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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: Jiachen Li <lijiachen.duke@gmail.com>
#
'''
Spin-unrestricted G0W0 method based on the fully analytic scheme.
This implementation has N^6 scaling, and is accurate for all states.
Reference:
J. Chem. Theory Comput. 9, 1, 232-246 (2013)
'''
import numpy as np
import scipy
import time
from pyscf import dft, scf
from pyscf.lib import einsum, logger, temporary_env
from pyscf.gw.ugw_ac import UGWAC, get_g0
from pyscf.gw.utils.ac_grid import _get_scaled_legendre_roots
def kernel(gw):
# local variables for convenience
nmo = gw.nmo[0]
nocc = gw.nocc
mf = gw._scf
mo_energy = gw.mo_energy
mo_coeff = gw.mo_coeff
mf_mo_energy = gw._scf.mo_energy
if gw.Lpq is None:
with temporary_env(gw.with_df, verbose=0), temporary_env(gw.mol, verbose=0):
gw.Lpq = gw.ao2mo(gw.mo_coeff)
# mean-field exchange-correlation matrix
with temporary_env(mf, verbose=0):
vxc_ao = mf.get_veff()
vj_ao = mf.get_j()
vxc_ao[0] = vxc_ao[0] - (vj_ao[0] + vj_ao[1])
vxc_ao[1] = vxc_ao[1] - (vj_ao[0] + vj_ao[1])
gw.vxc = vxc = np.asarray([mo_coeff[s].T @ vxc_ao[s] @ mo_coeff[s] for s in range(2)])
# exchange self-energy
if gw.vhf_df is False:
dm = mf.make_rdm1()
if (not isinstance(mf, dft.uks.UKS)) and isinstance(mf, scf.uhf.UHF):
uhf = mf
else:
uhf = scf.UHF(gw.mol)
vk_ao = uhf.get_veff(dm=dm)
vj_ao = uhf.get_j(dm=dm)
vk_ao[0] = vk_ao[0] - (vj_ao[0] + vj_ao[1])
vk_ao[1] = vk_ao[1] - (vj_ao[0] + vj_ao[1])
vk = np.zeros(shape=[2, nmo, nmo], dtype=np.double)
for s in range(2):
vk[s] = mo_coeff[s].T @ vk_ao[s] @ mo_coeff[s]
else:
vk = np.zeros(shape=[2, nmo, nmo], dtype=np.double)
for s in range(2):
vk[s] = -einsum('Lpi,Liq->pq', gw.Lpq[s, :, :, : nocc[s]], gw.Lpq[s, :, : nocc[s], :])
gw.vk = vk
# diagonalize the RPA matrix
gw.exci, _ = exci, xpy = diagonalize_phrpa(nocc=nocc, mo_energy=mo_energy, Lpq=gw.Lpq, RPAE=gw.RPAE)
# calculate the transition density
gw.rho = rho = get_transition_density(nocc=nocc, xpy=xpy, Lpq=gw.Lpq)
# calculate the self-energy
sigma = get_sigma(
nocc=nocc, mo_energy=mo_energy, mo_energy_prev=mo_energy, exci=exci, rho=rho, eta=gw.eta, fullsigma=False
)
# quasiparticle equation
if gw.qpe_linearized is True:
derivative = get_sigma_derivative(
nocc=nocc, mo_energy=mo_energy, mo_energy_prev=mo_energy, exci=exci, rho=rho, eta=gw.eta
)
z = 1.0 / (1.0 - derivative)
if gw.qpe_linearized_range is not None:
np.where((z < gw.qpe_linearized_range[0]) | (z > gw.qpe_linearized_range[1]), 1.0, z)
mo_energy = mf_mo_energy + z * (vk + sigma - vxc).diagonal(axis1=1, axis2=2)
else:
def quasiparticle(qp_energy):
sigma = get_sigma(
nocc=nocc, mo_energy=qp_energy, mo_energy_prev=mf_mo_energy, exci=exci, rho=rho, eta=gw.eta,
fullsigma=False)
return qp_energy - mf_mo_energy - (sigma + vk - vxc).diagonal(axis1=1, axis2=2)
try:
mo_energy = scipy.optimize.newton(
quasiparticle, mf_mo_energy, tol=gw.qpe_tol * nmo * 2.0, maxiter=gw.qpe_max_iter
)
except RuntimeError:
logger.warn(gw, 'quasiparticle equation fails to converge!')
gw.mo_energy = mo_energy
with np.printoptions(threshold=nmo):
logger.debug(gw, ' GW mo_energy spin-up =\n%s', mo_energy[0])
logger.debug(gw, ' GW mo_energy spin-down =\n%s', mo_energy[1])
logger.warn(gw, 'GW QP energies may not be sorted from min to max')
return
def diagonalize_phrpa(nocc, mo_energy, Lpq, RPAE=False):
r"""Diagonalize particle-hole RPA matrix.
The phRPA equation is solved for the response function for the screening interaction,
which is defined in equation 66-68 in doi.org/10.1021/ct300648t
The phRPA equation is reformulated as (A-B)(A+B)|X+Y>= \omega^2|X+Y>, which is equation 15 in 10.1063/1.477483
RPAE equations are defined as equation 2 and 3 in 10.1103/PhysRevA.85.042507
Parameters
----------
nocc : int 1d array
the number of occupied orbitals
mo_energy : double 2d array
orbital energy
Lpq : double 4d array
density-fitting matrix in the MO space
RPAE : bool, optional
add exchange in response, by default False
Returns
-------
w : double 1d array
excitation energy
v : double 2d array
X+Y eigenvector
"""
nspin, nmo = mo_energy.shape
nvir = [(nmo - nocc[i]) for i in range(nspin)]
rpa_dim = [(nocc[i] * nvir[i]) for i in range(nspin)]
full_dim = rpa_dim[0] + rpa_dim[1] if nspin == 2 else rpa_dim[0]
# direct ERI contribution
A = np.zeros(shape=[full_dim, full_dim], dtype=np.double)
for i in range(nspin):
for j in range(nspin):
A[i * rpa_dim[0] : i * rpa_dim[0] + rpa_dim[i], j * rpa_dim[0] : j * rpa_dim[0] + rpa_dim[j]] = einsum(
'Lia,Ljb->iajb', Lpq[i, :, : nocc[i], nocc[i] :], Lpq[j, :, : nocc[j], nocc[j] :]
).reshape(rpa_dim[i], rpa_dim[j])
B = np.array(A, copy=True)
# exchange ERI contribution
if RPAE is True:
for i in range(nspin):
A[i * rpa_dim[0] : i * rpa_dim[0] + rpa_dim[i], i * rpa_dim[0] : i * rpa_dim[0] + rpa_dim[i]] -= einsum(
'Lij,Lab->iajb', Lpq[i, :, : nocc[i], nocc[i] :], Lpq[i, :, : nocc[i], nocc[i] :]
).reshape(rpa_dim[i], rpa_dim[i])
B[i * rpa_dim[0] : i * rpa_dim[0] + rpa_dim[i], i * rpa_dim[0] : i * rpa_dim[0] + rpa_dim[i]] -= einsum(
'Lib,Lja->iajb', Lpq[i, :, : nocc[i], nocc[i] :], Lpq[i, :, : nocc[i], nocc[i] :]
).reshape(rpa_dim[i], rpa_dim[i])
# orbital energy contribution
orb_diff = []
for i in range(nspin):
orb_diff.append(np.asarray(mo_energy[i][None, nocc[i]:] - mo_energy[i][:nocc[i], None]).reshape(-1))
orb_diff = np.concatenate(orb_diff, axis=0)
np.fill_diagonal(A, A.diagonal() + orb_diff)
# A+B and A-B matrix
apb = A + B
amb = A - B
# equation 15 in doi/10.1063/1.477483, solved by LAPACK function dspgv
w, v = scipy.linalg.eigh(apb, amb, type=3)
w = np.sqrt(w)
v = v.T
# (A-B)^{-1/2} |X+Y> to |X+Y>
for i in range(w.shape[0]):
v[i, :] /= np.sqrt(w[i])
return w, v
def get_transition_density(nocc, xpy, Lpq):
"""Calculate the transition density.
Equation 85 in doi/abs/10.1021/ct300648t
Parameters
----------
nocc : int 1d array
the number of occupied orbitals
xpy : double 2d array
X+Y eigenvector
Lpq : double 4d ndarray
density-fitting matrix in the MO space
Returns
-------
rho : double 4d array
transition density
"""
nspin, naux, nmo, _ = Lpq.shape
nvir = [(nmo - nocc[i]) for i in range(nspin)]
rpa_dim = [(nocc[i] * nvir[i]) for i in range(nspin)]
full_dim = rpa_dim[0] + rpa_dim[1] if nspin == 2 else rpa_dim[0]
rho = np.zeros(shape=[nspin, full_dim, nmo, nmo], dtype=np.double)
t = np.zeros(shape=[full_dim, naux], dtype=np.double)
for i in range(nspin):
t += np.matmul(
xpy[:, i * rpa_dim[0] : i * rpa_dim[0] + rpa_dim[i]], Lpq[i, :, : nocc[i], nocc[i] :].reshape(naux, -1).T
)
for i in range(nspin):
rho[i] = np.matmul(t, Lpq[i].reshape(naux, -1)).reshape(-1, nmo, nmo)
return rho
def get_sigma(nocc, mo_energy, mo_energy_prev, exci, rho, eta=1.0e-5, fullsigma=False, mode='b'):
"""Get the real part of the GW correlation self-energy.
Equation 83 in doi.org/10.1021/ct300648t
mode 'a' and 'b' correspond to equation 10 and 11 in doi.org/10.1103/PhysRevB.76.165106
Parameters
----------
nocc : int 1d array
the number of occupied orbitals
mo_energy : double 2d array
orbital energy
mo_energy_prev : double 2d array
orbital energy in previous iteration
exci : double 1d array
phRPA excitation energy
rho : double 4d array
transition density
eta : double, optional
broadening parameter, by default 1.0e-5
fullsigma : bool, optional
calculate off-diagonal elements, by default False
mode : str, optional
mode for off-diagonal elements, by default "b"
Returns
-------
sigma : double 3d array
real part of the GW static correlation self-energy
"""
nspin, nmo = mo_energy.shape
eta2 = np.square(3.0 * eta)
homo = max(mo_energy[0][nocc[0] - 1], mo_energy[1][nocc[1] - 1])
lumo = min(mo_energy[0][nocc[0]], mo_energy[1][nocc[1]])
ef = (homo + lumo) / 2.0
sigma = np.zeros(shape=[nspin, nmo, nmo], dtype=np.double)
if fullsigma is False:
for s in range(nspin):
energy_occ = mo_energy[s][:, None, None] - mo_energy_prev[s][None, : nocc[s], None] + exci[None, None, :]
energy_vir = mo_energy[s][:, None, None] - mo_energy_prev[s][None, nocc[s] :, None] - exci[None, None, :]
energy = np.concatenate([energy_occ, energy_vir], axis=1)
energy = energy / (np.square(energy) + eta2)
sigma[s][np.diag_indices(nmo)] = einsum('mpr,prm->p', np.square(rho[s]), energy)
else:
assert mode in ['a', 'b']
for s in range(nspin):
if mode == 'a':
# mode A : off-diagonal evaluated as average, equation 11 in doi.org/10.1103/PhysRevB.76.165106
energy_occ = mo_energy[s, :, None, None] - mo_energy_prev[s, None, :nocc[s], None] + exci[None, None, :]
energy_vir = mo_energy[s, :, None, None] - mo_energy_prev[s, None, nocc[s]:, None] - exci[None, None, :]
energy = np.concatenate([energy_occ, energy_vir], axis=1)
energy = energy / (np.square(energy) + eta2)
sigma[s] = einsum('mpr,mqr,prm->pq', rho[s], rho[s], energy)
sigma[s] += einsum('mpr,mqr,qrm->pq', rho[s], rho[s], energy)
sigma[s] *= 0.5
elif mode == 'b':
# mode B: off-diagonal evaluated at Fermi-level, equation 10 in doi.org/10.1103/PhysRevB.76.165106
energy_occ = ef - mo_energy_prev[s][:nocc[s], None] + exci[None, :]
energy_vir = ef - mo_energy_prev[s][nocc[s]:, None] - exci[None, :]
energy = np.concatenate([energy_occ, energy_vir])
energy = energy / (np.square(energy) + eta2)
sigma[s] = einsum('mpr,mqr,rm->pq', rho[s], rho[s], energy)
energy_occ = mo_energy[s, :, None, None] - mo_energy_prev[s, None, :nocc[s], None] + exci[None, None, :]
energy_vir = mo_energy[s, :, None, None] - mo_energy_prev[s, None, nocc[s]:, None] - exci[None, None, :]
energy = np.concatenate([energy_occ, energy_vir], axis=1)
energy = energy / (np.square(energy) + eta2)
sigma_diag = einsum('mpr,prm->p', np.square(rho[s]), energy)
np.fill_diagonal(sigma[s], sigma_diag)
return sigma
def get_sigma_derivative(nocc, mo_energy, mo_energy_prev, exci, rho, eta=1.0e-5):
"""Get the first-order derivative of the self-energy to the frequency.
Equation.84 in doi.org/10.1021/ct300648t
Parameters
----------
nocc : int 1d array
the number of occupied orbitals
mo_energy : double 2d ndarray
orbital energy
mo_energy_prev : double 2d array
orbital energy in previous iteration
exci : double 1d array
phRPA excitation energy
rho : double 4d array
transition density
eta : double, optional
broadening parameter, by default 1.0e-5
Returns
-------
derivative : double 2d ndarray
first-order derivative of the correlation self-energy to the frequency
"""
nspin, nmo = mo_energy.shape
eta2 = np.square(3.0 * eta)
derivative = np.zeros(shape=[nspin, nmo], dtype=np.double)
for s in range(nspin):
# occupied part
energy = mo_energy[s][:, None, None] - mo_energy_prev[s][None, : nocc[s], None] + exci[None, None, :]
energy = np.square(energy)
energy = (eta2 - energy) / (energy + eta2) ** 2
derivative[s] += einsum('mpi,pim->p', np.square(rho[s][:, :, : nocc[s]]), energy)
# virtual part
energy = mo_energy[s][:, None, None] - mo_energy_prev[s][None, nocc[s] :, None] - exci[None, None, :]
energy = np.square(energy)
energy = (eta2 - energy) / (energy + eta2) ** 2
derivative[s] += einsum('mpa,pam->p', np.square(rho[s][:, :, nocc[s] :]), energy)
return derivative
def make_gf(gw, omega, eta, fullsigma=True, mode='linear'):
"""Get exact dynamical Green's function and self-energy.
Dynamical self-energy is evaluated as equation 78 in doi.org/10.1021/ct300648t
Two modes for solving Dyson equation
"dyson" for using inverse Dyson equation.
"linear" for G = G0 + G0 Sigma G0, as equation 16 in doi.org/10.1021/acs.jctc.0c01264
Parameters
----------
gw : UGWExact
gw object, provides attributes: nocc, nmo, _scf.mo_energy, exci, rho, vk, vxc
omega : double or complex array
frequency grids
eta : double
broadening parameter
fullsigma : bool, optional
calculate off-diagonal elements, by default True
mode : str, optional
mode for Dyson equation, 'linear' or 'dyson', by default 'linear'
Returns
-------
gf : complex 4d array
GW Green's function
gf0 : complex 4d array
non-interacting Green's function
sigma : complex 4d array
self-energy
"""
nmo = gw.nmo[0]
nocc = gw.nocc
exci = gw.exci
mo_energy = np.asarray(gw._scf.mo_energy)
# get self-energy
# NOTE: this is for G0W0 Green's function, so mo_energy is from the SCF calculation
sigma = np.zeros(shape=[2, nmo, nmo, len(omega)], dtype=np.complex128)
for s in range(2):
energy_occ = omega[:, None, None] - mo_energy[s, None, : nocc[s], None] + (exci[None, None, :] - 3.0 * 1j * eta)
energy_vir = omega[:, None, None] - mo_energy[s, None, nocc[s] :, None] - (exci[None, None, :] - 3.0 * 1j * eta)
energy = np.concatenate([energy_occ, energy_vir], axis=1)
energy = 1.0 / energy
if fullsigma is False:
sigma_diag = einsum('mpr,wrm->pw', np.square(gw.rho[s]), energy)
for iw in range(len(omega)):
sigma[s, ..., iw] = np.diag(sigma_diag[:, iw])
else:
sigma[s] = einsum('mpr,mqr,wrm->pqw', gw.rho[s], gw.rho[s], energy)
# Dyson equation for Green's function
gf0 = get_g0(omega=omega, mo_energy=gw._scf.mo_energy, eta=eta)
gf = np.zeros_like(gf0)
sigma_diff = np.array(sigma, copy=True)
if fullsigma is True:
for iw in range(len(omega)):
sigma_diff[:, :, :, iw] += gw.vk - gw.vxc
else:
for s in range(2):
for iw in range(len(omega)):
for i in range(nmo): # Apply the static correction to every orbital.
sigma_diff[s, i, i, iw] += gw.vk[s, i, i] - gw.vxc[s, i, i]
for s in range(2):
for iw in range(len(omega)):
if mode == 'linear':
gf[s, :, :, iw] = gf0[s, :, :, iw] + gf0[s, :, :, iw] @ sigma_diff[s, :, :, iw] @ gf0[s, :, :, iw]
elif mode == 'dyson':
gf[s, :, :, iw] = np.linalg.inv(np.linalg.inv(gf0[s, :, :, iw]) - sigma_diff[s, :, :, iw])
return gf, gf0, sigma
def make_diag_dos(gw, omega, eta):
"""Get density of states using diagonal self-energy.
Equation 7 in doi.org/10.1021/acs.jctc.2c00617
Parameters
----------
gw : UGWExact
gw object, provides attributes: nocc, nmo, _scf.mo_energy, exci, rho, vk, vxc
omega : complex 1d array
frequency grids
eta : double
broadening parameter
Returns
-------
dos : double 3d array
orbital-resolved density of states
"""
nmo = gw.nmo[0]
nocc = gw.nocc
mo_energy = np.asarray(gw._scf.mo_energy)
eta2 = np.square(3.0 * eta)
sigma_real = np.zeros(shape=[2, nmo, len(omega)], dtype=np.double)
sigma_imag = np.zeros_like(sigma_real)
for s in range(2):
energy_occ = omega[:, None, None] - mo_energy[s, None, : nocc[s], None] + gw.exci[None, None, :]
energy_vir = omega[:, None, None] - mo_energy[s, None, nocc[s] :, None] - gw.exci[None, None, :]
energy = np.concatenate([energy_occ, energy_vir], axis=1)
energy_real = energy / (np.square(energy) + eta2)
energy_imag = eta / (np.square(energy) + eta2)
energy_imag[:, nocc[s] :, :] *= -1.0
sigma_real[s] = einsum('mpr,wrm->pw', np.square(gw.rho[s]), energy_real) * 2.0
sigma_imag[s] = einsum('mpr,wrm->pw', np.square(gw.rho[s]), energy_imag) * 2.0
vk_minus_vxc = (gw.vk - gw.vxc).diagonal(axis1=1, axis2=2)
ereal = omega[None, None, :] - mo_energy[:, :, None] - (sigma_real + vk_minus_vxc[:, :, None])
dos = abs(sigma_imag) / ereal**2 + sigma_imag**2
dos /= np.pi
return dos
class UGWExactDF(UGWAC):
def __init__(self, mf, auxbasis=None):
UGWAC.__init__(self, mf, frozen=None, auxbasis=auxbasis)
# options
self.RPAE = False # exchange in RPA response
# matrices
self.exci = None # RPA excitation energy
self.rho = None # transition density
return
def dump_flags(self, verbose=None):
log = logger.Logger(self.stdout, self.verbose)
log.info('')
log.info('******** %s ********', self.__class__)
log.info('method = %s', self.__class__.__name__)
log.info('GW nmo = %s', self.nmo[0])
log.info('GW nocc = %s, nvir = %s', self.nocc, (self.nmo[s] - self.nocc[s] for s in range(2)))
log.info('broadening parameter = %.3e', self.eta)
log.info('RPA exchange response = %s', self.RPAE)
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
def kernel(self):
assert self.frozen is None or self.frozen == 0
assert self.orbs is None
assert self.outcore is False
assert hasattr(self._scf, 'sigma') is False
assert self.nmo[0] == self.nmo[1]
if self.Lpq is None:
self.initialize_df(auxbasis=self.auxbasis)
self.dump_flags()
cput0 = (time.process_time(), time.perf_counter())
kernel(self)
logger.timer(self, 'GW', *cput0)
return
make_gf = make_gf
make_diag_dos = make_diag_dos
def make_rdm1(self, nw=60, mode='linear', ao_repr=False):
r"""Get GW one-particle density matrix.
rdm1 = G(it=0) = \int dw G(iw)
Parameters
----------
nw : int, optional
number for imaginary frequency grids for integration, by default 60
mode : str, optional
mode for Dyson equation, by default "linear"
ao_repr : bool, optional
return density matrix in AO, by default False
Returns
-------
rdm1 : double ndarray
GW one-particle density matrix
"""
freqs, wts = _get_scaled_legendre_roots(nw)
homo = max(self._scf.mo_energy[0][self.nocc[0] - 1], self._scf.mo_energy[1][self.nocc[1] - 1])
lumo = min(self._scf.mo_energy[0][self.nocc[0]], self._scf.mo_energy[1][self.nocc[1]])
ef = (homo + lumo) / 2.0
omega = 1j * freqs + ef
gf = self.make_gf(omega=omega, eta=0, fullsigma=True, mode=mode)[0]
rdm1 = 1.0 / np.pi * einsum('sijw,w->sij', gf, wts).real + np.eye(self.nmo[0])[None, :, :] * 0.5
# symmetrize density matrix
for s in range(2):
rdm1[s] = 0.5 * (rdm1[s] + rdm1[s].T)
nelec = np.trace(rdm1, axis1=1, axis2=2)
logger.info(self, 'GW particle number up = %s, dn = %s, total = %s', nelec[0], nelec[1], nelec[0] + nelec[1])
if ao_repr is True:
for s in range(2):
rdm1[s] = self._scf.mo_coeff[s] @ rdm1[s] @ self._scf.mo_coeff[s].T
return rdm1
def energy_tot(self, nw=60):
"""Calculate GW total energy using Galitskii-Migdal formula.
V. M. Galitskii and A. B. Migdal, Zh. E ksp. Teor. Fiz. 34, 139~1958! @Sov. Phys. JETP 139, 96 ~1958!#
Working equation: equation A5 in Phys. Rev. B 88, 075105
Parameters
----------
nw : int, optional
number for imaginary frequency grids for integration, by default 60
Returns
-------
e_tot : double
GW total energy
e_hf : double
HF total energy
e_c : double
GW correlation energy
"""
# get correlation self-energy on the imaginary axis
freqs, wts = _get_scaled_legendre_roots(nw)
omega = 1j * freqs
# mode does not matter because gf is not used
_, gf0, sigma = self.make_gf(omega=omega, eta=0, fullsigma=True)
# GW correlation energy
g0_sigma_target = 1.0 / 2.0 / np.pi * einsum('sijw,sjiw,w->', gf0, sigma, wts)
e_c = g0_sigma_target.real
dm = self._scf.make_rdm1()
# keep this condition for embedding calculations
if (not isinstance(self._scf, dft.rks.RKS)) and isinstance(self._scf, scf.hf.RHF):
uhf = self._scf
else:
uhf = scf.UHF(self.mol)
with temporary_env(uhf, verbose=0):
e_hf = uhf.energy_elec(dm=dm)[0]
e_hf += self._scf.energy_nuc()
e_tot = e_hf + e_c
logger.info(self, 'HF energy@GW density = %.8f', e_hf)
logger.info(self, 'GW correlation energy = %.8f', e_c)
logger.info(self, 'GW total energy = %.8f', e_tot)
return e_tot, e_hf, e_c