#!/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 G0W0 approximation with analytic continuation
This implementation has N^4 scaling, and is faster than GW-CD (N^4)
and analytic GW (N^6) methods.
GW-AC is recommended for valence states only, and is inaccurate for core states.
Method:
See T. Zhu and G.K.-L. Chan, arxiv:2007.03148 (2020) for details
Compute Sigma on imaginary frequency with density fitting,
then analytically continued to real frequency
Useful References:
J. Chem. Theory Comput. 12, 3623-3635 (2016)
New J. Phys. 14, 053020 (2012)
'''
from functools import reduce
import numpy
import numpy as np
import h5py
from scipy.optimize import newton, least_squares
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
from pyscf import __config__
einsum = lib.einsum
[docs]
def kernel(gw, mo_energy, mo_coeff, Lpq=None, orbs=None,
nw=None, vhf_df=False, verbose=logger.NOTE):
'''
GW-corrected quasiparticle orbital energies
Returns:
A list : converged, mo_energy, mo_coeff
'''
mf = gw._scf
if gw.frozen is None:
frozen = 0
else:
frozen = gw.frozen
# only support frozen core
assert (isinstance(frozen, int))
assert (frozen < gw.nocc)
if Lpq is None:
Lpq = gw.ao2mo(mo_coeff)
if orbs is None:
orbs = range(gw.nmo)
else:
orbs = [x - frozen for x in orbs]
if orbs[0] < 0:
logger.warn(gw, 'GW orbs must be larger than frozen core!')
raise RuntimeError
# v_xc
v_mf = mf.get_veff() - mf.get_j()
v_mf = reduce(numpy.dot, (mo_coeff.T, v_mf, mo_coeff))
nocc = gw.nocc
nmo = gw.nmo
# v_hf from DFT/HF density
if vhf_df and frozen == 0:
# density fitting for vk
vk = -einsum('Lni,Lim->nm',Lpq[:,:,:nocc],Lpq[:,:nocc,:])
else:
# exact vk without density fitting
dm = mf.make_rdm1()
rhf = scf.RHF(gw.mol)
vk = rhf.get_veff(gw.mol,dm) - rhf.get_j(gw.mol,dm)
vk = reduce(numpy.dot, (mo_coeff.T, vk, mo_coeff))
# Grids for integration on imaginary axis
freqs,wts = _get_scaled_legendre_roots(nw)
# Compute self-energy on imaginary axis i*[0,iw_cutoff]
sigmaI,omega = get_sigma_diag(gw, orbs, Lpq, freqs, wts, iw_cutoff=5.)
# Analytic continuation
if gw.ac == 'twopole':
coeff = AC_twopole_diag(sigmaI, omega, orbs, nocc)
elif gw.ac == 'pade':
coeff,omega_fit = AC_pade_thiele_diag(sigmaI, omega)
conv = True
mf_mo_energy = mo_energy.copy()
ef = (mo_energy[nocc-1] + mo_energy[nocc])/2.
mo_energy = np.zeros_like(gw._scf.mo_energy)
for p in orbs:
if gw.linearized:
# linearized G0W0
de = 1e-6
ep = mf_mo_energy[p]
#TODO: analytic sigma derivative
if gw.ac == 'twopole':
sigmaR = two_pole(ep-ef, coeff[:,p-orbs[0]]).real
dsigma = two_pole(ep-ef+de, coeff[:,p-orbs[0]]).real - sigmaR.real
elif gw.ac == 'pade':
sigmaR = pade_thiele(ep-ef, omega_fit[p-orbs[0]], coeff[:,p-orbs[0]]).real
dsigma = pade_thiele(ep-ef+de, omega_fit[p-orbs[0]], coeff[:,p-orbs[0]]).real - sigmaR.real
zn = 1.0/(1.0-dsigma/de)
e = ep + zn*(sigmaR.real + vk[p,p] - v_mf[p,p])
mo_energy[p+frozen] = e
else:
# self-consistently solve QP equation
def quasiparticle(omega):
if gw.ac == 'twopole':
sigmaR = two_pole(omega-ef, coeff[:,p-orbs[0]]).real
elif gw.ac == 'pade':
sigmaR = pade_thiele(omega-ef, omega_fit[p-orbs[0]], coeff[:,p-orbs[0]]).real
return omega - mf_mo_energy[p] - (sigmaR.real + vk[p,p] - v_mf[p,p])
try:
e = newton(quasiparticle, mf_mo_energy[p], tol=1e-6, maxiter=100)
mo_energy[p+frozen] = e
except RuntimeError:
conv = False
if gw.verbose >= logger.DEBUG:
numpy.set_printoptions(threshold=nmo)
logger.debug(gw, ' GW mo_energy =\n%s', mo_energy)
numpy.set_printoptions(threshold=1000)
return conv, mo_energy, mo_coeff
[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)
Pia = einsum('Pia,ia->Pia',Lpq,eia)
# Response from both spin-up and spin-down density
Pi = 4. * einsum('Pia,Qia->PQ',Pia,Lpq)
return Pi
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def get_sigma_diag(gw, orbs, Lpq, freqs, wts, iw_cutoff=None):
'''
Compute GW correlation self-energy (diagonal elements)
in MO basis on imaginary axis
'''
mo_energy = _mo_energy_without_core(gw, gw._scf.mo_energy)
nocc = gw.nocc
nw = len(freqs)
naux = Lpq.shape[0]
norbs = len(orbs)
# TODO: Treatment of degeneracy
if (mo_energy[nocc] - mo_energy[nocc-1]) < 1e-3:
logger.warn(gw, 'GW not well-defined for degeneracy!')
ef = (mo_energy[nocc-1] + mo_energy[nocc])/2.
# Integration on numerical grids
if iw_cutoff is not None:
nw_sigma = sum(iw < iw_cutoff for iw in freqs) + 1
else:
nw_sigma = nw + 1
# Compute occ for -iw and vir for iw separately
# to avoid branch cuts in analytic continuation
omega_occ = np.zeros((nw_sigma),dtype=np.complex128)
omega_vir = np.zeros((nw_sigma),dtype=np.complex128)
omega_occ[1:] = -1j*freqs[:(nw_sigma-1)]
omega_vir[1:] = 1j*freqs[:(nw_sigma-1)]
orbs_occ = [i for i in orbs if i < nocc]
norbs_occ = len(orbs_occ)
emo_occ = omega_occ[None,:] + ef - mo_energy[:,None]
emo_vir = omega_vir[None,:] + ef - mo_energy[:,None]
sigma = np.zeros((norbs,nw_sigma),dtype=np.complex128)
omega = np.zeros((norbs,nw_sigma),dtype=np.complex128)
for p in range(norbs):
orbp = orbs[p]
if orbp < nocc:
omega[p] = omega_occ.copy()
else:
omega[p] = omega_vir.copy()
for w in range(nw):
Pi = get_rho_response(freqs[w], mo_energy, Lpq[:,:nocc,nocc:])
Pi_inv = np.linalg.inv(np.eye(naux)-Pi)-np.eye(naux)
g0_occ = wts[w] * emo_occ / (emo_occ**2+freqs[w]**2)
g0_vir = wts[w] * emo_vir / (emo_vir**2+freqs[w]**2)
Qnm = einsum('Pnm,PQ->Qnm',Lpq[:,orbs,:],Pi_inv)
Wmn = einsum('Qnm,Qmn->mn',Qnm,Lpq[:,:,orbs])
sigma[:norbs_occ] += -einsum('mn,mw->nw',Wmn[:,:norbs_occ],g0_occ)/np.pi
sigma[norbs_occ:] += -einsum('mn,mw->nw',Wmn[:,norbs_occ:],g0_vir)/np.pi
return sigma, omega
def _get_scaled_legendre_roots(nw):
"""
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 ndarray
wts : 1D ndarray
"""
freqs, wts = np.polynomial.legendre.leggauss(nw)
x0 = 0.5
freqs_new = x0*(1.+freqs)/(1.-freqs)
wts = wts*2.*x0/(1.-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 ndarray
wts : 1D ndarray
"""
freqs = np.zeros(nw)
wts = np.zeros(nw)
a = 0.2
for w in range(nw):
t = (w+1.0)/nw * np.pi/2.
freqs[w] = a / np.tan(t)
if w != nw-1:
wts[w] = a*np.pi/2./nw/(np.sin(t)**2)
else:
wts[w] = a*np.pi/4./nw/(np.sin(t)**2)
return freqs[::-1], wts[::-1]
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def two_pole_fit(coeff, omega, sigma):
cf = coeff[:5] + 1j*coeff[5:]
f = cf[0] + cf[1]/(omega+cf[3]) + cf[2]/(omega+cf[4]) - sigma
f[0] = f[0]/0.01
return np.array([f.real,f.imag]).reshape(-1)
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def two_pole(freqs, coeff):
cf = coeff[:5] + 1j*coeff[5:]
return cf[0] + cf[1]/(freqs+cf[3]) + cf[2]/(freqs+cf[4])
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def AC_twopole_diag(sigma, omega, orbs, nocc):
"""
Analytic continuation to real axis using a two-pole model
Returns:
coeff: 2D array (ncoeff, norbs)
"""
norbs, nw = sigma.shape
coeff = np.zeros((10,norbs))
for p in range(norbs):
# target = np.array([sigma[p].real,sigma[p].imag]).reshape(-1)
if orbs[p] < nocc:
x0 = np.array([0, 1, 1, 1, -1, 0, 0, 0, -1.0, -0.5])
else:
x0 = np.array([0, 1, 1, 1, -1, 0, 0, 0, 1.0, 0.5])
#TODO: analytic gradient
xopt = least_squares(two_pole_fit, x0, jac='3-point', method='trf', xtol=1e-10,
gtol = 1e-10, max_nfev=1000, verbose=0, args=(omega[p], sigma[p]))
if xopt.success is False:
print('WARN: 2P-Fit Orb %d not converged, cost function %e'%(p,xopt.cost))
coeff[:,p] = xopt.x.copy()
return coeff
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def thiele(fn,zn):
nfit = len(zn)
g = np.zeros((nfit,nfit),dtype=np.complex128)
g[:,0] = fn.copy()
for i in range(1,nfit):
g[i:,i] = (g[i-1,i-1]-g[i:,i-1])/((zn[i:]-zn[i-1])*g[i:,i-1])
a = g.diagonal()
return a
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def pade_thiele(freqs,zn,coeff):
nfit = len(coeff)
X = coeff[-1]*(freqs-zn[-2])
for i in range(nfit-1):
idx = nfit-i-1
X = coeff[idx]*(freqs-zn[idx-1])/(1.+X)
X = coeff[0]/(1.+X)
return X
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def AC_pade_thiele_diag(sigma, omega):
"""
Analytic continuation to real axis using a Pade approximation
from Thiele's reciprocal difference method
Reference: J. Low Temp. Phys. 29, 179 (1977)
Returns:
coeff: 2D array (ncoeff, norbs)
omega: 2D array (norbs, npade)
"""
idx = range(1,40,6)
sigma1 = sigma[:,idx].copy()
sigma2 = sigma[:,(idx[-1]+4)::4].copy()
sigma = np.hstack((sigma1,sigma2))
omega1 = omega[:,idx].copy()
omega2 = omega[:,(idx[-1]+4)::4].copy()
omega = np.hstack((omega1,omega2))
norbs, nw = sigma.shape
npade = nw // 2
coeff = np.zeros((npade*2,norbs),dtype=np.complex128)
for p in range(norbs):
coeff[:,p] = thiele(sigma[p,:npade*2], omega[p,:npade*2])
return coeff, omega[:,:npade*2]
def _mo_energy_without_core(gw, mo_energy):
return mo_energy[get_frozen_mask(gw)]
def _mo_without_core(gw, mo):
return mo[:,get_frozen_mask(gw)]
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class GWAC(lib.StreamObject):
linearized = getattr(__config__, 'gw_gw_GW_linearized', False)
# Analytic continuation: pade or twopole
ac = getattr(__config__, 'gw_gw_GW_ac', 'pade')
_keys = {
'linearized','ac', 'with_df', 'mol', 'frozen',
'mo_energy', 'mo_coeff', 'mo_occ', 'sigma',
}
def __init__(self, mf, frozen=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-GW 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)
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: GW quasiparticle energy, not scf mo_energy
self.mo_energy = None
self.mo_coeff = mf.mo_coeff
self.mo_occ = mf.mo_occ
self.sigma = None
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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('GW nocc = %d, nvir = %d', nocc, nvir)
if self.frozen is not None:
log.info('frozen = %s', self.frozen)
logger.info(self, 'use perturbative linearized QP eqn = %s', self.linearized)
logger.info(self, 'analytic continuation method = %s', self.ac)
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
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def kernel(self, mo_energy=None, mo_coeff=None, Lpq=None, orbs=None, nw=100, vhf_df=False):
"""
Input:
orbs: self-energy orbs
nw: grid number
vhf_df: whether using density fitting for HF exchange
Output:
mo_energy: GW quasiparticle 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.converged, self.mo_energy, self.mo_coeff = \
kernel(self, mo_energy, mo_coeff,
Lpq=Lpq, orbs=orbs, nw=nw, vhf_df=vhf_df, verbose=self.verbose)
logger.warn(self, 'GW QP energies may not be sorted from min to max')
logger.timer(self, 'GW', *cput0)
return self.mo_energy
[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 = numpy.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()
nocc = mol.nelectron//2
nmo = mf.mo_energy.size
nvir = nmo-nocc
gw = GWAC(mf)
gw.frozen = 0
gw.linearized = False
gw.ac = 'pade'
gw.kernel(orbs=range(nocc-3,nocc+3))
print(gw.mo_energy)
assert (abs(gw.mo_energy[nocc-1]- -0.412849230989) < 1e-5)
assert (abs(gw.mo_energy[nocc] -0.165745160102) < 1e-5)
