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1191 lines (1028 loc) · 42.9 KB
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#!/usr/bin/env python
# Copyright 2014-2026 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>
# Author: Christopher Hillenbrand <chillenbrand15@gmail.com>
# Author: Jiachen Li <lijiachen.duke@gmail.com>
#
'''
Spin-restricted G0W0 method based on the analytic continuation scheme.
This implementation has N^4 scaling,
and is faster than GW-CD (N^4~N^5) and fully analytic GW (N^6) methods.
GW-AC is recommended for valence states only, and is inaccurate for core states.
References:
T. Zhu and G.K.-L. Chan, J. Chem. Theory. Comput. 17, 727-741 (2021)
New J. Phys. 14 053020 (2012)
'''
import h5py
import numpy as np
from scipy.optimize import newton
import scipy.linalg as sla
import time
from pyscf import df, dft, lib, scf
from pyscf.ao2mo._ao2mo import nr_e2
from pyscf.lib import einsum, logger, temporary_env
from pyscf.mp.mp2 import get_nocc, get_nmo, get_frozen_mask
from pyscf.scf.addons import _fermi_smearing_occ, _gaussian_smearing_occ, _smearing_optimize
from pyscf.gw.utils.ac_grid import _get_scaled_legendre_roots, PadeAC, TwoPoleAC
from pyscf.gw.utils.gw_np_helper import get_id_minus_pi_inv, mkslice, addto_diagonal
def kernel(gw):
# local variables for convenience
mf = gw._scf
nocc = gw.nocc
# set frozen orbitals
set_frozen_orbs(gw)
orbs = gw.orbs
orbs_frz = gw.orbs_frz
# get non-frozen quantities
mo_energy_frz = _mo_energy_without_core(gw, gw.mo_energy)
mo_coeff_frz = _mo_without_core(gw, gw.mo_coeff)
if gw.Lpq is None and (not gw.outcore):
with temporary_env(gw.with_df, verbose=0), temporary_env(gw.mol, verbose=0):
gw.Lpq = gw.ao2mo(mo_coeff_frz)
Lpq = gw.Lpq
# mean-field exchange-correlation
with temporary_env(mf, verbose=0):
v_mf = mo_coeff_frz.T @ (mf.get_veff() - mf.get_j()) @ mo_coeff_frz
gw.vxc = v_mf
# exchange self-energy
if gw.vhf_df and gw.outcore is False:
# TODO: support smearing
vk = -einsum('Lpi,Liq->pq', Lpq[:, :, :nocc], Lpq[:, :nocc, :])
else:
vk = gw.get_sigma_exchange(mo_coeff=mo_coeff_frz)
gw.vk = vk
# set up Fermi level
gw.ef = ef = gw.get_ef(mo_energy=mf.mo_energy)
# grids for integration on imaginary axis
quad_freqs, quad_wts = _get_scaled_legendre_roots(gw.nw)
eval_freqs_with_zero = gw.setup_evaluation_grid(fallback_freqs=quad_freqs, fallback_wts=quad_wts)
# compute self-energy on imaginary axis
if gw.outcore:
sigmaI, omega = get_sigma_outcore(
gw, orbs_frz, quad_freqs, quad_wts, ef, mo_energy_frz, mo_coeff_frz, iw_cutoff=gw.ac_iw_cutoff,
eval_freqs=eval_freqs_with_zero, fullsigma=gw.fullsigma
)
else:
sigmaI, omega = get_sigma(
gw, orbs_frz, Lpq, quad_freqs, quad_wts, ef, mo_energy_frz, iw_cutoff=gw.ac_iw_cutoff,
eval_freqs=eval_freqs_with_zero, fullsigma=gw.fullsigma
)
# analytic continuation
if gw.ac == 'twopole':
acobj = TwoPoleAC(orbs_frz, nocc)
elif gw.ac == 'pade':
acobj = PadeAC(npts=gw.ac_pade_npts, step_ratio=gw.ac_pade_step_ratio)
if gw.ac_idx is not None:
acobj.idx = gw.ac_idx
acobj.ac_fit(sigmaI, omega, axis=-1)
# get GW quasiparticle energy
if gw.fullsigma:
diag_acobj = acobj.diagonal(axis1=0, axis2=1)
else:
diag_acobj = acobj
mo_energy = np.zeros_like(gw._scf.mo_energy)
for ip, (p_in_frz, p_in_all) in enumerate(zip(orbs_frz, orbs)):
if gw.qpe_linearized is True:
# linearized G0W0
de = 1e-6
sigmaR = diag_acobj[ip].ac_eval(gw._scf.mo_energy[p_in_all]).real
dsigma = diag_acobj[ip].ac_eval(gw._scf.mo_energy[p_in_all] + de).real - sigmaR.real
zn = 1.0 / (1.0 - dsigma / de)
if gw.qpe_linearized_range is not None:
zn = 1.0 if zn < gw.qpe_linearized_range[0] or zn > gw.qpe_linearized_range[1] else zn
mo_energy[p_in_all] = gw._scf.mo_energy[p_in_all] + \
zn * (sigmaR.real + vk[p_in_frz, p_in_frz] - v_mf[p_in_frz, p_in_frz])
else:
# self-consistently solve QP equation
def quasiparticle(omega):
sigmaR = diag_acobj[ip].ac_eval(omega).real
return omega - gw._scf.mo_energy[p_in_all] \
- (sigmaR + vk[p_in_frz, p_in_frz] - v_mf[p_in_frz, p_in_frz])
try:
mo_energy[p_in_all] = newton(
quasiparticle, gw._scf.mo_energy[p_in_all], tol=gw.qpe_tol, maxiter=gw.qpe_max_iter
)
except RuntimeError:
logger.warn(gw, 'QPE for orbital=%d not converged!', p_in_all)
# save GW results
gw.acobj = acobj
gw.mo_energy = mo_energy
with np.printoptions(threshold=len(mo_energy)):
logger.debug(gw, ' GW mo_energy =\n%s', mo_energy)
logger.warn(gw, 'GW QP energies may not be sorted from min to max')
if gw.writefile > 0:
fn = 'vxc.h5'
feri = h5py.File(fn, 'w')
feri['vk'] = np.asarray(vk)
feri['v_mf'] = np.asarray(v_mf)
feri.close()
fn = 'sigma_imag.h5'
feri = h5py.File(fn, 'w')
feri['sigmaI'] = np.asarray(sigmaI)
feri['omega'] = np.asarray(omega)
feri.close()
acobj.save('ac_coeff.h5')
return
def get_rho_response(omega, mo_energy, Lia, out=None):
"""Compute density-density response function in auxiliary basis at freq iw.
See equation 58 in 10.1088/1367-2630/14/5/053020,
and equation 24 in doi.org/10.1021/acs.jctc.0c00704.
Parameters
----------
omega : double
imaginary part of a frequency point
mo_energy : double 1d array
orbital energy
Lia : double 3d ndarray
occ-vir block of three-center density-fitting matrix
out : double 2d array, optional
a location into which the result is stored, by default None
Returns
-------
Pi : double 2d array
density-density response function in auxiliary basis at freq iw
"""
naux, nocc, nvir = Lia.shape
eia = mo_energy[:nocc, None] - mo_energy[None, nocc:]
# factor 4.0 comes from (1) spin-up and spin-down (2) conjugated term
eia = 4.0 * eia / (omega**2 + eia**2)
Pia = lib.broadcast_mul(Lia, eia)
Pi = np.matmul(Pia.reshape(naux, nocc * nvir), Lia.reshape(naux, nocc * nvir).T, out=out)
return Pi
def get_rho_response_metal(omega, mo_energy, mo_occ, Lpq, out=None):
"""Get response function in auxiliary basis for metallic systems.
Also applicable to molecular SCF with smearing.
Parameters
----------
omega : double
imaginary part of a frequency point
mo_energy : double 1d array
orbital energy
mo_occ : double 1d array
occupation number with a factor of 2
Lpq : double 3d array
three-center density-fitting matrix in MO
out : double ndarray, optional
a location into which the result is stored, by default None
Returns
-------
Pi : double 2d array
density-density response function in auxiliary basis at freq iw
"""
naux = Lpq.shape[0]
eia = mo_energy[:, None] - mo_energy[None, :]
fia = mo_occ[:, None] - mo_occ[None, :]
# factor 4.0 comes from (1) spin-up and spin-down (2) conjugated term
# both ia and ai are included, this gives a factor of 2.0
# restricted mo_occ gives another factor of 2.0
eia = eia * fia / (omega**2 + eia**2)
Pia = lib.broadcast_mul(Lpq, eia)
Pi = np.matmul(Pia.reshape(naux, -1), Lpq.reshape(naux, -1).T, out=out)
return Pi
def get_sigma(
gw, orbs, Lpq, quad_freqs, quad_wts, ef, mo_energy, mo_coeff=None, mo_occ=None, iw_cutoff=None, eval_freqs=None,
mo_energy_w=None, fullsigma=False):
"""Compute GW correlation self-energy on imaginary axis.
See equation 62 and 62 in 10.1088/1367-2630/14/5/053020,
and equation 27 in doi.org/10.1021/acs.jctc.0c00704.
Parameters
----------
gw : GWAC
GW object
orbs : list
list of orbital indexes
Lpq : double 3d array
three-center density-fitting matrix in MO space
quad_freqs : double 1d array
position of imaginary frequency grids used for integration
quad_wts : double 1d array
weight of imaginary frequency grids
ef : double
Fermi level
mo_energy : double 1d array
orbital energy in G
mo_coeff : double 2d array
coefficient from AO to MO
mo_occ : double 1d array, optional
occupation number, by default None
iw_cutoff : double, optional
imaginary grid cutoff for fitting, by default None
eval_freqs : double 1d array, optional
imaginary frequency points at which to evaluate self energy.
Self energy is evaluated at ef + 1j * eval_freqs.
mo_energy_w : double 1d array, optional
orbital energy in W, by default None
fullsigma : bool, optional
calculate off-diagonal elements, by default False
Returns
-------
sigma : complex 2d or 3d ndarray
self-energy on the imaginary axis
omega : complex 1d array
imaginary frequency grids of self-energy
"""
if eval_freqs is None:
eval_freqs = quad_freqs
nocc = gw.nocc
nquadfreqs = len(quad_freqs)
nevalfreqs = len(eval_freqs)
naux, nmo, _ = Lpq.shape
norbs = len(orbs)
mo_energy_g = mo_energy
if mo_energy_w is None:
mo_energy_w = mo_energy
if mo_coeff is None:
mo_coeff = _mo_without_core(gw, gw.mo_coeff)
if mo_occ is None:
mo_occ = _mo_energy_without_core(gw, gw.mo_occ)
# integration on numerical grids
if iw_cutoff is not None and gw.rdm is False:
nw_sigma = sum(eval_freqs < iw_cutoff)
else:
nw_sigma = nevalfreqs
nw_cutoff = nw_sigma if iw_cutoff is None else sum(eval_freqs < iw_cutoff)
omega = ef + 1j * eval_freqs[:nw_sigma]
emo = omega[None] - mo_energy_g[:, None]
if fullsigma is False:
sigma_real = np.zeros(shape=[nw_sigma, norbs], dtype=np.double)
sigma_imag = np.zeros(shape=[nw_sigma, norbs], dtype=np.double)
else:
sigma_real = np.zeros(shape=[nw_sigma, norbs, norbs], dtype=np.double)
sigma_imag = np.zeros(shape=[nw_sigma, norbs, norbs], dtype=np.double)
# make density-fitting matrix for contractions
if hasattr(gw._scf, 'sigma') is False:
Lia = np.ascontiguousarray(Lpq[:, :nocc, nocc:])
# assume Lpq = Lpq, so we don't generate Lpq[:, mkslice(orbs), :]
l_slice = Lpq[:, :, mkslice(orbs)].reshape(naux, -1)
# self-energy is calculated as equation 27 in doi.org/10.1021/acs.jctc.0c00704
logger.info(gw, 'Starting get_sigma_diag main loop with %d frequency points.', nquadfreqs)
Pi = None
if fullsigma is False:
Qmn = None
Wmn = None
naux_ones = np.ones(shape=[1, naux], dtype=np.double)
else:
Qmn = np.zeros(shape=[naux, norbs], dtype=np.double)
Wmn = np.zeros(shape=[nmo, norbs, norbs], dtype=np.double)
for w in range(nquadfreqs):
if gw.verbose >= 4:
gw.stdout.write(f'{w:4d} ')
if w % 12 == 11:
gw.stdout.write('\n')
gw.stdout.flush()
# Pi_inv = (I - Pi)^-1 - I.
if hasattr(gw._scf, 'sigma'):
Pi = get_rho_response_metal(quad_freqs[w], mo_energy_w, mo_occ, Lpq, out=Pi)
else:
Pi = get_rho_response(quad_freqs[w], mo_energy_w, Lia)
# Pi_inv contains (I - Pi)^-1.
Pi_inv = get_id_minus_pi_inv(Pi, overwrite_input=True)
# second line in equation 27
# Pi_inv now contains (I - Pi)^-1 - I.
addto_diagonal(Pi_inv, -1.0)
g0 = quad_wts[w] * emo / (emo**2 + quad_freqs[w] ** 2)
# split g0 into real and imag parts to avoid costly type conversions
g0r = np.ascontiguousarray(g0.real)
g0i = np.ascontiguousarray(g0.imag)
if fullsigma is False:
# n is the index of orbitals in orbs, m is the index of orbitals of nmo
# last line of equation 27, contraction from left to right
# Qmn = \sum_P V^{nm}_P (Pi_inv)_{PQ} = \sum_P V^{mn}_P (Pi_inv)_{PQ}
Qmn = np.matmul(Pi_inv, l_slice, out=Qmn)
# Qmn = Qmn v^{mn}_Q
Qmn *= l_slice
# Wmn = \sum_Q Qmn
Wmn = np.matmul(naux_ones, Qmn, out=Wmn)
# sigma -= einsum('mn, mw -> nw', Wmn, g0) / np.pi
sla.blas.dgemm(
alpha=-1.0 / np.pi,
a=Wmn.reshape(nmo, norbs).T,
b=g0r.T,
c=sigma_real.T,
trans_a=0,
trans_b=1,
beta=1.0,
overwrite_c=True,
)
sla.blas.dgemm(
alpha=-1.0 / np.pi,
a=Wmn.reshape(nmo, norbs).T,
b=g0i.T,
c=sigma_imag.T,
trans_a=0,
trans_b=1,
beta=1.0,
overwrite_c=True,
)
else:
# n and n' are the index of orbitals in orbs, m is the index of orbitals of nmo
# last line of equation 27, contraction from left to right
# Qmn = \sum_P V^{nm}_P (PiV)_{PQ} = \sum_P V^{mn}_P (PiV)_{PQ}
for orbm in range(nmo):
np.matmul(Pi_inv, l_slice.reshape(naux, nmo, norbs)[:, orbm, :], out=Qmn)
# Wmn is actually Wmnn'
# Wmnn' = \sum_Q Qmn v^{mn'}_Q
np.matmul(Qmn.T, l_slice.reshape(naux, nmo, norbs)[:, orbm, :], out=Wmn[orbm])
# sigma -= einsum('mnl,mw->wnl', Wmn, g0)/np.pi
sla.blas.dgemm(
alpha=-1.0 / np.pi,
a=Wmn.reshape(nmo, norbs * norbs).T,
b=g0r.T,
c=sigma_real.reshape(nw_sigma, norbs * norbs).T,
trans_a=0,
trans_b=1,
beta=1.0,
overwrite_c=True,
)
sla.blas.dgemm(
alpha=-1.0 / np.pi,
a=Wmn.reshape(nmo, norbs * norbs).T,
b=g0i.T,
c=sigma_imag.reshape(nw_sigma, norbs * norbs).T,
trans_a=0,
trans_b=1,
beta=1.0,
overwrite_c=True,
)
sigma = sigma_real + 1.0j * sigma_imag
if fullsigma is False:
sigma = np.ascontiguousarray(sigma.transpose(1, 0))
else:
sigma = np.ascontiguousarray(sigma.transpose(1, 2, 0))
logger.info(gw, '\nFinished get_sigma_diag main loop.')
if gw.rdm is True:
gw.sigmaI = sigma
return sigma[..., :nw_cutoff], omega[:nw_cutoff]
def get_sigma_outcore(gw, orbs, quad_freqs, quad_wts, ef, mo_energy, mo_coeff, mo_occ=None,
iw_cutoff=None, e
10000
val_freqs=None, mo_energy_w=None, fullsigma=False):
"""Low-memory routine to compute GW correlation self-energy on imaginary axis.
See equation 62 and 62 in 10.1088/1367-2630/14/5/053020,
and equation 27 in doi.org/10.1021/acs.jctc.0c00704.
Parameters
----------
gw : GWAC
GW object
orbs : list
list of orbital indexes
quad_freqs : double 1d array
position of imaginary frequency grids used for integration
quad_wts : double 1d array
weight of imaginary frequency grids
ef : double
Fermi level
mo_energy : double 1d array
orbital energy in G
mo_coeff : double 2d array
coefficient from AO to MO
mo_occ : double 2d array, optional
occupation number, by default None
iw_cutoff : double, optional
imaginary grid cutoff for fitting, by default None
eval_freqs : double 1d array, optional
imaginary frequency points at which to evaluate self energy.
Self energy is evaluated at ef + 1j * eval_freqs.
mo_energy_w : double array, optional
orbital energy in W, by default None
fullsigma : bool, optional
calculate off-diagonal elements, by default False
Returns
-------
sigma : complex 2d or 3d array
self-energy on the imaginary axis
omega : complex 1d array
imaginary frequency grids of self-energy
"""
if eval_freqs is None:
eval_freqs = quad_freqs
nocc = gw.nocc
nmo = gw.nmo
nw = len(quad_freqs)
nw2 = len(eval_freqs)
norbs = len(orbs)
with temporary_env(gw.with_df, verbose=0), temporary_env(gw.mol, verbose=0):
naux = gw.with_df.get_naoaux()
mo_energy_g = mo_energy
if mo_energy_w is None:
mo_energy_w = mo_energy
if mo_occ is None:
mo_occ = _mo_energy_without_core(gw, gw.mo_occ)
# integration on numerical grids
if iw_cutoff is not None and gw.rdm is False:
nw_sigma = sum(eval_freqs < iw_cutoff)
else:
nw_sigma = nw2
nw_cutoff = nw_sigma if iw_cutoff is None else sum(eval_freqs < iw_cutoff)
omega = ef + 1j * eval_freqs[:nw_sigma]
Pi = np.zeros(shape=[nw, naux, naux], dtype=np.double)
if hasattr(gw._scf, 'sigma'):
nseg = (nmo + gw.segsize - 1) // gw.segsize # No empty trailing block.
else:
nseg = (nocc + gw.segsize - 1) // gw.segsize
for i in range(nseg):
if hasattr(gw._scf, 'sigma'):
orb_start = i * gw.segsize
orb_end = min((i + 1) * gw.segsize, nmo)
ijslice = (orb_start, orb_end, 0, nmo)
else:
orb_start = i * gw.segsize
orb_end = min((i + 1) * gw.segsize, nocc)
ijslice = (orb_start, orb_end, nocc, nmo)
Lia = gw.loop_ao2mo(mo_coeff=mo_coeff, ijslice=ijslice)
for w in range(nw):
if hasattr(gw._scf, 'sigma'):
eia = mo_energy_w[orb_start:orb_end, None] - mo_energy_w[None, :]
fia = mo_occ[orb_start:orb_end, None] - mo_occ[None, :]
eia = eia * fia / (quad_freqs[w] ** 2 + eia**2)
Pia = lib.broadcast_mul(Lia, eia)
else:
eia = mo_energy_w[orb_start:orb_end, None] - mo_energy_w[None, nocc:]
eia = 4.0 * eia / (quad_freqs[w] ** 2 + eia**2)
Pia = lib.broadcast_mul(Lia, eia)
Pi[w] += np.matmul(Pia.reshape(naux, -1), Lia.reshape(naux, -1).T)
del Pia
del Lia
for w in range(nw):
Pi[w] = np.linalg.inv(np.eye(naux) - Pi[w]) - np.eye(naux)
Pi_inv = Pi
logger.info(gw, f'Starting get_sigma_diag_outcore main loop with {nseg} segments.')
if fullsigma is False:
sigma_real = np.zeros(shape=[nw_sigma, norbs], dtype=np.double)
sigma_imag = np.zeros(shape=[nw_sigma, norbs], dtype=np.double)
else:
sigma_real = np.zeros(shape=[nw_sigma, norbs, norbs], dtype=np.double)
sigma_imag = np.zeros(shape=[nw_sigma, norbs, norbs], dtype=np.double)
if fullsigma is False:
Qmn = None
Wmn = None
naux_ones = np.ones((1, naux))
emo = omega[None] - mo_energy_g[:, None]
nseg = (nmo + gw.segsize - 1) // gw.segsize # No empty trailing block.
for i in range(nseg):
if gw.verbose >= 4:
gw.stdout.write(f'{i:4d} ')
if i % 12 == 11:
gw.stdout.write('\n')
gw.stdout.flush()
orb_start = i * gw.segsize
orb_end = min((i + 1) * gw.segsize, nmo)
ijslice = (orb_start, orb_end, 0, nmo)
Lpq = gw.loop_ao2mo(mo_coeff=mo_coeff, ijslice=ijslice)
l_slice = np.ascontiguousarray(Lpq[:, :, mkslice(orbs)].reshape(naux, -1))
del Lpq
for w in range(nw):
g0 = quad_wts[w] * emo[orb_start:orb_end] / (emo[orb_start:orb_end] ** 2 + quad_freqs[w] ** 2)
# split g0 into real and imag parts to avoid costly type conversions
g0r = np.ascontiguousarray(g0.real)
g0i = np.ascontiguousarray(g0.imag)
if fullsigma is False:
Qmn = np.matmul(Pi_inv[w].T, l_slice)
Qmn *= l_slice
Wmn = np.matmul(naux_ones, Qmn)
# sigma -= einsum('mn,mw->wn', Wmn, g0) / np.pi
sla.blas.dgemm(
alpha=-1.0 / np.pi,
a=Wmn.reshape(orb_end - orb_start, norbs).T,
b=g0r.T,
c=sigma_real.T,
trans_a=0,
trans_b=1,
beta=1.0,
overwrite_c=True,
)
sla.blas.dgemm(
alpha=-1.0 / np.pi,
a=Wmn.reshape(orb_end - orb_start, norbs).T,
b=g0i.T,
c=sigma_imag.T,
trans_a=0,
trans_b=1,
beta=1.0,
overwrite_c=True,
)
else:
Wmn = np.zeros(shape=[orb_end - orb_start, norbs, norbs], dtype=np.double)
for orbm in range(orb_end - orb_start):
Qmn = np.matmul(Pi_inv[w], l_slice.reshape(naux, orb_start - orb_end, norbs)[:, orbm, :])
np.matmul(Qmn.T, l_slice.reshape(naux, orb_start - orb_end, norbs)[:, orbm, :], out=Wmn[orbm])
# sigma -= einsum('mnl,mw->wnl', Wmn, g0)/np.pi
sla.blas.dgemm(
alpha=-1.0 / np.pi,
a=Wmn.reshape(orb_end - orb_start, norbs * norbs).T,
b=g0r.T,
c=sigma_real.reshape(nw_sigma, norbs * norbs).T,
trans_a=0,
trans_b=1,
beta=1.0,
overwrite_c=True,
)
sla.blas.dgemm(
alpha=-1.0 / np.pi,
a=Wmn.reshape(orb_end - orb_start, norbs * norbs).T,
b=g0i.T,
c=sigma_imag.reshape(nw_sigma, norbs * norbs).T,
trans_a=0,
trans_b=1,
beta=1.0,
overwrite_c=True,
)
sigma = sigma_real + 1.0j * sigma_imag
if fullsigma is False:
sigma = np.ascontiguousarray(sigma.transpose(1, 0))
else:
sigma = np.ascontiguousarray(sigma.transpose(1, 2, 0))
logger.info(gw, '\nFinished get_sigma_diag_outcore main loop.')
if gw.rdm is True:
gw.sigmaI = sigma
return sigma[..., :nw_cutoff], omega[:nw_cutoff]
def get_g0(omega, mo_energy, eta):
"""Get non-interacting Green's function.
Parameters
----------
omega : double or complex array
frequency grids
mo_energy : double 1d array
orbital energy
eta : double
broadening parameter
Returns
-------
gf0 : complex 3d array
non-interacting Green's function
"""
nmo = len(mo_energy)
nw = len(omega)
gf0 = np.zeros(shape=[nmo, nmo, nw], dtype=np.complex128)
gf0[np.arange(nmo), np.arange(nmo), :] = 1.0 / (omega[np.newaxis, :] + 1j * eta - mo_energy[:, np.newaxis])
return gf0
def _mo_energy_without_core(gw, mo_energy):
"""Get non-frozen orbital energy.
Parameters
----------
gw : GWAC
GW object, provides attributes: frozen, mo_occ, _nmo
mo_energy : double 1d array
full orbital energy
Returns
-------
mo_energy : double 1d array
non-frozen orbital energy
"""
return mo_energy[get_frozen_mask(gw)]
def _mo_without_core(gw, mo):
"""Get non-frozen orbital coefficient.
Parameters
----------
gw : GWAC
GW object, provides attributes: frozen, mo_occ, _nmo
mo : double 3d array
full orbital coefficient
Returns
-------
mo : double 3d darray
non-frozen orbital coefficient
"""
return mo[:, get_frozen_mask(gw)]
def set_frozen_orbs(gw):
"""Set orbs and orbs_frz attributes from frozen mask.
orbs: list of orbital index in all orbitals
orbs_frz: list of orbital index in non-frozen orbitals
Parameters
----------
gw : GWAC
restricted GW object
"""
if gw.frozen is not None:
if gw.orbs is not None:
if isinstance(gw.frozen, (int, np.int64)):
# frozen core
gw.orbs_frz = [x - gw.frozen for x in gw.orbs]
else:
# frozen list
assert isinstance(gw.frozen[0], (int, np.int64))
gw.orbs_frz = []
for orbi in gw.orbs:
count = len([p for p in gw.frozen if p <= orbi])
gw.orbs_frz.append(orbi - count)
if any(np.array(gw.orbs_frz) < 0):
raise RuntimeError('GW orbs must be larger than frozen core!')
else:
gw.orbs_frz = range(gw.nmo)
gw.orbs = range(len(gw._scf.mo_energy))
if isinstance(gw.frozen, (int, np.int64)):
gw.orbs = list(set(gw.orbs) - set(range(gw.frozen)))
else:
assert isinstance(gw.frozen[0], (int, np.int64))
gw.orbs = list(set(gw.orbs) - set(gw.frozen))
else:
if gw.orbs is None:
gw.orbs = gw.orbs_frz = range(len(gw._scf.mo_energy))
else:
gw.orbs_frz = gw.orbs
return
class GWAC(lib.StreamObject):
def __init__(self, mf, frozen=None, auxbasis=None):
self.mol = mf.mol # mol object
self._scf = mf # mean-field object
self.verbose = self.mol.verbose # verbose level
self.stdout = self.mol.stdout # standard output
self.max_memory = mf.max_memory # max memory in MB
self.auxbasis = auxbasis # auxiliary basis set for density fitting
# options
self.frozen = frozen # frozen orbital options
self.orbs = None # list of orbital index in full nmo
self.fullsigma = False # calculate off-diagonal self-energy
self.rdm = False # calculate GW density matrix
self.vhf_df = False # use density-fitting for exchange self-energy
self.outcore = False # low-memory routine to calculate self-energy
self.segsize = 100 # number of orbitals in one segment for outcore
self.eta = 5.0e-3 # broadening parameter
self.nw = 100 # number of quadrature points for integration
self.nw2 = None # number of points at which to evaluate self-energy
self.ac = 'pade' # analytical continuation method
self.ac_iw_cutoff = 5.0 # imaginary frequency cutting for fitting self-energy
self.ac_pade_npts = 18 # number of selected points for Pade approximation
self.ac_pade_step_ratio = 2.0 / 3.0 # final/initial step size for Pade approximation
self.ac_pes_mmax = 6
self.ac_pes_maxiter = 200 # max iteration in PES fitting
self.ac_pes_disp = False
self.ac_idx = None # indices of iw to use for AC fitting
self.qpe_max_iter = 100 # max iteration in iteratively solving quasiparticle equation
self.qpe_tol = 1.0e-6 # tolerance in Newton method for iteratively quasiparticle equation
self.qpe_linearized = False # use linearized quasiparticle equation
self.qpe_linearized_range = [0.5, 1.5] # Z-shot factor range, if not in this range, z=1
self.writefile = 0 # write file level
# don't modify the following attributes, they are not input options
self._nocc = None # number of occupied orbitals
self._nmo = None # number of orbitals (exclude frozen orbitals)
self.orbs_frz = None # list of orbital index in non-frozen orbitals
self.mo_energy = np.array(mf.mo_energy, copy=True) # quasiparticle energy
self.mo_coeff = np.array(mf.mo_coeff, copy=True) # quasiparticle orbital coefficient
self.mo_occ = np.array(mf.mo_occ, copy=True) # quasiparticle orbital occupation
self.Lpq = None # three-center density-fitting matrix in MO
# results
self.vk = None # exchange self-energy matrix
self.vxc = None # mean-field vxc matrix
self.freqs = None # frequency grids, size=nw2
self.wts = None # weights of frequency grids, size=nw2
self.ef = None # Fermi level
self.sigmaI = None # self-energy in the imaginary axis
self.acobj = None # analytical continuation object
return
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)
log.info('frozen orbitals = %s', self.frozen)
log.info('off-diagonal self-energy = %s', self.fullsigma)
log.info('GW density matrix = %s', self.rdm)
log.info('density-fitting for exchange = %s', self.vhf_df)
log.info('outcore for self-energy= %s', self.outcore)
if self.outcore is True:
log.info('outcore segment size = %d', self.segsize)
log.info('broadening parameter = %.3e', self.eta)
if self.nw2 is None:
log.info('number of grids = %d', self.nw)
else:
log.info('grid size for W is %d', self.nw)
log.info('grid size for self-energy is %d', self.nw2)
log.info('analytic continuation method = %s', self.ac)
log.info('imaginary frequency cutoff = %.1f', self.ac_iw_cutoff)
if self.ac == 'pade':
log.info('Pade points = %d', self.ac_pade_npts)
log.info('Pade step ratio = %.3f', self.ac_pade_step_ratio)
log.info('use perturbative linearized QP eqn = %s', self.qpe_linearized)
if self.qpe_linearized is True:
log.info('linearized factor range = %s', self.qpe_linearized_range)
else:
log.info('QPE max iter = %d', self.qpe_max_iter)
log.info('QPE tolerance = %.1e', self.qpe_tol)
log.info('')
return
@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
def kernel(self):
"""Do one-shot GW calculation using analytical continuation."""
# smeared GW needs denser grids to be accurate
if hasattr(self._scf, 'sigma'):
assert self.frozen is None or self.nmo == self.mo_occ.size
self.nw = max(400, self.nw)
self.ac_pade_npts = 18
self.ac_pade_step_ratio = 5.0 / 6.0
if self.Lpq is None:
self.initialize_df(auxbasis=self.auxbasis)
cput0 = (time.process_time(), time.perf_counter())
self.dump_flags()
kernel(self)
logger.timer(self, 'GW', *cput0)
return
def initialize_df(self, auxbasis=None):
"""Initialize density fitting.
Parameters
----------
auxbasis : str, optional
name of auxiliary basis set, by default None
"""
if getattr(self._scf, 'with_df', None):
self.with_df = self._scf.with_df
else:
self.with_df = df.DF(self._scf.mol)
if auxbasis is not None:
self.with_df.auxbasis = auxbasis
else:
try:
self.with_df.auxbasis = df.make_auxbasis(self._scf.mol, mp2fit=True)
except RuntimeError:
self.with_df.auxbasis = df.make_auxbasis(self._scf.mol, mp2fit=False)
self._keys.update(['with_df'])
return
def ao2mo(self, mo_coeff=None):
"""Transform density-fitting integral from AO to MO.
Parameters
----------
mo_coeff : double 2d array, optional
coefficient from AO to MO, by default None
Returns
-------
Lpq : double 3d array
three-center density-fitting matrix in MO
"""
if mo_coeff is None:
mo_coeff = self.mo_coeff
nmo = mo_coeff.shape[1]
nao = mo_coeff.shape[0]
naux = self.with_df.get_naoaux()
nmo_pair = nmo * (nmo + 1) // 2
nao_pair = nao * (nao + 1) // 2
mem_incore = (max(nao_pair * naux, nmo**2 * naux) + nmo_pair * naux) * 8 / 1e6
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 < self.max_memory) or self.mol.incore_anyway:
Lpq = nr_e2(self.with_df._cderi, mo, ijslice, aosym='s2', out=Lpq)
return Lpq.reshape(naux, nmo, nmo)
else:
raise MemoryError
def loop_ao2mo(self, mo_coeff=None, ijslice=None, blksize=None):
"""Transform density-fitting integral from AO to MO by block.
Parameters
----------
mo_coeff : double 2d array, optional
coefficient from AO to MO, by default None
ijslice : tuple, optional
tuples for (1st idx start, 1st idx end, 2nd idx start, 2nd idx end), by default None
Returns
-------
eri_3d : double 3d array
three-center density-fitting matrix in MO in a block
"""
nmo = self.nmo
naux = self.with_df.get_naoaux()
mo = np.asarray(mo_coeff, order='F')
if ijslice is None:
ijslice = (0, nmo, 0, nmo)
nislice = ijslice[1] - ijslice[0]
njslice = ijslice[3] - ijslice[2]
with_df = self.with_df
if blksize is None:
mem_now = lib.current_memory()[0]
max_memory = max(2000, (self.max_memory - mem_now) * 0.3)
blksize = int(min(naux, max(with_df.blockdim, (max_memory * 1e6 / 8) / (nmo * nmo))))
eri_3d = np.empty((naux, nislice * njslice))
for eri1, (p0, p1) in zip(with_df.loop(blksize=blksize), lib.prange(0, naux, blksize)):
nr_e2(eri1, mo, ijslice, aosym='s2', mosym='s1', out=eri_3d[p0:p1])
return eri_3d.reshape(naux, nislice, njslice)
def get_ef(self, mo_energy=None):
"""Get Fermi level.
For gapped systems, Fermi level is computed as the average between HOMO and LUMO.
For metallic systems, Fermi level is optmized according to mo_energy.
Parameters
----------
mo_energy : double 1d array, optional
orbital energy, by default None
Returns
-------
ef : double
Fermi level
"""
if mo_energy is None:
mo_energy = self.mo_energy
if hasattr(self._scf, 'sigma'):
f_occ = _fermi_smearing_occ if self._scf.smearing_method.lower() == 'fermi' else _gaussian_smearing_occ
ef = _smearing_optimize(f_occ, mo_energy, self._scf.mol.nelectron / 2, self._scf.sigma)[0]
else:
# working with full space mo_energy and nocc here
nocc = self._scf.mol.nelectron // 2
if (mo_energy[nocc] - mo_energy[nocc - 1]) < 1e-3:
logger.warn(self, 'GW not well-defined for degeneracy!')
ef = (mo_energy[nocc - 1] + mo_energy[nocc]) * 0.5
return ef
def energy_tot(self):
"""Compute GW total energy according to Galitskii-Migdal formula.
See equation 6 in doi.org/10.1103/PhysRevB.86.081102,
and equation 11 in doi.org/10.1021/acs.jctc.0c01264.
NOTE: self-energy evaluted by numerical integration at large frequency is difficult to be accurate,
so this function is numerically unstable.
Returns
-------
E_tot : double
GW total energy
E_hf : double
Hartree-Fock total energy
Ec : double
GW correlation energy