#!/usr/bin/env python
# Copyright 2014-2021 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: Tianyu Zhu <zhutianyu1991@gmail.com>
#
"""
Spin-restricted random phase approximation (direct RPA/dRPA in chemistry)
with N^4 scaling
Method:
Main routines are based on GW-AC method descirbed in:
T. Zhu and G.K.-L. Chan, J. Chem. Theory. Comput. 17, 727-741 (2021)
X. Ren et al., New J. Phys. 14, 053020 (2012)
"""
import numpy as np
from pyscf import lib
from pyscf.lib import logger
from pyscf.ao2mo import _ao2mo
from pyscf import df, scf
from pyscf.mp.mp2 import get_nocc, get_nmo, get_frozen_mask
einsum = lib.einsum
# ****************************************************************************
# core routines, kernel, rpa_ecorr, rho_response
# ****************************************************************************
[docs]
def kernel(rpa, mo_energy, mo_coeff, Lpq=None, nw=40, x0=0.5, verbose=logger.NOTE):
"""
RPA correlation and total energy
Args:
Lpq : density fitting 3-center integral in MO basis.
nw : number of frequency point on imaginary axis.
x0: scaling factor for frequency grid.
Returns:
e_tot : RPA total energy
e_hf : EXX energy
e_corr : RPA correlation energy
"""
mf = rpa._scf
# only support frozen core
if rpa.frozen is not None:
assert isinstance(rpa.frozen, int)
assert rpa.frozen < rpa.nocc
if Lpq is None:
Lpq = rpa.ao2mo(mo_coeff)
# Grids for integration on imaginary axis
freqs, wts = _get_scaled_legendre_roots(nw, x0)
# Compute HF exchange energy (EXX)
dm = mf.make_rdm1()
rhf = scf.RHF(rpa.mol)
e_hf = rhf.energy_elec(dm=dm)[0]
e_hf += mf.energy_nuc()
# Compute RPA correlation energy
e_corr = get_rpa_ecorr(rpa, Lpq, freqs, wts)
# Compute total energy
e_tot = e_hf + e_corr
logger.debug(rpa, ' RPA total energy = %s', e_tot)
logger.debug(rpa, ' EXX energy = %s, RPA corr energy = %s', e_hf, e_corr)
return e_tot, e_hf, e_corr
[docs]
def get_rpa_ecorr(rpa, Lpq, freqs, wts):
"""
Compute RPA correlation energy
"""
mo_energy = _mo_energy_without_core(rpa, rpa._scf.mo_energy)
nocc = rpa.nocc
nw = len(freqs)
naux = Lpq.shape[0]
if (mo_energy[nocc] - mo_energy[nocc-1]) < 1e-3:
logger.warn(rpa, 'Current RPA code not well-defined for degeneracy!')
e_corr = 0.
for w in range(nw):
Pi = get_rho_response(freqs[w], mo_energy, Lpq[:, :nocc, nocc:])
ec_w = np.log(np.linalg.det(np.eye(naux) - Pi))
ec_w += np.trace(Pi)
e_corr += 1./(2.*np.pi) * ec_w * wts[w]
return e_corr
[docs]
def get_rho_response(omega, mo_energy, Lpq):
"""
Compute density response function in auxiliary basis at freq iw.
"""
naux, nocc, nvir = Lpq.shape
eia = mo_energy[:nocc, None] - mo_energy[None, nocc:]
eia = eia / (omega**2 + eia * eia)
# Response from both spin-up and spin-down density
Pia = Lpq * (eia * 4.0)
Pi = einsum('Pia, Qia -> PQ', Pia, Lpq)
return Pi
# ****************************************************************************
# frequency integral quadrature, legendre, clenshaw_curtis
# ****************************************************************************
def _get_scaled_legendre_roots(nw, x0=0.5):
"""
Scale nw Legendre roots, which lie in the
interval [-1, 1], so that they lie in [0, inf)
Ref: www.cond-mat.de/events/correl19/manuscripts/ren.pdf
Returns:
freqs : 1D array
wts : 1D array
"""
freqs, wts = np.polynomial.legendre.leggauss(nw)
freqs_new = x0 * (1.0 + freqs) / (1.0 - freqs)
wts = wts * 2.0 * x0 / (1.0 - freqs)**2
return freqs_new, wts
def _get_clenshaw_curtis_roots(nw):
"""
Clenshaw-Curtis qaudrature on [0,inf)
Ref: J. Chem. Phys. 132, 234114 (2010)
Returns:
freqs : 1D array
wts : 1D array
"""
freqs = np.zeros(nw)
wts = np.zeros(nw)
a = 0.2
for w in range(nw):
t = (w + 1.0) / nw * np.pi * 0.5
freqs[w] = a / np.tan(t)
if w != nw - 1:
wts[w] = a*np.pi * 0.5 / nw / (np.sin(t)**2)
else:
wts[w] = a*np.pi * 0.25 / nw / (np.sin(t)**2)
return freqs[::-1], wts[::-1]
def _mo_energy_without_core(rpa, mo_energy):
return mo_energy[get_frozen_mask(rpa)]
def _mo_without_core(rpa, mo):
return mo[:,get_frozen_mask(rpa)]
[docs]
class RPA(lib.StreamObject):
_keys = set((
'mol', 'frozen',
'with_df', 'mo_energy', 'mo_coeff', 'mo_occ', 'e_corr', 'e_hf', 'e_tot',
))
def __init__(self, mf, frozen=None, auxbasis=None):
self.mol = mf.mol
self._scf = mf
self.verbose = self.mol.verbose
self.stdout = self.mol.stdout
self.max_memory = mf.max_memory
self.frozen = frozen
# DF-RPA must use density fitting integrals
if getattr(mf, 'with_df', None):
self.with_df = mf.with_df
else:
self.with_df = df.DF(mf.mol)
if auxbasis:
self.with_df.auxbasis = auxbasis
else:
self.with_df.auxbasis = df.make_auxbasis(mf.mol, mp2fit=True)
##################################################
# don't modify the following attributes, they are not input options
self._nocc = None
self._nmo = None
self.mo_energy = mf.mo_energy
self.mo_coeff = mf.mo_coeff
self.mo_occ = mf.mo_occ
self.e_corr = None
self.e_hf = None
self.e_tot = None
[docs]
def dump_flags(self):
log = logger.Logger(self.stdout, self.verbose)
log.info('')
log.info('******** %s ********', self.__class__)
log.info('method = %s', self.__class__.__name__)
nocc = self.nocc
nvir = self.nmo - nocc
log.info('RPA nocc = %d, nvir = %d', nocc, nvir)
if self.frozen is not None:
log.info('frozen orbitals = %d', self.frozen)
return self
@property
def nocc(self):
return self.get_nocc()
@nocc.setter
def nocc(self, n):
self._nocc = n
@property
def nmo(self):
return self.get_nmo()
@nmo.setter
def nmo(self, n):
self._nmo = n
get_nocc = get_nocc
get_nmo = get_nmo
get_frozen_mask = get_frozen_mask
[docs]
def kernel(self, mo_energy=None, mo_coeff=None, Lpq=None, nw=40, x0=0.5):
"""
Args:
mo_energy : 1D array (nmo), mean-field mo energy
mo_coeff : 2D array (nmo, nmo), mean-field mo coefficient
Lpq : 3D array (naux, nmo, nmo), 3-index ERI
nw: integer, grid number
x0: real, scaling factor for frequency grid
Returns:
self.e_tot : RPA total eenrgy
self.e_hf : EXX energy
self.e_corr : RPA correlation energy
"""
if mo_coeff is None:
mo_coeff = _mo_without_core(self, self._scf.mo_coeff)
if mo_energy is None:
mo_energy = _mo_energy_without_core(self, self._scf.mo_energy)
cput0 = (logger.process_clock(), logger.perf_counter())
self.dump_flags()
self.e_tot, self.e_hf, self.e_corr = \
kernel(self, mo_energy, mo_coeff, Lpq=Lpq, nw=nw, x0=x0, verbose=self.verbose)
logger.timer(self, 'RPA', *cput0)
return self.e_corr
[docs]
def ao2mo(self, mo_coeff=None):
if mo_coeff is None:
mo_coeff = self.mo_coeff
nmo = self.nmo
naux = self.with_df.get_naoaux()
mem_incore = (2 * nmo**2*naux) * 8 / 1e6
mem_now = lib.current_memory()[0]
mo = np.asarray(mo_coeff, order='F')
ijslice = (0, nmo, 0, nmo)
Lpq = None
if (mem_incore + mem_now < 0.99 * self.max_memory) or self.mol.incore_anyway:
Lpq = _ao2mo.nr_e2(self.with_df._cderi, mo, ijslice, aosym='s2', out=Lpq)
return Lpq.reshape(naux, nmo, nmo)
else:
logger.warn(self, 'Memory may not be enough!')
raise NotImplementedError
if __name__ == '__main__':
from pyscf import gto, dft
mol = gto.Mole()
mol.verbose = 4
mol.atom = [
[8 , (0. , 0. , 0.)],
[1 , (0. , -0.7571 , 0.5861)],
[1 , (0. , 0.7571 , 0.5861)]]
mol.basis = 'def2-svp'
mol.build()
mf = dft.RKS(mol)
mf.xc = 'pbe'
mf.kernel()
rpa = RPA(mf)
rpa.kernel()
print ('RPA e_tot, e_hf, e_corr = ', rpa.e_tot, rpa.e_hf, rpa.e_corr)
assert (abs(rpa.e_corr- -0.30783004035780076) < 1e-6)
assert (abs(rpa.e_tot- -76.26428191794182) < 1e-6)