*pow(pi,1.25)/(alphac+alphad)\
*exp(-alphac*alphad/(alphac+alphad)*rcd2)
rpq2 = norm2(xyzp-xyzq)
T = zeta*eta/zpe*rpq2
mtot = sum(lmna)+sum(lmnc)
vrr_terms = {}#zeros((la+1,ma+1,na+1,lc+1,mc+1,nc+1,mtot+1),'d')
for m in xrange(mtot+1):
vrr_terms[0,0,0,0,0,0,m] = Fgamma(m,T)*Kab*Kcd/sqrt(zpe)
for i in xrange(la):
for m in xrange(mtot-i):
vrr_terms[i+1,0,0, 0,0,0, m] = (px-xa)*vrr_terms[i,0,0, 0,0,0, m] +\
(wx-px)*vrr_terms[i,0,0, 0,0,0, m+1]
if i>0:
vrr_terms[i+1,0,0, 0,0,0, m] += i/2./zeta*(
vrr_terms[i-1,0,0, 0,0,0, m]
- eta/zpe*vrr_terms[i-1,0,0, 0,0,0, m+1] )
for i,j in product(xrange(la+1),xrange(ma)):
for m in range(mtot-i-j):
vrr_terms[i,j+1,0, 0,0,0, m] = (py-ya)*vrr_terms[i,j,0, 0,0,0, m] +\
(wy-py)*vrr_terms[i,j,0, 0,0,0, m+1]
if j>0:
vrr_terms[i,j+1,0, 0,0,0, m] += j/2./zeta*(
vrr_terms[i,j-1,0, 0,0,0, m]
- eta/zpe*vrr_terms[i,j-1,0, 0,0,0, m+1] )
for i,j,k in product(xrange(la+1),xrange(ma+1),xrange(na)):
for m in range(mtot-i-j-k):
vrr_terms[i,j,k+1, 0,0,0, m] = (pz-za)*vrr_terms[i,j,k, 0,0,0, m] +\
(wz-pz)*vrr_terms[i,j,k, 0,0,0, m+1]
if k>0:
vrr_terms[i,j,k+1, 0,0,0, m] += k/2./zeta*(
vrr_terms[i,j,k-1, 0,0,0, m]
- eta/zpe*vrr_terms[i,j,k-1, 0,0,0, m+1])
for i,j,k, q in product(xrange(la+1),xrange(ma+1),xrange(na+1),xrange(lc)):
for m in range(mtot-i-j-k-q):
vrr_terms[i,j,k, q+1,0,0, m] = (qx-xc)*vrr_terms[i,j,k, q,0,0, m] +\
(wx-qx)*vrr_terms[i,j,k, q,0,0, m+1]
if q>0:
vrr_terms[i,j,k, q+1,0,0, m] += q/2./eta*(
vrr_terms[i,j,k, q-1,0,0, m]
- zeta/zpe*vrr_terms[i,j,k, q-1,0,0, m+1])
if i>0:
vrr_terms[i,j,k, q+1,0,0, m] += i/2./zpe*vrr_terms[i-1,j,k, q,0,0, m+1]
for i,j,k, q,r in product(xrange(la+1),xrange(ma+1),xrange(na+1),
xrange(lc+1),xrange(mc)):
for m in range(mtot-i-j-k-q-r):
vrr_terms[i,j,k, q,r+1,0, m] = (qy-yc)*vrr_terms[i,j,k, q,r,0, m] +\
(wy-qy)*vrr_terms[i,j,k, q,r,0, m+1]
if r>0:
vrr_terms[i,j,k, q,r+1,0, m] += r/2./eta*(
vrr_terms[i,j,k, q,r-1,0, m]
- zeta/zpe*vrr_terms[i,j,k, q,r-1,0, m+1])
if j>0:
vrr_terms[i,j,k, q,r+1,0, m] += j/2./zpe*vrr_terms[i,j-1,k,q,r,0,m+1]
for i,j,k, q,r,s in product(xrange(la+1),xrange(ma+1),xrange(na+1),
xrange(lc+1),xrange(mc+1),xrange(nc)):
for m in range(mtot-i-j-k-q-r-s):
vrr_terms[i,j,k,q,r,s+1,m] = (qz-zc)*vrr_terms[i,j,k,q,r,s,m] +\
(wz-qz)*vrr_terms[i,j,k,q,r,s,m+1]
if s>0:
vrr_terms[i,j,k,q,r,s+1,m] += s/2./eta*(
vrr_terms[i,j,k,q,r,s-1,m]
- zeta/zpe*vrr_terms[i,j,k,q,r,s-1,m+1])
if k>0:
vrr_terms[i,j,k,q,r,s+1,m] += k/2./zpe*vrr_terms[i,j,k-1,q,r,s,m+1]
return vrr_terms
def vrr_array(xyza, lmna, alphaa, xyzb, alphab, xyzc, lmnc, alphac, xyzd,
alphad):
la, ma, na = lmna
lc, mc, nc = lmnc
xa, ya, za = xyza
xb, yb, zb = xyzb
xc, yc, zc = xyzc
xd, yd, zd = xyzd
px, py, pz = xyzp = gaussian_product_center(alphaa, xyza, alphab, xyzb)
qx, qy, qz = xyzq = gaussian_product_center(alphac, xyzc, alphad, xyzd)
zeta, eta = float(alphaa + alphab), float(alphac + alphad)
wx, wy, wz = xyzw = gaussian_product_center(zeta, xyzp, eta, xyzq)
zpe = zeta + eta
rab2 = norm2(xyza - xyzb)
Kab = sqrt(2)*pow(pi,1.25)/(alphaa+alphab)\
*exp(-alphaa*alphab/(alphaa+alphab)*rab2)
rcd2 = norm2(xyzc - xyzd)
Kcd = sqrt(2)*pow(pi,1.25)/(alphac+alphad)\
*exp(-alphac*alphad/(alphac+alphad)*rcd2)
rpq2 = norm2(xyzp - xyzq)
T = zeta * eta / zpe * rpq2
mtot = sum(lmna) + sum(lmnc)
vrr_terms = {} #zeros((la+1,ma+1,na+1,lc+1,mc+1,nc+1,mtot+1),'d')
for m in xrange(mtot + 1):
vrr_terms[0, 0, 0, 0, 0, 0, m] = Fgamma(m, T) * Kab * Kcd / sqrt(zpe)
for i in xrange(la):
for m in xrange(mtot - i):
vrr_terms[i+1,0,0, 0,0,0, m] = (px-xa)*vrr_terms[i,0,0, 0,0,0, m] +\
(wx-px)*vrr_terms[i,0,0, 0,0,0, m+1]
if i > 0:
vrr_terms[i + 1, 0, 0, 0, 0, 0, m] += i / 2. / zeta * (
vrr_terms[i - 1, 0, 0, 0, 0, 0, m] -
eta / zpe * vrr_terms[i - 1, 0, 0, 0, 0, 0, m + 1])
for i, j in product(xrange(la + 1), xrange(ma)):
for m in range(mtot - i - j):
vrr_terms[i,j+1,0, 0,0,0, m] = (py-ya)*vrr_terms[i,j,0, 0,0,0, m] +\
(wy-py)*vrr_terms[i,j,0, 0,0,0, m+1]
if j > 0:
vrr_terms[i, j + 1, 0, 0, 0, 0, m] += j / 2. / zeta * (
vrr_terms[i, j - 1, 0, 0, 0, 0, m] -
eta / zpe * vrr_terms[i, j - 1, 0, 0, 0, 0, m + 1])
for i, j, k in product(xrange(la + 1), xrange(ma + 1), xrange(na)):
for m in range(mtot - i - j - k):
vrr_terms[i,j,k+1, 0,0,0, m] = (pz-za)*vrr_terms[i,j,k, 0,0,0, m] +\
(wz-pz)*vrr_terms[i,j,k, 0,0,0, m+1]
if k > 0:
vrr_terms[i, j, k + 1, 0, 0, 0, m] += k / 2. / zeta * (
vrr_terms[i, j, k - 1, 0, 0, 0, m] -
eta / zpe * vrr_terms[i, j, k - 1, 0, 0, 0, m + 1])
for i, j, k, q in product(xrange(la + 1), xrange(ma + 1), xrange(na + 1),
xrange(lc)):
for m in range(mtot - i - j - k - q):
vrr_terms[i,j,k, q+1,0,0, m] = (qx-xc)*vrr_terms[i,j,k, q,0,0, m] +\
(wx-qx)*vrr_terms[i,j,k, q,0,0, m+1]
if q > 0:
vrr_terms[i, j, k, q + 1, 0, 0, m] += q / 2. / eta * (
vrr_terms[i, j, k, q - 1, 0, 0, m] -
zeta / zpe * vrr_terms[i, j, k, q - 1, 0, 0, m + 1])
if i > 0:
vrr_terms[i, j, k, q + 1, 0, 0,
m] += i / 2. / zpe * vrr_terms[i - 1, j, k, q, 0, 0,
m + 1]
for i, j, k, q, r in product(xrange(la + 1), xrange(ma + 1),
xrange(na + 1), xrange(lc + 1), xrange(mc)):
for m in range(mtot - i - j - k - q - r):
vrr_terms[i,j,k, q,r+1,0, m] = (qy-yc)*vrr_terms[i,j,k, q,r,0, m] +\
(wy-qy)*vrr_terms[i,j,k, q,r,0, m+1]
if r > 0:
vrr_terms[i, j, k, q, r + 1, 0, m] += r / 2. / eta * (
vrr_terms[i, j, k, q, r - 1, 0, m] -
zeta / zpe * vrr_terms[i, j, k, q, r - 1, 0, m + 1])
if j > 0:
vrr_terms[i, j, k, q, r + 1, 0,
m] += j / 2. / zpe * vrr_terms[i, j - 1, k, q, r, 0,
m + 1]
for i, j, k, q, r, s in product(xrange(la + 1), xrange(ma + 1),
xrange(na + 1), xrange(lc + 1),
xrange(mc + 1), xrange(nc)):
for m in range(mtot - i - j - k - q - r - s):
vrr_terms[i,j,k,q,r,s+1,m] = (qz-zc)*vrr_terms[i,j,k,q,r,s,m] +\
(wz-qz)*vrr_terms[i,j,k,q,r,s,m+1]
if s > 0:
vrr_terms[i, j, k, q, r, s + 1, m] += s / 2. / eta * (
vrr_terms[i, j, k, q, r, s - 1, m] -
zeta / zpe * vrr_terms[i, j, k, q, r, s - 1, m + 1])
if k > 0:
vrr_terms[i, j, k, q, r, s + 1,
m] += k / 2. / zpe * vrr_terms[i, j, k - 1, q, r, s,
m + 1]
return vrr_terms
