#!/usr/bin/env python
# Copyright 2014-2020 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: Qiming Sun <osirpt.sun@gmail.com>
#
'''
Integral transformation with analytic Fourier transformation
'''
import numpy
from pyscf import lib
from pyscf import ao2mo
from pyscf.ao2mo import _ao2mo
from pyscf.ao2mo.incore import iden_coeffs, _conc_mos
from pyscf.pbc.df.df_jk import zdotNC
from pyscf.pbc.df.fft_ao2mo import _format_kpts, _iskconserv
from pyscf.pbc.df.df_ao2mo import _mo_as_complex, _dtrans, _ztrans
from pyscf.pbc.df.df_ao2mo import warn_pbc2d_eri
from pyscf.pbc.lib import kpts_helper
from pyscf.pbc.lib.kpts_helper import is_zero, gamma_point, unique
from pyscf import __config__
[docs]
def get_eri(mydf, kpts=None,
compact=getattr(__config__, 'pbc_df_ao2mo_get_eri_compact', True)):
cell = mydf.cell
nao = cell.nao_nr()
kptijkl = _format_kpts(kpts)
if not _iskconserv(cell, kptijkl):
lib.logger.warn(cell, 'aft_ao2mo: momentum conservation not found in '
'the given k-points %s', kptijkl)
return numpy.zeros((nao,nao,nao,nao))
kpti, kptj, kptk, kptl = kptijkl
q = kptj - kpti
mesh = mydf.mesh
coulG = mydf.weighted_coulG(q, False, mesh)
nao_pair = nao * (nao+1) // 2
max_memory = max(2000, (mydf.max_memory - lib.current_memory()[0]) * .8)
####################
# gamma point, the integral is real and with s4 symmetry
if gamma_point(kptijkl):
eriR = numpy.zeros((nao_pair,nao_pair))
for pqkR, pqkI, p0, p1 \
in mydf.pw_loop(mesh, kptijkl[:2], q, max_memory=max_memory,
aosym='s2'):
lib.ddot(pqkR*coulG[p0:p1], pqkR.T, 1, eriR, 1)
lib.ddot(pqkI*coulG[p0:p1], pqkI.T, 1, eriR, 1)
pqkR = pqkI = None
if not compact:
eriR = ao2mo.restore(1, eriR, nao).reshape(nao**2,-1)
return eriR
####################
# (kpt) i == j == k == l != 0
# (kpt) i == l && j == k && i != j && j != k =>
#
# complex integrals, N^4 elements
elif is_zero(kpti-kptl) and is_zero(kptj-kptk):
eriR = numpy.zeros((nao**2,nao**2))
eriI = numpy.zeros((nao**2,nao**2))
for pqkR, pqkI, p0, p1 \
in mydf.pw_loop(mesh, kptijkl[:2], q, max_memory=max_memory):
# rho_pq(G+k_pq) * conj(rho_rs(G-k_rs))
zdotNC(pqkR*coulG[p0:p1], pqkI*coulG[p0:p1], pqkR.T, pqkI.T,
1, eriR, eriI, 1)
pqkR = pqkI = None
pqkR = pqkI = coulG = None
# transpose(0,1,3,2) because
# j == k && i == l =>
# (L|ij).transpose(0,2,1).conj() = (L^*|ji) = (L^*|kl) => (M|kl)
# rho_rs(-G+k_rs) = conj(transpose(rho_sr(G+k_sr), (0,2,1)))
eri = lib.transpose((eriR+eriI*1j).reshape(-1,nao,nao), axes=(0,2,1))
return eri.reshape(nao**2,-1)
####################
# aosym = s1, complex integrals
#
# If kpti == kptj, (kptl-kptk)*a has to be multiples of 2pi because of the wave
# vector symmetry. k is a fraction of reciprocal basis, 0 < k/b < 1, by definition.
# So kptl/b - kptk/b must be -1 < k/b < 1. => kptl == kptk
#
else:
eriR = numpy.zeros((nao**2,nao**2))
eriI = numpy.zeros((nao**2,nao**2))
#
# (pq|rs) = \sum_G 4\pi rho_pq rho_rs / |G+k_{pq}|^2
# rho_pq = 1/N \sum_{Tp,Tq} \int exp(-i(G+k_{pq})*r) p(r-Tp) q(r-Tq) dr
# = \sum_{Tq} exp(i k_q*Tq) \int exp(-i(G+k_{pq})*r) p(r) q(r-Tq) dr
# Note the k-point wrap-around for rho_rs, which leads to G+k_{pq} in FT
# rho_rs = 1/N \sum_{Tr,Ts} \int exp( i(G+k_{pq})*r) r(r-Tr) s(r-Ts) dr
# = \sum_{Ts} exp(i k_s*Ts) \int exp( i(G+k_{pq})*r) r(r) s(r-Ts) dr
# rho_pq can be directly evaluated by AFT (function pw_loop)
# rho_pq = pw_loop(k_q, G+k_{pq})
# Assuming r(r) and s(r) are real functions, rho_rs is evaluated
# rho_rs = 1/N \sum_{Tr,Ts} \int exp( i(G+k_{pq})*r) r(r-Tr) s(r-Ts) dr
# = conj(\sum_{Ts} exp(-i k_s*Ts) \int exp(-i(G+k_{pq})*r) r(r) s(r-Ts) dr)
# = conj( pw_loop(-k_s, G+k_{pq}) )
#
# TODO: For complex AO function r(r) and s(r), pw_loop function needs to be
# extended to include Gv vector in the arguments
for (pqkR, pqkI, p0, p1), (rskR, rskI, q0, q1) in \
lib.izip(mydf.pw_loop(mesh, kptijkl[:2], q, max_memory=max_memory*.5),
mydf.pw_loop(mesh,-kptijkl[2:], q, max_memory=max_memory*.5)):
pqkR *= coulG[p0:p1]
pqkI *= coulG[p0:p1]
zdotNC(pqkR, pqkI, rskR.T, rskI.T, 1, eriR, eriI, 1)
pqkR = pqkI = rskR = rskI = None
return (eriR+eriI*1j)
[docs]
def general(mydf, mo_coeffs, kpts=None,
compact=getattr(__config__, 'pbc_df_ao2mo_general_compact', True)):
warn_pbc2d_eri(mydf)
cell = mydf.cell
kptijkl = _format_kpts(kpts)
kpti, kptj, kptk, kptl = kptijkl
if isinstance(mo_coeffs, numpy.ndarray) and mo_coeffs.ndim == 2:
mo_coeffs = (mo_coeffs,) * 4
if not _iskconserv(cell, kptijkl):
lib.logger.warn(cell, 'aft_ao2mo: momentum conservation not found in '
'the given k-points %s', kptijkl)
return numpy.zeros([mo.shape[1] for mo in mo_coeffs])
q = kptj - kpti
mesh = mydf.mesh
coulG = mydf.weighted_coulG(q, False, mesh)
all_real = not any(numpy.iscomplexobj(mo) for mo in mo_coeffs)
max_memory = max(2000, (mydf.max_memory - lib.current_memory()[0]) * .5)
####################
# gamma point, the integral is real and with s4 symmetry
if gamma_point(kptijkl) and all_real:
ijmosym, nij_pair, moij, ijslice = _conc_mos(mo_coeffs[0], mo_coeffs[1], compact)
klmosym, nkl_pair, mokl, klslice = _conc_mos(mo_coeffs[2], mo_coeffs[3], compact)
eri_mo = numpy.zeros((nij_pair,nkl_pair))
sym = (iden_coeffs(mo_coeffs[0], mo_coeffs[2]) and
iden_coeffs(mo_coeffs[1], mo_coeffs[3]))
ijR = ijI = klR = klI = buf = None
for pqkR, pqkI, p0, p1 \
in mydf.pw_loop(mesh, kptijkl[:2], q, max_memory=max_memory,
aosym='s2'):
buf = lib.transpose(pqkR, out=buf)
ijR, klR = _dtrans(buf, ijR, ijmosym, moij, ijslice,
buf, klR, klmosym, mokl, klslice, sym)
lib.ddot(ijR.T, klR*coulG[p0:p1,None], 1, eri_mo, 1)
buf = lib.transpose(pqkI, out=buf)
ijI, klI = _dtrans(buf, ijI, ijmosym, moij, ijslice,
buf, klI, klmosym, mokl, klslice, sym)
lib.ddot(ijI.T, klI*coulG[p0:p1,None], 1, eri_mo, 1)
pqkR = pqkI = None
return eri_mo
####################
# (kpt) i == j == k == l != 0
# (kpt) i == l && j == k && i != j && j != k =>
#
elif is_zero(kpti-kptl) and is_zero(kptj-kptk):
mo_coeffs = _mo_as_complex(mo_coeffs)
nij_pair, moij, ijslice = _conc_mos(mo_coeffs[0], mo_coeffs[1])[1:]
nlk_pair, molk, lkslice = _conc_mos(mo_coeffs[3], mo_coeffs[2])[1:]
eri_mo = numpy.zeros((nij_pair,nlk_pair), dtype=numpy.complex128)
sym = (iden_coeffs(mo_coeffs[0], mo_coeffs[3]) and
iden_coeffs(mo_coeffs[1], mo_coeffs[2]))
zij = zlk = buf = None
for pqkR, pqkI, p0, p1 \
in mydf.pw_loop(mesh, kptijkl[:2], q, max_memory=max_memory):
buf = lib.transpose(pqkR+pqkI*1j, out=buf)
zij, zlk = _ztrans(buf, zij, moij, ijslice,
buf, zlk, molk, lkslice, sym)
lib.dot(zij.T, zlk.conj()*coulG[p0:p1,None], 1, eri_mo, 1)
pqkR = pqkI = None
nmok = mo_coeffs[2].shape[1]
nmol = mo_coeffs[3].shape[1]
eri_mo = lib.transpose(eri_mo.reshape(-1,nmol,nmok), axes=(0,2,1))
return eri_mo.reshape(nij_pair,nlk_pair)
####################
# aosym = s1, complex integrals
#
# If kpti == kptj, (kptl-kptk)*a has to be multiples of 2pi because of the wave
# vector symmetry. k is a fraction of reciprocal basis, 0 < k/b < 1, by definition.
# So kptl/b - kptk/b must be -1 < k/b < 1. => kptl == kptk
#
else:
mo_coeffs = _mo_as_complex(mo_coeffs)
nij_pair, moij, ijslice = _conc_mos(mo_coeffs[0], mo_coeffs[1])[1:]
nkl_pair, mokl, klslice = _conc_mos(mo_coeffs[2], mo_coeffs[3])[1:]
eri_mo = numpy.zeros((nij_pair,nkl_pair), dtype=numpy.complex128)
tao = []
ao_loc = None
zij = zkl = buf = None
for (pqkR, pqkI, p0, p1), (rskR, rskI, q0, q1) in \
lib.izip(mydf.pw_loop(mesh, kptijkl[:2], q, max_memory=max_memory*.5),
mydf.pw_loop(mesh,-kptijkl[2:], q, max_memory=max_memory*.5)):
buf = lib.transpose(pqkR+pqkI*1j, out=buf)
zij = _ao2mo.r_e2(buf, moij, ijslice, tao, ao_loc, out=zij)
buf = lib.transpose(rskR-rskI*1j, out=buf)
zkl = _ao2mo.r_e2(buf, mokl, klslice, tao, ao_loc, out=zkl)
zij *= coulG[p0:p1,None]
lib.dot(zij.T, zkl, 1, eri_mo, 1)
pqkR = pqkI = rskR = rskI = None
return eri_mo
[docs]
def get_ao_pairs_G(mydf, kpts=numpy.zeros((2,3)), q=None, shls_slice=None,
compact=getattr(__config__, 'pbc_df_ao_pairs_compact', False)):
'''Calculate forward Fourier transform (G|ij) of all AO pairs.
Returns:
ao_pairs_G : 2D complex array
For gamma point, the shape is (ngrids, nao*(nao+1)/2); otherwise the
shape is (ngrids, nao*nao)
'''
if kpts is None: kpts = numpy.zeros((2,3))
cell = mydf.cell
kpts = numpy.asarray(kpts)
q = kpts[1] - kpts[0]
coords = cell.gen_uniform_grids(mydf.mesh)
ngrids = len(coords)
max_memory = max(2000, (mydf.max_memory - lib.current_memory()[0]) * .5)
if shls_slice is None:
shls_slice = (0, cell.nbas, 0, cell.nbas)
ish0, ish1, jsh0, jsh1 = shls_slice
ao_loc = cell.ao_loc_nr()
i0 = ao_loc[ish0]
i1 = ao_loc[ish1]
j0 = ao_loc[jsh0]
j1 = ao_loc[jsh1]
compact = compact and (i0 == j0) and (i1 == j1)
if compact and gamma_point(kpts): # gamma point
aosym = 's2'
else:
aosym = 's1'
ao_pairs_G = numpy.empty((ngrids,(i1-i0)*(j1-j0)), dtype=numpy.complex128)
for pqkR, pqkI, p0, p1 \
in mydf.pw_loop(mydf.mesh, kpts, q, shls_slice,
max_memory=max_memory, aosym=aosym):
ao_pairs_G[p0:p1] = pqkR.T + pqkI.T * 1j
return ao_pairs_G
[docs]
def get_mo_pairs_G(mydf, mo_coeffs, kpts=numpy.zeros((2,3)), q=None,
compact=getattr(__config__, 'pbc_df_mo_pairs_compact', False)):
'''Calculate forward fourier transform (G|ij) of all MO pairs.
Args:
mo_coeff: length-2 list of (nao,nmo) ndarrays
The two sets of MO coefficients to use in calculating the
product |ij).
Returns:
mo_pairs_G : (ngrids, nmoi*nmoj) ndarray
The FFT of the real-space MO pairs.
'''
if kpts is None: kpts = numpy.zeros((2,3))
cell = mydf.cell
kpts = numpy.asarray(kpts)
q = kpts[1] - kpts[0]
coords = cell.gen_uniform_grids(mydf.mesh)
nmoi = mo_coeffs[0].shape[1]
nmoj = mo_coeffs[1].shape[1]
ngrids = len(coords)
max_memory = max(2000, (mydf.max_memory - lib.current_memory()[0]) * .5)
mo_pairs_G = numpy.empty((ngrids,nmoi,nmoj), dtype=numpy.complex128)
nao = cell.nao
for pqkR, pqkI, p0, p1 \
in mydf.pw_loop(mydf.mesh, kpts, q,
max_memory=max_memory, aosym='s2'):
pqk = lib.unpack_tril(pqkR + pqkI*1j, axis=0).reshape(nao,nao,-1)
mo_pairs_G[p0:p1] = lib.einsum('pqk,pi,qj->kij', pqk, *mo_coeffs[:2])
return mo_pairs_G.reshape(ngrids,nmoi*nmoj)
[docs]
def ao2mo_7d(mydf, mo_coeff_kpts, kpts=None, factor=1, out=None):
cell = mydf.cell
if kpts is None:
kpts = mydf.kpts
nkpts = len(kpts)
if isinstance(mo_coeff_kpts, numpy.ndarray) and mo_coeff_kpts.ndim == 3:
mo_coeff_kpts = [mo_coeff_kpts] * 4
else:
mo_coeff_kpts = list(mo_coeff_kpts)
# Shape of the orbitals can be different on different k-points. The
# orbital coefficients must be formatted (padded by zeros) so that the
# shape of the orbital coefficients are the same on all k-points. This can
# be achieved by calling pbc.mp.kmp2.padded_mo_coeff function
nmoi, nmoj, nmok, nmol = [x.shape[2] for x in mo_coeff_kpts]
eri_shape = (nkpts, nkpts, nkpts, nmoi, nmoj, nmok, nmol)
if gamma_point(kpts):
dtype = numpy.result_type(*mo_coeff_kpts)
else:
dtype = numpy.complex128
if out is None:
out = numpy.empty(eri_shape, dtype=dtype)
else:
assert (out.shape == eri_shape)
kptij_lst = numpy.array([(ki, kj) for ki in kpts for kj in kpts])
kptis_lst = kptij_lst[:,0]
kptjs_lst = kptij_lst[:,1]
kpt_ji = kptjs_lst - kptis_lst
uniq_kpts, uniq_index, uniq_inverse = unique(kpt_ji)
ngrids = numpy.prod(mydf.mesh)
nao = cell.nao
max_memory = max(2000, mydf.max_memory-lib.current_memory()[0]-nao**4*16/1e6) * .5
# To hold intermediates
fswap = lib.H5TmpFile()
tao = []
ao_loc = None
kconserv = kpts_helper.get_kconserv(cell, kpts)
for uniq_id, kpt in enumerate(uniq_kpts):
q = uniq_kpts[uniq_id]
adapted_ji_idx = numpy.where(uniq_inverse == uniq_id)[0]
kptjs = kptjs_lst[adapted_ji_idx]
coulG = mydf.weighted_coulG(q, False, mydf.mesh)
coulG *= factor
moij_list = []
ijslice_list = []
for ji, ji_idx in enumerate(adapted_ji_idx):
ki = ji_idx // nkpts
kj = ji_idx % nkpts
moij, ijslice = _conc_mos(mo_coeff_kpts[0][ki], mo_coeff_kpts[1][kj])[2:]
moij_list.append(moij)
ijslice_list.append(ijslice)
fswap.create_dataset('zij/'+str(ji), (ngrids,nmoi*nmoj), 'D')
for aoaoks, p0, p1 in mydf.ft_loop(mydf.mesh, q, kptjs,
max_memory=max_memory):
for ji, aoao in enumerate(aoaoks):
ki = adapted_ji_idx[ji] // nkpts
kj = adapted_ji_idx[ji] % nkpts
buf = aoao.transpose(1,2,0).reshape(nao**2,p1-p0)
zij = _ao2mo.r_e2(lib.transpose(buf), moij_list[ji],
ijslice_list[ji], tao, ao_loc)
zij *= coulG[p0:p1,None]
fswap['zij/'+str(ji)][p0:p1] = zij
buf = zij = None
mokl_list = []
klslice_list = []
for kk in range(nkpts):
kl = kconserv[ki, kj, kk]
mokl, klslice = _conc_mos(mo_coeff_kpts[2][kk], mo_coeff_kpts[3][kl])[2:]
mokl_list.append(mokl)
klslice_list.append(klslice)
fswap.create_dataset('zkl/'+str(kk), (ngrids,nmok*nmol), 'D')
ki = adapted_ji_idx[0] // nkpts
kj = adapted_ji_idx[0] % nkpts
kptls = kpts[kconserv[ki, kj, :]]
for aoaoks, p0, p1 in mydf.ft_loop(mydf.mesh, q, -kptls,
max_memory=max_memory):
for kk, aoao in enumerate(aoaoks):
buf = aoao.conj().transpose(1,2,0).reshape(nao**2,p1-p0)
zkl = _ao2mo.r_e2(lib.transpose(buf), mokl_list[kk],
klslice_list[kk], tao, ao_loc)
fswap['zkl/'+str(kk)][p0:p1] = zkl
buf = zkl = None
for ji, ji_idx in enumerate(adapted_ji_idx):
ki = ji_idx // nkpts
kj = ji_idx % nkpts
for kk in range(nkpts):
zij = numpy.asarray(fswap['zij/'+str(ji)])
zkl = numpy.asarray(fswap['zkl/'+str(kk)])
tmp = lib.dot(zij.T, zkl)
if dtype == numpy.double:
tmp = tmp.real
out[ki,kj,kk] = tmp.reshape(eri_shape[3:])
del (fswap['zij'])
del (fswap['zkl'])
return out
if __name__ == '__main__':
from pyscf.pbc import gto as pgto
from pyscf.pbc.df import AFTDF
L = 5.
n = 11
cell = pgto.Cell()
cell.a = numpy.diag([L,L,L])
cell.mesh = numpy.array([n,n,n])
cell.atom = '''He 3. 2. 3.
He 1. 1. 1.'''
#cell.basis = {'He': [[0, (1.0, 1.0)]]}
#cell.basis = '631g'
#cell.basis = {'He': [[0, (2.4, 1)], [1, (1.1, 1)]]}
cell.basis = 'ccpvdz'
cell.verbose = 0
cell.build(0,0)
nao = cell.nao_nr()
numpy.random.seed(1)
kpts = numpy.random.random((4,3))
kpts[3] = -numpy.einsum('ij->j', kpts[:3])
with_df = AFTDF(cell, kpts)
with_df.mesh = [n] * 3
mo =(numpy.random.random((nao,nao)) +
numpy.random.random((nao,nao))*1j)
eri = with_df.get_eri(kpts).reshape((nao,)*4)
eri0 = numpy.einsum('pjkl,pi->ijkl', eri , mo.conj())
eri0 = numpy.einsum('ipkl,pj->ijkl', eri0, mo )
eri0 = numpy.einsum('ijpl,pk->ijkl', eri0, mo.conj())
eri0 = numpy.einsum('ijkp,pl->ijkl', eri0, mo )
eri1 = with_df.ao2mo(mo, kpts)
print(abs(eri1-eri0).sum())
