def transform_(cint_coeff, orig_bas, bas_to_append, env, pexp, pcoeff):
np1, nc1 = cint_coeff.shape
l = orig_bas[gto.ANG_OF]
if cell.cart:
degen = (l + 1) * (l + 2) // 2
else:
degen = 2 * l + 1
if np1 >= nc1:
bas = orig_bas.copy()
bas[NPRIM_OF] = np1
bas[PTR_EXP] = pexp
bas[PTR_COEFF] = pcoeff
bas_to_append.append(bas)
coeff = np.eye(nc1 * degen)
else:
bas = np.repeat(orig_bas.copy()[None, :], np1, axis=0)
bas[:, NPRIM_OF] = 1
bas[:, NCTR_OF] = 1
bas[:, PTR_EXP] = pexp + np.arange(np1)
bas[:, PTR_COEFF] = pcoeff + np.arange(np1)
bas_to_append.extend(bas)
exps = _env[pexp:pexp + np1]
cs = gto.gto_norm(l, exps)
_env[pcoeff:pcoeff + np1] = cs
unit_coeff = np.eye(degen)
coeff = np.einsum('p,mn,pc->pmcn', 1 / cs, unit_coeff, cint_coeff)
coeff = coeff.reshape(np1 * degen, nc1 * degen)
return coeff
def integral_one_gaussian_from_overlap(mol):
"""
Outputs the integral of only one Gaussian
using as second Gaussian in <i|j> an 1s
basis funtion of an He atom with
alpha=1e-15 and coeff=1.00000. The result
is divided by the norm of the He function.
"""
# Creating auxiliar Molecule
intmol = gto.Mole()
intmol.atom = '''He 0. 0. 0.'''
intmol.basis = He_flat_basis
intmol.build()
# Merging molecules
atm_x, bas_x, env_x = gto.conc_env(intmol._atm, intmol._bas, intmol._env,
mol._atm, mol._bas, mol._env)
# Computing the overlap
HF_partial = gto.moleintor.getints('cint1e_ovlp_sph',
atm_x,
bas_x,
env_x,
comp=1,
hermi=0,
aosym='s1',
out=None)
return np.hstack(HF_partial[:, :1])[1:] / gto.gto_norm(0, He_exp_flt)
def hellmann_feynman_df(mol, ia):
"""
Outputs the integral
Z_I*<i|(r-R_I)/|r-R_I|^3|j>
of only one Gaussian using as second Gaussian
an 1s basis funtion of an He atom with
alpha=1e-15 and coeff=1.00000. The result
is divided by the norm of the He function.
"""
# Creating auxiliar Molecule
intmol = gto.Mole()
intmol.atom = '''He 0. 0. 0.'''
intmol.basis = He_flat_basis
intmol.build()
# Merging molecules
atm_x, bas_x, env_x = gto.conc_env(intmol._atm, intmol._bas, intmol._env,\
mol._atm, mol._bas, mol._env)
PTR_RINV_ORIG = 4
env_x[PTR_RINV_ORIG:PTR_RINV_ORIG + 3] = mol.atom_coord(ia)
HF_partial = gto.moleintor.getints('cint1e_drinv_sph',
atm_x,
bas_x,
env_x,
comp=3,
hermi=0,
aosym='s1',
out=None)
return -mol.atom_charge(ia)*\
np.hstack(HF_partial[:,:,:1])[1:]/gto.gto_norm(0,He_exp_flt)
def integral_one_gaussian_polarization(mol):
"""
Outputs the integral <i|r|j> with the only-one-Gaussian trick
using as second Gaussian an 1s basis funtion of an He atom with
alpha=1e-15 and coeff=1.00000. The result is divided by the
norm of the 'dummy' He function.
"""
# Creating auxiliar Molecule
intmol = gto.Mole()
intmol.atom = '''He 0. 0. 0.'''
intmol.basis = He_flat_basis
intmol.build()
# Merging molecules
atm_x, bas_x, env_x = gto.conc_env(intmol._atm, intmol._bas, intmol._env,
mol._atm, mol._bas, mol._env)
# Computing the overlap
PTR_COMMON_ORIG = 1
env_x[PTR_COMMON_ORIG:PTR_COMMON_ORIG + 3] = (
0., 0., 0.) #(mol.atom_coord(1)-mol.atom_coord(0))/2.
#mol.set_common_orig((1.,0.,0.))
HF_partial = gto.moleintor.getints('cint1e_r_sph',
atm_x,
bas_x,
env_x,
comp=3,
hermi=0,
aosym='s1',
out=None)
return np.hstack(HF_partial[:, :, :1])[1:] / gto.gto_norm(0, He_exp_flt)
def fake_cell_vnl(cell):
'''Generate fake cell for V_{nl}.
gaussian function p_i^l Y_{lm}
'''
fake_env = [cell.atom_coords().ravel()]
fake_atm = cell._atm.copy()
fake_atm[:, gto.PTR_COORD] = numpy.arange(0, cell.natm * 3, 3)
ptr = cell.natm * 3
fake_bas = []
hl_blocks = []
for ia in range(cell.natm):
if cell.atom_charge(ia) == 0: # pass ghost atoms
continue
symb = cell.atom_symbol(ia)
if symb in cell._pseudo:
pp = cell._pseudo[symb]
# nproj_types = pp[4]
for l, (rl, nl, hl) in enumerate(pp[5:]):
if nl > 0:
alpha = .5 / rl**2
norm = gto.gto_norm(l, alpha)
fake_env.append([alpha, norm])
fake_bas.append([ia, l, 1, 1, 0, ptr, ptr + 1, 0])
#
# Function p_i^l (PRB, 58, 3641 Eq 3) is (r^{2(i-1)})^2 square normalized to 1.
# But here the fake basis is square normalized to 1. A factor ~ p_i^l / p_1^l
# is attached to h^l_ij (for i>1,j>1) so that (factor * fake-basis * r^{2(i-1)})
# is normalized to 1. The factor is
# r_l^{l+(4-1)/2} sqrt(Gamma(l+(4-1)/2)) sqrt(Gamma(l+3/2))
# ------------------------------------------ = ----------------------------------
# r_l^{l+(4i-1)/2} sqrt(Gamma(l+(4i-1)/2)) sqrt(Gamma(l+2i-1/2)) r_l^{2i-2}
#
fac = numpy.array(
[_PLI_FAC[l, i] / rl**(i * 2) for i in range(nl)])
hl = numpy.einsum('i,ij,j->ij', fac, numpy.asarray(hl),
fac)
hl_blocks.append(hl)
ptr += 2
fakecell = copy.copy(cell)
fakecell._atm = numpy.asarray(fake_atm, dtype=numpy.int32)
fakecell._bas = numpy.asarray(fake_bas, dtype=numpy.int32)
fakecell._env = numpy.asarray(numpy.hstack(fake_env), dtype=numpy.double)
return fakecell, hl_blocks
def fake_cell_vnl(cell):
'''Generate fake cell for V_{nl}.
gaussian function p_i^l Y_{lm}
'''
fake_env = [cell.atom_coords().ravel()]
fake_atm = cell._atm.copy()
fake_atm[:,gto.PTR_COORD] = numpy.arange(0, cell.natm*3, 3)
ptr = cell.natm * 3
fake_bas = []
hl_blocks = []
for ia in range(cell.natm):
if cell.atom_charge(ia) == 0: # pass ghost atoms
continue
symb = cell.atom_symbol(ia)
if symb in cell._pseudo:
pp = cell._pseudo[symb]
nproj_types = pp[4]
for l, (rl, nl, hl) in enumerate(pp[5:]):
if nl > 0:
alpha = .5 / rl**2
norm = gto.gto_norm(l, alpha)
fake_env.append([alpha, norm])
fake_bas.append([ia, l, 1, 1, 0, ptr, ptr+1, 0])
# Function p_i^l (PRB, 58, 3641 Eq 3) is (r^{2(i-1)})^2 square normalized to 1.
# But here the fake basis is square normalized to 1. A factor ~ p_i^l / p_1^l
# is attached to h^l_ij (for i>1,j>1) so that (factor * fake-basis * r^{2(i-1)})
# is normalized to 1. The factor is
# r_l^{l+(4-1)/2} sqrt(Gamma(l+(4-1)/2)) sqrt(Gamma(l+3/2))
# ------------------------------------------ = ----------------------------------
# r_l^{l+(4i-1)/2} sqrt(Gamma(l+(4i-1)/2)) sqrt(Gamma(l+2i-1/2)) r_l^{2i-2}
fac = numpy.array([_PLI_FAC[l,i]/rl**(i*2) for i in range(nl)])
hl = numpy.einsum('i,ij,j->ij', fac, numpy.asarray(hl), fac)
hl_blocks.append(hl)
ptr += 2
fakecell = copy.copy(cell)
fakecell._atm = numpy.asarray(fake_atm, dtype=numpy.int32)
fakecell._bas = numpy.asarray(fake_bas, dtype=numpy.int32)
fakecell._env = numpy.asarray(numpy.hstack(fake_env), dtype=numpy.double)
#fakecell.nimgs = [2]*3
return fakecell, hl_blocks
mol.atom =molstr
mol.basis = '631g'
mol.build()
F11=force_en_1Gz(2.02525091,mol.atom_coord(0),
mol.atom_coord(1),1)*1.000000
F2=force_nn_1Gz(mol.atom_coord(0),
mol.atom_coord(1),1)
F1=0.3570601*F11
F_Total=F1+F2
print '\nF_EN = %12.6f' % (F1)
print 'F_NN = %12.6f' % (F2)
print 'F_Tot = %12.6f' % (F_Total)
print 'Norm = %12.6f' % (np.power(np.pi/2.0252509,3/2.)*0.3570601*3.5447)
print 'Norm = %12.6f' % (gto.gto_norm(0,2.0252509)*0.3570601*3.5447)
print '''
=====Example 2:=====
'''
mol.basis = 'sto3g'
F11=force_en_1Gz(15.6752927,mol.atom_coord(0),
mol.atom_coord(1),1)*0.0186886
F12=force_en_1Gz(3.6063578 ,mol.atom_coord(0),
mol.atom_coord(1),1)*0.0631670
F13=force_en_1Gz(1.2080016 ,mol.atom_coord(0),
mol.atom_coord(1),1)*0.1204609
The integral cint1e_ovlp_sph <He_aux|H_orbital> have to
give N/M^2, where N and M are defined below
in the example (M1, M2, M3, N1, N2 and N3)
N is the norm of the Gaussian \int exp(-2*alpha*r^2)dr.
The basis xxx set for He is
H S
1e-15 1.00000000
The sto-3g basis set for H is
H S
3.42525091 0.15432897
0.62391373 0.53532814
0.16885540 0.44463454
'''
# Computing the norms:
N1 = gto.gto_norm(0, 3.42525091)
M1 = gto.gto_norm(0, 3.42525091 / 2)
N2 = gto.gto_norm(0, 0.62391373)
M2 = gto.gto_norm(0, 0.62391373 / 2)
N3 = gto.gto_norm(0, 0.16885540)
M3 = gto.gto_norm(0, 0.16885540 / 2)
print 'From the Norms M,N:', 0.15432897 * N1 / M1 / M1 + 0.53532814 * N2 / M2 / M2 + 0.44463454 * N3 / M3 / M3
print 'From the integrals:', integral_one_gaussian_from_overlap(mol)[0]
print ''
print '''
=====Example 3:=====
'''
# Creating auxiliar Molecule
intmol = gto.Mole()
def write_mo(fout, mol, mo_coeff, mo_energy=None, mo_occ=None):
nmo = mo_coeff.shape[1]
mo_cart = []
centers = []
types = []
exps = []
p0 = 0
for ib in range(mol.nbas):
ia = mol.bas_atom(ib)
l = mol.bas_angular(ib)
es = mol.bas_exp(ib)
c = numpy.einsum('pi,p->pi', mol.bas_ctr_coeff(ib),
1 / gto.gto_norm(l, es))
np, nc = c.shape
nd = nc * (2 * l + 1)
mosub = mo_coeff[p0:p0 + nd].reshape(-1, nc, nmo)
c2s = gto.cart2sph(l)
mosub = numpy.einsum('yki,cy,pk->pci', mosub, c2s, c)
mo_cart.append(mosub.reshape(-1, nmo))
for t in TYPE_MAP[l]:
types.append([t] * np)
ncart = gto.len_cart(l)
exps.extend([es] * ncart)
centers.extend([ia + 1] * (np * ncart))
p0 += nd
mo_cart = numpy.vstack(mo_cart)
centers = numpy.hstack(centers)
types = numpy.hstack(types)
exps = numpy.hstack(exps)
nprim, nmo = mo_cart.shape
fout.write('title line\n')
fout.write(
'GAUSSIAN %3d MOL ORBITALS %3d PRIMITIVES %3d NUCLEI\n'
% (mo_cart.shape[1], mo_cart.shape[0], mol.natm))
for ia in range(mol.natm):
x, y, z = mol.atom_coord(ia)
fout.write(
'%-4s %-4d (CENTRE%3d) %12.8f %12.8f %12.8f CHARGE = %.1f\n' %
(mol.atom_pure_symbol(ia), ia, ia, x, y, z, mol.atom_charge(ia)))
for i0, i1 in lib.prange(0, nprim, 20):
fout.write('CENTRE ASSIGNMENTS %s\n' %
''.join('%3d' % x for x in centers[i0:i1]))
for i0, i1 in lib.prange(0, nprim, 20):
fout.write('TYPE ASSIGNMENTS %s\n' % ''.join('%3d' % x
for x in types[i0:i1]))
for i0, i1 in lib.prange(0, nprim, 5):
fout.write('EXPONENTS %s\n' % ' '.join('%14.7E' % x
for x in exps[i0:i1]))
for k in range(nmo):
mo = mo_cart[:, k]
if mo_energy is None or mo_occ is None:
fout.write('CANMO %d\n' (k, mo_occ[k], mo_energy[k]))
else:
fout.write(
'MO %-4d OCC NO = %12.8f ORB. ENERGY = %12.8f\n'
% (k, mo_occ[k], mo_energy[k]))
for i0, i1 in lib.prange(0, nprim, 5):
fout.write('%s\n' % ' '.join('%15.8E' % x for x in mo[i0:i1]))
