#!/usr/bin/env python
# Copyright 2017-2021 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.
#
# Authors: James D. McClain <jmcclain@princeton.edu>
#
"""Module for running k-point ccsd(t)"""
import numpy as np
import pyscf.pbc.cc.kccsd
from pyscf import lib
from pyscf.lib import logger
from pyscf.lib.misc import tril_product
from pyscf.pbc import scf
from pyscf.pbc.lib import kpts_helper
from pyscf.lib.numpy_helper import cartesian_prod
from pyscf.lib.parameters import LOOSE_ZERO_TOL, LARGE_DENOM # noqa
from pyscf.pbc.cc.kccsd_t_rhf import _get_epqr
from pyscf.pbc.mp.kmp2 import (get_frozen_mask, get_nocc, get_nmo,
padded_mo_coeff, padding_k_idx) # noqa
#einsum = np.einsum
einsum = lib.einsum
# CCSD(T) equations taken from Tu, Yang, Wang, and Guo JPC (135), 2011
#
# There are some complex conjugates not included in the equations
# by Watts, Gauss, Bartlett JCP (98), 1993
[docs]
def kernel(mycc, eris, t1=None, t2=None, max_memory=2000, verbose=logger.INFO):
'''Returns the CCSD(T) for general spin-orbital integrals with k-points.
Note:
Returns real part of the CCSD(T) energy, raises warning if there is
a complex part.
Args:
mycc (:class:`GCCSD`): Coupled-cluster object storing results of
a coupled-cluster calculation.
eris (:class:`_ERIS`): Integral object holding the relevant electron-
repulsion integrals and Fock matrix elements
t1 (:obj:`ndarray`): t1 coupled-cluster amplitudes
t2 (:obj:`ndarray`): t2 coupled-cluster amplitudes
max_memory (float): Maximum memory used in calculation (NOT USED)
verbose (int, :class:`Logger`): verbosity of calculation
Returns:
energy_t (float): The real-part of the k-point CCSD(T) energy.
'''
assert isinstance(mycc, pyscf.pbc.cc.kccsd.GCCSD)
if isinstance(verbose, logger.Logger):
log = verbose
else:
log = logger.Logger(mycc.stdout, verbose)
if t1 is None: t1 = mycc.t1
if t2 is None: t2 = mycc.t2
if t1 is None or t2 is None:
raise TypeError('Must pass in t1/t2 amplitudes to k-point CCSD(T)! (Maybe '
'need to run `.ccsd()` on the ccsd object?)')
cell = mycc._scf.cell
kpts = mycc.kpts
# The dtype of any local arrays that will be created
dtype = t1.dtype
nkpts, nocc, nvir = t1.shape
mo_energy_occ = [eris.mo_energy[ki][:nocc] for ki in range(nkpts)]
mo_energy_vir = [eris.mo_energy[ki][nocc:] for ki in range(nkpts)]
fov = eris.fock[:, :nocc, nocc:]
mo_e_o = mo_energy_occ
mo_e_v = mo_energy_vir
# Set up class for k-point conservation
kconserv = kpts_helper.get_kconserv(cell, kpts)
# Get location of padded elements in occupied and virtual space
nonzero_opadding, nonzero_vpadding = padding_k_idx(mycc, kind="split")
energy_t = 0.0
for ki in range(nkpts):
for kj in range(ki + 1):
for kk in range(kj + 1):
# eigenvalue denominator: e(i) + e(j) + e(k)
eijk = _get_epqr([0,nocc,ki,mo_e_o,nonzero_opadding],
[0,nocc,kj,mo_e_o,nonzero_opadding],
[0,nocc,kk,mo_e_o,nonzero_opadding])
# Factors to include for permutational symmetry among k-points for occupied space
if ki == kj and kj == kk:
symm_ijk_kpt = 1. # only one degeneracy
elif ki == kj or kj == kk:
symm_ijk_kpt = 3. # 3 degeneracies when only one k-point is unique
else:
symm_ijk_kpt = 6. # 3! combinations of arranging 3 distinct k-points
for ka in range(nkpts):
for kb in range(ka + 1):
# Find momentum conservation condition for triples
# amplitude t3ijkabc
kc = kpts_helper.get_kconserv3(cell, kpts, [ki, kj, kk, ka, kb])
if kc not in range(kb + 1):
continue
# -1.0 so the LARGE_DENOM does not cancel with the one from eijk
eabc = _get_epqr([0,nvir,ka,mo_e_v,nonzero_vpadding],
[0,nvir,kb,mo_e_v,nonzero_vpadding],
[0,nvir,kc,mo_e_v,nonzero_vpadding],
fac=[-1.,-1.,-1.])
# Factors to include for permutational symmetry among k-points for virtual space
if ka == kb and kb == kc:
symm_abc_kpt = 1. # one unique combination of (ka, kb, kc)
elif ka == kb or kb == kc:
symm_abc_kpt = 3. # 3 unique combinations of (ka, kb, kc)
else:
symm_abc_kpt = 6. # 6 unique combinations of (ka, kb, kc)
# Determine the a, b, c indices we will loop over as
# determined by the k-point symmetry.
abc_indices = cartesian_prod([range(nvir)] * 3)
symm_3d = symm_2d_ab = symm_2d_bc = False
if ka == kc: # == kb from lower triangular summation
abc_indices = tril_product(range(nvir), repeat=3, tril_idx=[0, 1, 2]) # loop a >= b >= c
symm_3d = True
elif ka == kb:
abc_indices = tril_product(range(nvir), repeat=3, tril_idx=[0, 1]) # loop a >= b
symm_2d_ab = True
elif kb == kc:
abc_indices = tril_product(range(nvir), repeat=3, tril_idx=[1, 2]) # loop b >= c
symm_2d_bc = True
for a, b, c in abc_indices:
# See symm_3d and abc_indices above for description of factors
symm_abc = 1.
if symm_3d:
if a == b == c:
symm_abc = 1.
elif a == b or b == c:
symm_abc = 3.
else:
symm_abc = 6.
elif symm_2d_ab:
if a == b:
symm_abc = 1.
else:
symm_abc = 2.
elif symm_2d_bc:
if b == c:
symm_abc = 1.
else:
symm_abc = 2.
# Form energy denominator
eijkabc = (eijk[:,:,:] + eabc[a,b,c])
# Form connected triple excitation amplitude
t3c = np.zeros((nocc, nocc, nocc), dtype=dtype)
# First term: 1 - p(ij) - p(ik)
ke = kconserv[kj, ka, kk]
t3c = t3c + einsum('jke,ie->ijk', t2[kj, kk, ka, :, :, a, :], -eris.ovvv[ki, ke, kc, :, :, c, b].conj())
ke = kconserv[ki, ka, kk]
t3c = t3c - einsum('ike,je->ijk', t2[ki, kk, ka, :, :, a, :], -eris.ovvv[kj, ke, kc, :, :, c, b].conj())
ke = kconserv[kj, ka, ki]
t3c = t3c - einsum('jie,ke->ijk', t2[kj, ki, ka, :, :, a, :], -eris.ovvv[kk, ke, kc, :, :, c, b].conj())
km = kconserv[kb, ki, kc]
t3c = t3c - einsum('mi,jkm->ijk', t2[km, ki, kb, :, :, b, c], eris.ooov[kj, kk, km, :, :, :, a].conj())
km = kconserv[kb, kj, kc]
t3c = t3c + einsum('mj,ikm->ijk', t2[km, kj, kb, :, :, b, c], eris.ooov[ki, kk, km, :, :, :, a].conj())
km = kconserv[kb, kk, kc]
t3c = t3c + einsum('mk,jim->ijk', t2[km, kk, kb, :, :, b, c], eris.ooov[kj, ki, km, :, :, :, a].conj())
# Second term: - p(ab) + p(ab) p(ij) + p(ab) p(ik)
ke = kconserv[kj, kb, kk]
t3c = t3c - einsum('jke,ie->ijk', t2[kj, kk, kb, :, :, b, :], -eris.ovvv[ki, ke, kc, :, :, c, a].conj())
ke = kconserv[ki, kb, kk]
t3c = t3c + einsum('ike,je->ijk', t2[ki, kk, kb, :, :, b, :], -eris.ovvv[kj, ke, kc, :, :, c, a].conj())
ke = kconserv[kj, kb, ki]
t3c = t3c + einsum('jie,ke->ijk', t2[kj, ki, kb, :, :, b, :], -eris.ovvv[kk, ke, kc, :, :, c, a].conj())
km = kconserv[ka, ki, kc]
t3c = t3c + einsum('mi,jkm->ijk', t2[km, ki, ka, :, :, a, c], eris.ooov[kj, kk, km, :, :, :, b].conj())
km = kconserv[ka, kj, kc]
t3c = t3c - einsum('mj,ikm->ijk', t2[km, kj, ka, :, :, a, c], eris.ooov[ki, kk, km, :, :, :, b].conj())
km = kconserv[ka, kk, kc]
t3c = t3c - einsum('mk,jim->ijk', t2[km, kk, ka, :, :, a, c], eris.ooov[kj, ki, km, :, :, :, b].conj())
# Third term: - p(ac) + p(ac) p(ij) + p(ac) p(ik)
ke = kconserv[kj, kc, kk]
t3c = t3c - einsum('jke,ie->ijk', t2[kj, kk, kc, :, :, c, :], -eris.ovvv[ki, ke, ka, :, :, a, b].conj())
ke = kconserv[ki, kc, kk]
t3c = t3c + einsum('ike,je->ijk', t2[ki, kk, kc, :, :, c, :], -eris.ovvv[kj, ke, ka, :, :, a, b].conj())
ke = kconserv[kj, kc, ki]
t3c = t3c + einsum('jie,ke->ijk', t2[kj, ki, kc, :, :, c, :], -eris.ovvv[kk, ke, ka, :, :, a, b].conj())
km = kconserv[kb, ki, ka]
t3c = t3c + einsum('mi,jkm->ijk', t2[km, ki, kb, :, :, b, a], eris.ooov[kj, kk, km, :, :, :, c].conj())
km = kconserv[kb, kj, ka]
t3c = t3c - einsum('mj,ikm->ijk', t2[km, kj, kb, :, :, b, a], eris.ooov[ki, kk, km, :, :, :, c].conj())
km = kconserv[kb, kk, ka]
t3c = t3c - einsum('mk,jim->ijk', t2[km, kk, kb, :, :, b, a], eris.ooov[kj, ki, km, :, :, :, c].conj())
# Form disconnected triple excitation amplitude contribution
t3d = np.zeros((nocc, nocc, nocc), dtype=dtype)
# First term: 1 - p(ij) - p(ik)
if ki == ka:
t3d = t3d + einsum('i,jk->ijk', t1[ki, :, a], -eris.oovv[kj, kk, kb, :, :, b, c].conj())
t3d = t3d + einsum('i,jk->ijk',-fov[ki, :, a], t2[kj, kk, kb, :, :, b, c])
if kj == ka:
t3d = t3d - einsum('j,ik->ijk', t1[kj, :, a], -eris.oovv[ki, kk, kb, :, :, b, c].conj())
t3d = t3d - einsum('j,ik->ijk',-fov[kj, :, a], t2[ki, kk, kb, :, :, b, c])
if kk == ka:
t3d = t3d - einsum('k,ji->ijk', t1[kk, :, a], -eris.oovv[kj, ki, kb, :, :, b, c].conj())
t3d = t3d - einsum('k,ji->ijk',-fov[kk, :, a], t2[kj, ki, kb, :, :, b, c])
# Second term: - p(ab) + p(ab) p(ij) + p(ab) p(ik)
if ki == kb:
t3d = t3d - einsum('i,jk->ijk', t1[ki, :, b], -eris.oovv[kj, kk, ka, :, :, a, c].conj())
t3d = t3d - einsum('i,jk->ijk',-fov[ki, :, b], t2[kj, kk, ka, :, :, a, c])
if kj == kb:
t3d = t3d + einsum('j,ik->ijk', t1[kj, :, b], -eris.oovv[ki, kk, ka, :, :, a, c].conj())
t3d = t3d + einsum('j,ik->ijk',-fov[kj, :, b], t2[ki, kk, ka, :, :, a, c])
if kk == kb:
t3d = t3d + einsum('k,ji->ijk', t1[kk, :, b], -eris.oovv[kj, ki, ka, :, :, a, c].conj())
t3d = t3d + einsum('k,ji->ijk',-fov[kk, :, b], t2[kj, ki, ka, :, :, a, c])
# Third term: - p(ac) + p(ac) p(ij) + p(ac) p(ik)
if ki == kc:
t3d = t3d - einsum('i,jk->ijk', t1[ki, :, c], -eris.oovv[kj, kk, kb, :, :, b, a].conj())
t3d = t3d - einsum('i,jk->ijk',-fov[ki, :, c], t2[kj, kk, kb, :, :, b, a])
if kj == kc:
t3d = t3d + einsum('j,ik->ijk', t1[kj, :, c], -eris.oovv[ki, kk, kb, :, :, b, a].conj())
t3d = t3d + einsum('j,ik->ijk',-fov[kj, :, c], t2[ki, kk, kb, :, :, b, a])
if kk == kc:
t3d = t3d + einsum('k,ji->ijk', t1[kk, :, c], -eris.oovv[kj, ki, kb, :, :, b, a].conj())
t3d = t3d + einsum('k,ji->ijk',-fov[kk, :, c], t2[kj, ki, kb, :, :, b, a])
t3c_plus_d = t3c + t3d
t3c_plus_d /= eijkabc
energy_t += symm_abc_kpt * symm_ijk_kpt * symm_abc * einsum('ijk,ijk', t3c, t3c_plus_d.conj())
energy_t = (1. / 36) * energy_t / nkpts
if abs(energy_t.imag) > 1e-4:
log.warn('Non-zero imaginary part of CCSD(T) energy was found %s',
energy_t.imag)
log.note('CCSD(T) correction per cell = %.15g', energy_t.real)
return energy_t.real
# Gamma point calculation
#
# Parameters
# ----------
# mesh : [24, 24, 24]
# kpt : [1, 1, 2]
# Returns
# -------
# SCF : -8.65192329453 Hartree per cell
# CCSD : -0.15529836941 Hartree per cell
# CCSD(T) : -0.00191451068 Hartree per cell
if __name__ == '__main__':
from pyscf.pbc import gto
from pyscf.pbc import cc
cell = gto.Cell()
cell.atom = '''
C 0.000000000000 0.000000000000 0.000000000000
C 1.685068664391 1.685068664391 1.685068664391
'''
cell.basis = 'gth-szv'
cell.pseudo = 'gth-pade'
cell.a = '''
0.000000000, 3.370137329, 3.370137329
3.370137329, 0.000000000, 3.370137329
3.370137329, 3.370137329, 0.000000000'''
cell.unit = 'B'
cell.verbose = 5
cell.mesh = [24, 24, 24]
cell.build()
kpts = cell.make_kpts([1, 1, 3])
kpts -= kpts[0]
kmf = scf.KRHF(cell, kpts=kpts, exxdiv=None)
ehf = kmf.kernel()
mycc = cc.KGCCSD(kmf)
ecc, t1, t2 = mycc.kernel()
energy_t = kernel(mycc)