#!/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 described 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, scipy
from pyscf import lib
from pyscf.lib import logger
from pyscf.ao2mo import _ao2mo
from pyscf import df, scf
from pyscf.mp import dfmp2
einsum = lib.einsum
# ****************************************************************************
# core routines kernel
# ****************************************************************************
[docs]
def kernel(rpa, eris=None, nw=40, x0=0.5, verbose=None):
'''
RPA correlation and total energy
Args:
cderi_ov:
Array-like object, Cholesky decomposed ERI in OV subspace.
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
'''
log = logger.new_logger(rpa, verbose)
if eris is None:
eris = rpa.ao2mo()
naux = eris.naux
cput1 = (logger.process_clock(), logger.perf_counter())
# Compute exact exchange energy (EXX)
e_hf = rpa.get_e_hf()
cput2 = cput1 = log.timer('EXX energy', *cput1)
# Compute RPA correlation energy
e_corr = 0
e_ov = rpa.make_e_ov()
f_ov = rpa.make_f_ov()
for igrid,(omega,weigh) in enumerate(zip(*_get_scaled_legendre_roots(nw, x0))):
diel = rpa.make_dielectric_matrix(omega, e_ov, f_ov, eris)
factor = weigh / (2.0 * np.pi)
e_corr += factor * np.log(np.linalg.det(np.eye(naux) - diel))
e_corr += factor * np.trace(diel)
diel = None
cput2 = log.timer_debug1('RPA corr grid %d/%d'%(igrid+1,nw), *cput2)
log.timer('RPA corr', *cput1)
if abs(e_corr.imag) > 1e-4:
log.warn('Non-zero imaginary part found in %s energy %s', rpa.__class__.__name__, e_corr)
e_corr = e_corr.real
return e_hf, e_corr
# ****************************************************************************
# frequency integral quadrature, legendre, clenshaw_curtis
# ****************************************************************************
[docs]
def make_dielectric_matrix(omega, e_ov, f_ov, eris, blksize=None):
'''
Compute dielectric matrix at a given frequency omega
Args:
omega : float, frequency
e_ov : 1D array (nocc * nvir), orbital energy differences
eris : DF ERI object
Returns:
diel : 2D array (naux, naux), dielectric matrix
'''
assert blksize is not None
nocc, nvir, naux = eris.nocc, eris.nvir, eris.naux
isreal = eris.dtype == np.float64
chi0 = (2.0 * e_ov * f_ov / (omega ** 2 + e_ov ** 2)).ravel()
diel = np.zeros((naux, naux), dtype=eris.dtype)
for p0,p1 in lib.prange(0, nocc*nvir, blksize):
ovL = eris.get_ov_blk(p0,p1)
ovL_chi = (ovL.T * chi0[p0:p1]).T
if isreal:
lib.ddot(ovL_chi.T, ovL, c=diel, beta=1)
else:
lib.dot(ovL_chi.T, ovL.conj(), c=diel, beta=1)
ovL = ovL_chi = None
return diel
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 quadrature 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 * 0.50 / nw / (np.sin(t)**2)
else:
wts[w] = a * np.pi * 0.25 / nw / (np.sin(t)**2)
return freqs[::-1], wts[::-1]
[docs]
class RPA(dfmp2.DFMP2):
_keys = {
'mol', 'frozen',
'with_df', 'mo_energy',
'mo_coeff', 'mo_occ',
'e_corr', 'e_hf', 'e_tot',
}
[docs]
def dump_flags(self, verbose=None):
log = logger.new_logger(self, verbose)
log.info('')
log.info('******** %s ********', self.__class__)
log.info('method = %s', self.__class__.__name__)
log.info('nocc = %d nmo = %d', self.nocc, self.nmo)
if self.frozen is not None:
log.info('frozen orbitals = %s', self.frozen)
return self
[docs]
def kernel(self, eris=None, nw=40, x0=0.5):
'''
The kernel function for direct RPA
'''
if np.iscomplexobj(self.mo_coeff):
''' The code runs for complex-valued orbitals but the results ain't quite right
(very close though...). Throw an exception for now.
'''
raise NotImplementedError
log = logger.new_logger(self)
cput0 = (logger.process_clock(), logger.perf_counter())
self.dump_flags()
res = kernel(
self, eris=eris, nw=nw, x0=x0, verbose=self.verbose)
self.e_hf, self.e_corr = res
log.timer(self.__class__.__name__, *cput0)
self._finalize()
return self.e_corr
def _finalize(self):
'''Hook for dumping results and clearing up the object.'''
log = logger.new_logger(self)
log.note('E(%s) = %.15g E_corr = %.15g E_hf = %.15g',
self.__class__.__name__, self.e_tot, self.e_corr, self.e_hf)
[docs]
def get_e_hf(self):
if self.e_hf is None:
mf = self._scf
if isinstance(mf, scf.hf.KohnShamDFT):
mf = mf.to_hf()
if getattr(mf, 'with_df', None) is None:
mf = mf.density_fit(with_df=self.with_df)
dm = mf.make_rdm1()
self.e_hf = mf.energy_elec(dm=dm)[0] + mf.energy_nuc()
return self.e_hf
[docs]
def make_e_ov(self):
'''
Compute orbital energy differences
'''
log = logger.new_logger(self)
moeocc, moevir = self.split_mo_energy()[1:3]
e_ov = (moeocc[:,None] - moevir).ravel()
gap = (-e_ov.max(), )
log.info('Lowest orbital energy difference: % 6.4e', np.min(gap))
if (np.min(gap) < 1e-3):
log.warn('RPA code is not well-defined for degenerate systems!')
log.warn('Lowest orbital energy difference: % 6.4e', np.min(gap))
return e_ov
[docs]
def make_f_ov(self):
'''
Compute orbital occupation number differences
'''
focc, fvir = self.split_mo_occ()[1:3]
return (focc[:,None] - fvir).ravel()
[docs]
def make_dielectric_matrix(self, omega, e_ov=None, f_ov=None, eris=None,
max_memory=None, blksize=None):
'''
Args:
omega : float, frequency
e_ov : 1D array (nocc * nvir), orbital energy differences
mo_coeff : (nao, nmo), mean-field mo coefficient
cderi_ov : (naux, nocc, nvir), Cholesky decomposed ERI in OV subspace.
Returns:
diel : 2D array (naux, naux), dielectric matrix
'''
if e_ov is None: e_ov = self.make_e_ov()
if f_ov is None: f_ov = self.make_f_ov()
if eris is None: eris = self.ao2mo()
if max_memory is None: max_memory = self.max_memory
if blksize is None:
mem_avail = max_memory - lib.current_memory()[0]
nocc, nvir, naux = eris.nocc, eris.nvir, eris.naux
dsize = eris.dsize
mem_blk = 2*naux * dsize/1e6 # ovL and ovL*chi0
blksize = max(1, min(nocc*nvir, int(np.floor(mem_avail*0.7 / mem_blk))))
else:
blksize = min(blksize, e_ov.size)
diel = make_dielectric_matrix(omega, e_ov, f_ov, eris, blksize=blksize)
return diel
dRPA = DirectRPA = RPA
if __name__ == '__main__':
from pyscf import gto, dft
mol = gto.Mole()
mol.atom = [
[8 , (0. , 0. , 0.)],
[1 , (0. , -0.7571 , 0.5861)],
[1 , (0. , 0.7571 , 0.5861)]]
mol.basis = 'def2svp'
mol.build()
mol.verbose = 4
mf = dft.RKS(mol)
mf.xc = 'pbe'
mf.kernel()
rpa = RPA(mf)
rpa.verbose = 4
eris = rpa.ao2mo()
e_corr_0 = rpa.kernel(eris=eris)
assert (abs(rpa.e_corr - -0.307830040357800) < 1e-6)
assert (abs(rpa.e_tot - -76.26651423730257) < 1e-6)
# Another implementation of direct RPA N^6
e_ov = -rpa.make_e_ov()
nov = e_ov.size
v_ov = np.asarray(eris.get_ov_blk(0,nov).T, order='C')
a = e_ov * np.eye(nov) + 2 * np.dot(v_ov.T, v_ov)
b = 2 * np.dot(v_ov.T, v_ov)
cmat = np.block([[a,b],[-b.conj(),-a.conj()]])
ev = scipy.linalg.eig(cmat)[0]
ev = ev.real
ev = ev[ev>0]
e_corr_1 = 0.5 * (np.sum(ev) - np.trace(a))
assert abs(e_corr_0 - e_corr_1) < 1e-8
