#!/usr/bin/env python
# Copyright 2014-2018,2021 The PySCF Developers. All Rights Reserved.
#
# Licensed under the Apache License, Version 2.0 (the "License");
# you may not use this file except in compliance with the License.
# You may obtain a copy of the License at
#
# http://www.apache.org/licenses/LICENSE-2.0
#
# Unless required by applicable law or agreed to in writing, software
# distributed under the License is distributed on an "AS IS" BASIS,
# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
# See the License for the specific language governing permissions and
# limitations under the License.
#
# Author: Timothy Berkelbach <tim.berkelbach@gmail.com>
#
'''PP with numeric integration. See also pyscf/pbc/gto/pesudo/pp_int.py
For GTH/HGH PPs, see:
Goedecker, Teter, Hutter, PRB 54, 1703 (1996)
Hartwigsen, Goedecker, and Hutter, PRB 58, 3641 (1998)
'''
import numpy as np
import scipy.linalg
import scipy.special
from pyscf import lib
from pyscf.gto import mole
from pyscf.pbc.gto.pseudo import pp_int
[docs]
def get_alphas(cell):
'''alpha parameters from the non-divergent Hartree+Vloc G=0 term.
See ewald.pdf
Returns:
alphas : (natm,) ndarray
'''
return get_alphas_gth(cell)
[docs]
def get_alphas_gth(cell):
'''alpha parameters for the local GTH pseudopotential.'''
G0 = np.zeros((1,3))
return -get_gth_vlocG(cell, G0)
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def get_vlocG(cell, Gv=None):
'''Local PP kernel in G space: Vloc(G)
Returns:
(natm, ngrids) ndarray
'''
if Gv is None: Gv = cell.Gv
vlocG = get_gth_vlocG(cell, Gv)
return vlocG
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def get_gth_vlocG(cell, Gv):
'''Local part of the GTH pseudopotential.
See MH (4.79).
Args:
Gv : (ngrids,3) ndarray
Returns:
(natm, ngrids) ndarray
'''
vlocG = pp_int.get_gth_vlocG_part1(cell, Gv)
# Add the C1, C2, C3, C4 contributions
G2 = np.einsum('ix,ix->i', Gv, Gv)
for ia in range(cell.natm):
symb = cell.atom_symbol(ia)
if symb not in cell._pseudo:
continue
pp = cell._pseudo[symb]
rloc, nexp, cexp = pp[1:3+1]
G2_red = G2 * rloc**2
cfacs = 0
if nexp >= 1:
cfacs += cexp[0]
if nexp >= 2:
cfacs += cexp[1] * (3 - G2_red)
if nexp >= 3:
cfacs += cexp[2] * (15 - 10*G2_red + G2_red**2)
if nexp >= 4:
cfacs += cexp[3] * (105 - 105*G2_red + 21*G2_red**2 - G2_red**3)
vlocG[ia,:] -= (2*np.pi)**(3/2.)*rloc**3*np.exp(-0.5*G2_red) * cfacs
return vlocG
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def get_projG(cell, kpt=np.zeros(3)):
'''PP weight and projector for the nonlocal PP in G space.
Returns:
hs : list( list( np.array( , ) ) )
- hs[atm][l][i,j]
projs : list( list( list( list( np.array(ngrids) ) ) ) )
- projs[atm][l][m][i][ngrids]
'''
return get_gth_projG(cell, kpt+cell.Gv)
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def get_gth_projG(cell, Gvs):
r'''G space projectors from the FT of the real-space projectors.
\int e^{iGr} p_j^l(r) Y_{lm}^*(theta,phi)
= i^l p_j^l(G) Y_{lm}^*(thetaG, phiG)
See MH Eq.(4.80)
'''
Gs,thetas,phis = cart2polar(Gvs)
hs = []
projs = []
for ia in range(cell.natm):
symb = cell.atom_symbol(ia)
pp = cell._pseudo[symb]
# nproj_types = pp[4]
h_ia = []
proj_ia = []
for l,proj in enumerate(pp[5:]):
rl, nl, hl = proj
h_ia.append( np.array(hl) )
proj_ia_l = []
for m in range(-l,l+1):
projG_ang = Ylm(l,m,thetas,phis).conj()
proj_ia_lm = []
for i in range(nl):
projG_radial = projG_li(Gs,l,i,rl)
proj_ia_lm.append( (1j)**l * projG_radial*projG_ang )
proj_ia_l.append(proj_ia_lm)
proj_ia.append(proj_ia_l)
hs.append(h_ia)
projs.append(proj_ia)
return hs, projs
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def projG_li(G, l, i, rl):
G = np.array(G)
G_red = G*rl
# MH Eq. (4.81)
return (_qli(G_red,l,i) * np.pi**(5/4.) * G**l * np.sqrt(rl**(2*l+3))
/ np.exp(0.5*G_red**2) )
def _qli(x,l,i):
# MH Eqs. (4.82)-(4.93) :: beware typos!
# Mathematica formulas:
# p[l_, i_, r_] = Sqrt[2] r^(l + 2 (i - 1)) Exp[-r^2/(2 R^2)]/(R^(l + (4 i - 1)/2) Sqrt[Gamma[l + (4 i - 1)/2]])
# pG[l_, i_, G_] = Integrate[p[l, i, r] 4 Pi r^2 SphericalBesselJ[l, G r], {r, 0, Infinity}]
# qG[l_, i_, G_] := pG[l, i, G]/(Pi^(5/4) G^l Sqrt[R^(2 l + 3)]/Exp[(G R)^2/2])
# FullSimplify[qG[4, 3, G], R > 0 && G > 0]
sqrt = np.sqrt
if l==0 and i==0:
return 4*sqrt(2.)
elif l==0 and i==1:
return 8*sqrt(2/15.)*(3-x**2) # MH & GTH (right)
#return sqrt(8*2/15.)*(3-x**2) # HGH (wrong)
elif l==0 and i==2:
#return 16/3.*sqrt(2/105.)*(15-20*x**2+4*x**4) # MH (wrong)
return 16/3.*sqrt(2/105.)*(15-10*x**2+x**4) # HGH (right)
elif l==1 and i==0:
return 8*sqrt(1/3.)
elif l==1 and i==1:
return 16*sqrt(1/105.)*(5-x**2)
elif l==1 and i==2:
#return 32/3.*sqrt(1/1155.)*(35-28*x**2+4*x**4) # MH (wrong)
return 32/3.*sqrt(1/1155.)*(35-14*x**2+x**4) # HGH (right)
elif l==2 and i==0:
return 8*sqrt(2/15.)
elif l==2 and i==1:
return 16/3.*sqrt(2/105.)*(7-x**2)
elif l==2 and i==2:
#return 32/3.*sqrt(2/15015.)*(63-36*x**2+4*x**4) # MH (wrong I think)
return 32/3.*sqrt(2/15015.)*(63-18*x**2+x**4) # TCB
elif l==3 and i==0:
return 16*sqrt(1/105.)
elif l==3 and i==1:
return 32/3.*sqrt(1/1155.)*(9-x**2)
elif l==3 and i==2:
return 64/45.*sqrt(1/1001.)*(99-22*x**2+x**4)
elif l==4 and i==0:
return 16/3.*sqrt(2/105.)
elif l==4 and i==1:
return 32/3.*sqrt(2/15015.)*(11-x**2)
elif l==4 and i==2:
return 64/45.*sqrt(2/17017.)*(143-26*x**2+x**4)
else:
print("*** WARNING *** l =", l, ", i =", i, "not yet implemented for NL PP!")
return 0.
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def Ylm_real(l,m,theta,phi):
'''Real spherical harmonics, if desired.'''
Ylabsm = Ylm(l,np.abs(m),theta,phi)
if m < 0:
return np.sqrt(2.) * Ylabsm.imag
elif m > 0:
return np.sqrt(2.) * Ylabsm.real
else: # m == 0
return Ylabsm.real
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def Ylm(l,m,theta,phi):
'''
Spherical harmonics; returns a complex number
Note the "convention" for theta and phi:
http://docs.scipy.org/doc/scipy-0.14.0/reference/generated/scipy.special.sph_harm.html
'''
#return scipy.special.sph_harm(m=m,n=l,theta=phi,phi=theta)
return scipy.special.sph_harm(m,l,phi,theta)
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def cart2polar(rvec):
# The rows of rvec are the 3-component vectors
# i.e. rvec is N x 3
x,y,z = rvec.T
r = lib.norm(rvec, axis=1)
# theta is the polar angle, 0 < theta < pi
# catch possible 0/0
theta = np.arccos(z/(r+1e-200))
# phi is the azimuthal angle, 0 < phi < 2pi (or -pi < phi < pi)
phi = np.arctan2(y,x)
return r, theta, phi
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def get_pp(cell, kpt=np.zeros(3)):
'''Get the periodic pseudotential nuc-el AO matrix
'''
from pyscf.pbc import tools
coords = cell.get_uniform_grids()
aoR = cell.pbc_eval_gto('GTOval', coords, kpt=kpt)
nao = cell.nao_nr()
SI = cell.get_SI()
vlocG = get_vlocG(cell)
vpplocG = -np.sum(SI * vlocG, axis=0)
vpplocG[0] = np.sum(get_alphas(cell)) # from get_jvloc_G0 function
# vpploc evaluated in real-space
vpplocR = tools.ifft(vpplocG, cell.mesh).real
vpploc = np.dot(aoR.T.conj(), vpplocR.reshape(-1,1)*aoR)
# vppnonloc evaluated in reciprocal space
aokG = tools.fftk(np.asarray(aoR.T, order='C'),
cell.mesh, np.exp(-1j*np.dot(coords, kpt))).T
ngrids = len(aokG)
fakemol = mole.Mole()
fakemol._atm = np.zeros((1,mole.ATM_SLOTS), dtype=np.int32)
fakemol._bas = np.zeros((1,mole.BAS_SLOTS), dtype=np.int32)
ptr = mole.PTR_ENV_START
fakemol._env = np.zeros(ptr+10)
fakemol._bas[0,mole.NPRIM_OF ] = 1
fakemol._bas[0,mole.NCTR_OF ] = 1
fakemol._bas[0,mole.PTR_EXP ] = ptr+3
fakemol._bas[0,mole.PTR_COEFF] = ptr+4
Gv = np.asarray(cell.Gv+kpt)
G_rad = lib.norm(Gv, axis=1)
vppnl = np.zeros((nao,nao), dtype=np.complex128)
for ia in range(cell.natm):
symb = cell.atom_symbol(ia)
if symb not in cell._pseudo:
continue
pp = cell._pseudo[symb]
for l, proj in enumerate(pp[5:]):
rl, nl, hl = proj
if nl > 0:
hl = np.asarray(hl)
fakemol._bas[0,mole.ANG_OF] = l
fakemol._env[ptr+3] = .5*rl**2
fakemol._env[ptr+4] = rl**(l+1.5)*np.pi**1.25
pYlm_part = fakemol.eval_gto('GTOval', Gv)
pYlm = np.empty((nl,l*2+1,ngrids))
for k in range(nl):
qkl = _qli(G_rad*rl, l, k)
pYlm[k] = pYlm_part.T * qkl
# pYlm is real
SPG_lmi = np.einsum('g,nmg->nmg', SI[ia].conj(), pYlm)
SPG_lm_aoG = np.einsum('nmg,gp->nmp', SPG_lmi, aokG)
tmp = np.einsum('ij,jmp->imp', hl, SPG_lm_aoG)
vppnl += np.einsum('imp,imq->pq', SPG_lm_aoG.conj(), tmp)
vppnl *= (1./ngrids**2)
if aoR.dtype == np.double:
return vpploc.real + vppnl.real
else:
return vpploc + vppnl
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def get_jvloc_G0(cell, kpt=np.zeros(3)):
'''Get the (separately divergent) Hartree + Vloc G=0 contribution.
'''
ovlp = cell.pbc_intor('int1e_ovlp', hermi=1, kpts=kpt)
return 1./cell.vol * np.sum(get_alphas(cell)) * ovlp