def make_L1(pcmobj, r_vdw, ylm_1sph, fi):
# See JCTC, 9, 3637, Eq (18)
mol = pcmobj.mol
natm = mol.natm
lmax = pcmobj.lmax
eta = pcmobj.eta
nlm = (lmax + 1)**2
coords_1sph, weights_1sph = ddcosmo.make_grids_one_sphere(
pcmobj.lebedev_order)
ngrid_1sph = weights_1sph.size
atom_coords = mol.atom_coords()
ylm_1sph = ylm_1sph.reshape(nlm, ngrid_1sph)
Lmat = numpy.zeros((natm, 3, natm, nlm, natm, nlm))
fi1 = make_fi1(pcmobj, pcmobj.get_atomic_radii())
for ja in range(natm):
part_weights = weights_1sph.copy()
part_weights[fi[ja] > 1] /= fi[ja, fi[ja] > 1]
part_weights1 = numpy.zeros((natm, 3, ngrid_1sph))
tmp = part_weights[fi[ja] > 1] / fi[ja, fi[ja] > 1]
part_weights1[:, :, fi[ja] > 1] = -tmp * fi1[:, :, ja, fi[ja] > 1]
for ka in ddcosmo.atoms_with_vdw_overlap(ja, atom_coords, r_vdw):
vjk = r_vdw[ja] * coords_1sph + atom_coords[ja] - atom_coords[ka]
rv = lib.norm(vjk, axis=1)
tjk = rv / r_vdw[ka]
wjk0 = pcmobj.regularize_xt(tjk, eta, r_vdw[ka])
wjk1 = regularize_xt1(tjk, eta * r_vdw[ka])
sjk = vjk.T / rv
wjk1 = 1. / r_vdw[ka] * wjk1 * sjk
wjk01 = wjk0 * part_weights1
wjk0 *= part_weights
wjk1 *= part_weights
pol0 = sph.multipoles(vjk, lmax)
pol1 = multipoles1(vjk, lmax)
p1 = 0
for l in range(lmax + 1):
fac = 4 * numpy.pi / (l * 2 + 1) / r_vdw[ka]**(l + 1)
p0, p1 = p1, p1 + (l * 2 + 1)
a = numpy.einsum('xn,zn,mn->zxm', ylm_1sph, wjk1, pol0[l])
a += numpy.einsum('xn,n,zmn->zxm', ylm_1sph, wjk0, pol1[l])
Lmat[ja, :, ja, :, ka, p0:p1] += -fac * a
Lmat[ka, :, ja, :, ka, p0:p1] -= -fac * a
a = numpy.einsum('xn,azn,mn->azxm', ylm_1sph, wjk01, pol0[l])
Lmat[:, :, ja, :, ka, p0:p1] += -fac * a
return Lmat
def make_L1(pcmobj, r_vdw, ylm_1sph, fi):
# See JCTC, 9, 3637, Eq (18)
mol = pcmobj.mol
natm = mol.natm
lmax = pcmobj.lmax
eta = pcmobj.eta
nlm = (lmax+1)**2
coords_1sph, weights_1sph = ddcosmo.make_grids_one_sphere(pcmobj.lebedev_order)
ngrid_1sph = weights_1sph.size
atom_coords = mol.atom_coords()
ylm_1sph = ylm_1sph.reshape(nlm,ngrid_1sph)
Lmat = numpy.zeros((natm,3,natm,nlm,natm,nlm))
fi1 = make_fi1(pcmobj, pcmobj.get_atomic_radii())
for ja in range(natm):
part_weights = weights_1sph.copy()
part_weights[fi[ja]>1] /= fi[ja,fi[ja]>1]
part_weights1 = numpy.zeros((natm,3,ngrid_1sph))
tmp = part_weights[fi[ja]>1] / fi[ja,fi[ja]>1]
part_weights1[:,:,fi[ja]>1] = -tmp * fi1[:,:,ja,fi[ja]>1]
for ka in ddcosmo.atoms_with_vdw_overlap(ja, atom_coords, r_vdw):
vjk = r_vdw[ja] * coords_1sph + atom_coords[ja] - atom_coords[ka]
rv = lib.norm(vjk, axis=1)
tjk = rv / r_vdw[ka]
wjk0 = pcmobj.regularize_xt(tjk, eta, r_vdw[ka])
wjk1 = regularize_xt1(tjk, eta*r_vdw[ka])
sjk = vjk.T / rv
wjk1 = 1./r_vdw[ka] * wjk1 * sjk
wjk01 = wjk0 * part_weights1
wjk0 *= part_weights
wjk1 *= part_weights
pol0 = sph.multipoles(vjk, lmax)
pol1 = multipoles1(vjk, lmax)
p1 = 0
for l in range(lmax+1):
fac = 4*numpy.pi/(l*2+1) / r_vdw[ka]**(l+1)
p0, p1 = p1, p1 + (l*2+1)
a = numpy.einsum('xn,zn,mn->zxm', ylm_1sph, wjk1, pol0[l])
a+= numpy.einsum('xn,n,zmn->zxm', ylm_1sph, wjk0, pol1[l])
Lmat[ja,:,ja,:,ka,p0:p1] += -fac * a
Lmat[ka,:,ja,:,ka,p0:p1] -= -fac * a
a = numpy.einsum('xn,azn,mn->azxm', ylm_1sph, wjk01, pol0[l])
Lmat[:,:,ja,:,ka,p0:p1] += -fac * a
return Lmat
def make_L(pcmobj, r_vdw, ylm_1sph, fi):
# See JCTC, 9, 3637, Eq (18)
mol = pcmobj.mol
natm = mol.natm
lmax = pcmobj.lmax
eta = pcmobj.eta
nlm = (lmax + 1)**2
coords_1sph, weights_1sph = make_grids_one_sphere(pcmobj.lebedev_order)
ngrid_1sph = weights_1sph.size
atom_coords = mol.atom_coords()
ylm_1sph = ylm_1sph.reshape(nlm, ngrid_1sph)
# JCP, 141, 184108 Eq (9), (12) is incorrect
# L_diag = <lm|(1/|s-s'|)|l'm'>
# Using Laplace expansion for electrostatic potential 1/r
# L_diag = 4pi/(2l+1)/|s| <lm|l'm'>
L_diag = numpy.zeros((natm, nlm))
p1 = 0
for l in range(lmax + 1):
p0, p1 = p1, p1 + (l * 2 + 1)
L_diag[:, p0:p1] = 4 * numpy.pi / (l * 2 + 1)
L_diag *= 1. / r_vdw[:natm].reshape(-1, 1)
Lmat = numpy.diag(L_diag.ravel()).reshape(natm, nlm, natm, nlm)
for ja in range(natm):
# scale the weight, precontract d_nj and w_n
# see JCTC 9, 3637, Eq (16) - (18)
# Note all values are scaled by 1/r_vdw to make the formulas
# consistent to Psi in JCP, 141, 184108
part_weights = weights_1sph.copy()
part_weights[fi[ja] > 1] /= fi[ja, fi[ja] > 1]
for ka in atoms_with_vdw_overlap(ja, atom_coords, r_vdw[:natm]):
vjk = r_vdw[ja] * coords_1sph + atom_coords[ja] - atom_coords[ka]
tjk = lib.norm(vjk, axis=1) / r_vdw[ka]
wjk = pcmobj.regularize_xt(tjk, eta, r_vdw[ka])
wjk *= part_weights
pol = sph.multipoles(vjk, lmax)
p1 = 0
for l in range(lmax + 1):
fac = 4 * numpy.pi / (l * 2 + 1) / r_vdw[ka]**(l + 1)
p0, p1 = p1, p1 + (l * 2 + 1)
a = numpy.einsum('xn,n,mn->xm', ylm_1sph, wjk, pol[l])
Lmat[ja, :, ka, p0:p1] += -fac * a
return Lmat
def make_L(pcmobj, r_vdw, ylm_1sph, fi):
# See JCTC, 9, 3637, Eq (18)
mol = pcmobj.mol
natm = mol.natm
lmax = pcmobj.lmax
eta = pcmobj.eta
nlm = (lmax+1)**2
coords_1sph, weights_1sph = make_grids_one_sphere(pcmobj.lebedev_order)
ngrid_1sph = weights_1sph.size
atom_coords = mol.atom_coords()
ylm_1sph = ylm_1sph.reshape(nlm,ngrid_1sph)
# JCP, 141, 184108 Eq (9), (12) is incorrect
# L_diag = <lm|(1/|s-s'|)|l'm'>
# Using Laplace expansion for electrostatic potential 1/r
# L_diag = 4pi/(2l+1)/|s| <lm|l'm'>
L_diag = numpy.zeros((natm,nlm))
p1 = 0
for l in range(lmax+1):
p0, p1 = p1, p1 + (l*2+1)
L_diag[:,p0:p1] = 4*numpy.pi/(l*2+1)
L_diag *= 1./r_vdw.reshape(-1,1)
Lmat = numpy.diag(L_diag.ravel()).reshape(natm,nlm,natm,nlm)
for ja in range(natm):
# scale the weight, precontract d_nj and w_n
# see JCTC 9, 3637, Eq (16) - (18)
# Note all values are scaled by 1/r_vdw to make the formulas
# consistent to Psi in JCP, 141, 184108
part_weights = weights_1sph.copy()
part_weights[fi[ja]>1] /= fi[ja,fi[ja]>1]
for ka in atoms_with_vdw_overlap(ja, atom_coords, r_vdw):
vjk = r_vdw[ja] * coords_1sph + atom_coords[ja] - atom_coords[ka]
tjk = lib.norm(vjk, axis=1) / r_vdw[ka]
wjk = pcmobj.regularize_xt(tjk, eta, r_vdw[ka])
wjk *= part_weights
pol = sph.multipoles(vjk, lmax)
p1 = 0
for l in range(lmax+1):
fac = 4*numpy.pi/(l*2+1) / r_vdw[ka]**(l+1)
p0, p1 = p1, p1 + (l*2+1)
a = numpy.einsum('xn,n,mn->xm', ylm_1sph, wjk, pol[l])
Lmat[ja,:,ka,p0:p1] += -fac * a
return Lmat
def cache_fake_multipoles(grids, r_vdw, lmax):
# For each type of atoms, cache the product of last two terms in
# JCP, 141, 184108, Eq (31):
# x_{<}^{l} / x_{>}^{l+1} Y_l^m
mol = grids.mol
atom_grids_tab = grids.gen_atomic_grids(mol)
r_vdw_type = {}
for ia in range(mol.natm):
symb = mol.atom_symbol(ia)
if symb not in r_vdw_type:
r_vdw_type[symb] = r_vdw[ia]
cached_pol = {}
for symb in atom_grids_tab:
x_nj, w = atom_grids_tab[symb]
r = lib.norm(x_nj, axis=1)
# Different equations are used in JCTC, 9, 3637. r*Ys (the fake_pole)
# is computed as r^l/r_vdw. "leak_idx" is not needed.
# Here, the implementation is based on JCP, 141, 184108
leak_idx = r > r_vdw_type[symb]
pol = sph.multipoles(x_nj, lmax)
fak_pol = []
for l in range(lmax + 1):
# x_{<}^{l} / x_{>}^{l+1} Y_l^m in JCP, 141, 184108, Eq (31)
#:Ys = sph.real_sph_vec(x_nj/r.reshape(-1,1), lmax, True)
#:rr = numpy.zeros_like(r)
#:rr[r<=r_vdw[ia]] = r[r<=r_vdw[ia]]**l / r_vdw[ia]**(l+1)
#:rr[r> r_vdw[ia]] = r_vdw[ia]**l / r[r>r_vdw[ia]]**(l+1)
#:xx_ylm = numpy.einsum('n,mn->mn', rr, Ys[l])
xx_ylm = pol[l] * (1. / r_vdw_type[symb]**(l + 1))
# The line below is not needed for JCTC, 9, 3637
xx_ylm[:,
leak_idx] *= (r_vdw_type[symb] / r[leak_idx])**(2 * l + 1)
fak_pol.append(xx_ylm)
cached_pol[symb] = (fak_pol, leak_idx)
return cached_pol
def make_A(pcmobj, r_vdw, ylm_1sph, ui):
# Part of A matrix defined in JCP, 144, 054101, Eq (43), (44)
mol = pcmobj.mol
natm = mol.natm
lmax = pcmobj.lmax
eta = pcmobj.eta
nlm = (lmax + 1)**2
coords_1sph, weights_1sph = ddcosmo.make_grids_one_sphere(
pcmobj.lebedev_order)
ngrid_1sph = weights_1sph.size
atom_coords = mol.atom_coords()
ylm_1sph = ylm_1sph.reshape(nlm, ngrid_1sph)
Amat = numpy.zeros((natm, nlm, natm, nlm))
for ja in range(natm):
# w_u = precontract w_n U_j
w_u = weights_1sph * ui[ja]
p1 = 0
for l in range(lmax + 1):
fac = 2 * numpy.pi / (l * 2 + 1)
p0, p1 = p1, p1 + (l * 2 + 1)
a = numpy.einsum('xn,n,mn->xm', ylm_1sph, w_u, ylm_1sph[p0:p1])
Amat[ja, :, ja, p0:p1] += -fac * a
for ka in ddcosmo.atoms_with_vdw_overlap(ja, atom_coords, r_vdw):
vjk = r_vdw[ja] * coords_1sph + atom_coords[ja] - atom_coords[ka]
rjk = lib.norm(vjk, axis=1)
pol = sph.multipoles(vjk, lmax)
p1 = 0
weights = w_u / rjk**(l * 2 + 1)
for l in range(lmax + 1):
fac = 4 * numpy.pi * l / (l * 2 + 1) * r_vdw[ka]**(l + 1)
p0, p1 = p1, p1 + (l * 2 + 1)
a = numpy.einsum('xn,n,mn->xm', ylm_1sph, weights, pol[l])
Amat[ja, :, ka, p0:p1] += -fac * a
return Amat
def cache_fake_multipoles(grids, r_vdw, lmax):
# For each type of atoms, cache the product of last two terms in
# JCP, 141, 184108, Eq (31):
# x_{<}^{l} / x_{>}^{l+1} Y_l^m
mol = grids.mol
atom_grids_tab = grids.gen_atomic_grids(mol)
r_vdw_type = {}
for ia in range(mol.natm):
symb = mol.atom_symbol(ia)
if symb not in r_vdw_type:
r_vdw_type[symb] = r_vdw[ia]
cached_pol = {}
for symb in atom_grids_tab:
x_nj, w = atom_grids_tab[symb]
r = lib.norm(x_nj, axis=1)
# Different equations are used in JCTC, 9, 3637. r*Ys (the fake_pole)
# is computed as r^l/r_vdw. "leak_idx" is not needed.
# Here, the implementation is based on JCP, 141, 184108
leak_idx = r > r_vdw_type[symb]
pol = sph.multipoles(x_nj, lmax)
fak_pol = []
for l in range(lmax+1):
# x_{<}^{l} / x_{>}^{l+1} Y_l^m in JCP, 141, 184108, Eq (31)
#:Ys = sph.real_sph_vec(x_nj/r.reshape(-1,1), lmax, True)
#:rr = numpy.zeros_like(r)
#:rr[r<=r_vdw[ia]] = r[r<=r_vdw[ia]]**l / r_vdw[ia]**(l+1)
#:rr[r> r_vdw[ia]] = r_vdw[ia]**l / r[r>r_vdw[ia]]**(l+1)
#:xx_ylm = numpy.einsum('n,mn->mn', rr, Ys[l])
xx_ylm = pol[l] * (1./r_vdw_type[symb]**(l+1))
# The line below is not needed for JCTC, 9, 3637
xx_ylm[:,leak_idx] *= (r_vdw_type[symb]/r[leak_idx])**(2*l+1)
fak_pol.append(xx_ylm)
cached_pol[symb] = (fak_pol, leak_idx)
return cached_pol
def make_A(pcmobj, r_vdw, ylm_1sph, ui):
# Part of A matrix defined in JCP, 144, 054101, Eq (43), (44)
mol = pcmobj.mol
natm = mol.natm
lmax = pcmobj.lmax
eta = pcmobj.eta
nlm = (lmax+1)**2
coords_1sph, weights_1sph = ddcosmo.make_grids_one_sphere(pcmobj.lebedev_order)
ngrid_1sph = weights_1sph.size
atom_coords = mol.atom_coords()
ylm_1sph = ylm_1sph.reshape(nlm,ngrid_1sph)
Amat = numpy.zeros((natm,nlm,natm,nlm))
for ja in range(natm):
# w_u = precontract w_n U_j
w_u = weights_1sph * ui[ja]
p1 = 0
for l in range(lmax+1):
fac = 2*numpy.pi/(l*2+1)
p0, p1 = p1, p1 + (l*2+1)
a = numpy.einsum('xn,n,mn->xm', ylm_1sph, w_u, ylm_1sph[p0:p1])
Amat[ja,:,ja,p0:p1] += -fac * a
for ka in ddcosmo.atoms_with_vdw_overlap(ja, atom_coords, r_vdw):
vjk = r_vdw[ja] * coords_1sph + atom_coords[ja] - atom_coords[ka]
rjk = lib.norm(vjk, axis=1)
pol = sph.multipoles(vjk, lmax)
p1 = 0
weights = w_u / rjk**(l*2+1)
for l in range(lmax+1):
fac = 4*numpy.pi*l/(l*2+1) * r_vdw[ka]**(l+1)
p0, p1 = p1, p1 + (l*2+1)
a = numpy.einsum('xn,n,mn->xm', ylm_1sph, weights, pol[l])
Amat[ja,:,ka,p0:p1] += -fac * a
return Amat