#!/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>
#
'''
PBC spin-restricted G0W0-CD QP eigenvalues with k-point sampling
This implementation has the same scaling (N^4) as GW-AC, more robust but slower.
GW-CD is particularly recommended for accurate core and high-energy states.
Method:
See T. Zhu and G.K.-L. Chan, arxiv:2007.03148 (2020) for details
Compute Sigma directly on real axis with density fitting
through a contour deformation method
'''
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.ao2mo.incore import _conc_mos
from pyscf.pbc import df, dft, scf
from pyscf.pbc.mp.kmp2 import get_nocc, get_nmo, get_frozen_mask
from pyscf import __config__
einsum = lib.einsum
[docs]
def kernel(gw, mo_energy, mo_coeff, orbs=None,
kptlist=None, nw=None, 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
assert (frozen == 0)
if orbs is None:
orbs = range(gw.nmo)
if kptlist is None:
kptlist = range(gw.nkpts)
nkpts = gw.nkpts
nklist = len(kptlist)
# v_xc
dm = np.array(mf.make_rdm1())
v_mf = np.array(mf.get_veff()) - np.array(mf.get_j(dm_kpts=dm))
for k in range(nkpts):
v_mf[k] = reduce(numpy.dot, (mo_coeff[k].T.conj(), v_mf[k], mo_coeff[k]))
nocc = gw.nocc
nmo = gw.nmo
# v_hf from DFT/HF density
if gw.fc:
exxdiv = 'ewald'
else:
exxdiv = None
rhf = scf.KRHF(gw.mol, gw.kpts, exxdiv=exxdiv)
rhf.with_df = gw.with_df
if getattr(gw.with_df, '_cderi', None) is None:
raise RuntimeError('Found incompatible integral scheme %s.'
'KGWCD can be only used with GDF integrals' %
gw.with_df.__class__)
if rhf.with_df._j_only:
logger.debug(gw, 'Rebuild CDERI for exchange integrals')
rhf.with_df.build(j_only=False)
vk = rhf.get_veff(gw.mol,dm_kpts=dm) - rhf.get_j(gw.mol,dm_kpts=dm)
for k in range(nkpts):
vk[k] = reduce(numpy.dot, (mo_coeff[k].T.conj(), vk[k], mo_coeff[k]))
# Grids for integration on imaginary axis
freqs,wts = _get_scaled_legendre_roots(nw)
logger.debug(gw, "Computing the imaginary part")
Wmn, Del_00, Del_P0, qij, q_abs = get_WmnI_diag(gw, orbs, kptlist, freqs)
conv = True
mo_energy = np.zeros_like(np.array(mf.mo_energy))
for k in range(nklist):
kn = kptlist[k]
for p in orbs:
if p < nocc:
delta = -2e-2
else:
delta = 2e-2
if gw.linearized:
# FIXME
logger.warn(gw,'linearization with CD leads to wrong quasiparticle energy')
raise NotImplementedError
else:
# self-consistently solve QP equation
def quasiparticle(omega):
if gw.fc:
sigmaR = get_sigma_diag(gw, omega, kn, p, Wmn[:,k,:,p-orbs[0],:],
Del_00, Del_P0[k,p-orbs[0],:], freqs, wts, qij, q_abs).real
else:
sigmaR = get_sigma_diag(gw, omega, kn, p, Wmn[:,k,:,p-orbs[0],:],
Del_00, Del_P0, freqs, wts, qij, q_abs).real
return omega - mf.mo_energy[kn][p] - (sigmaR.real + vk[kn,p,p].real - v_mf[kn,p,p].real)
try:
e = newton(quasiparticle, mf.mo_energy[kn][p]+delta, tol=1e-6, maxiter=50)
logger.debug(gw, "Computing poles for QP (k: %s, orb: %s)"%(kn,p))
mo_energy[kn,p] = e
except RuntimeError:
conv = False
mo_coeff = mf.mo_coeff
if gw.verbose >= logger.DEBUG:
numpy.set_printoptions(threshold=nmo)
for k in range(nkpts):
logger.debug(gw, ' GW mo_energy @ k%d =\n%s', k,mo_energy[k])
numpy.set_printoptions(threshold=1000)
return conv, mo_energy, mo_coeff
[docs]
def get_sigma_diag(gw, ep, kp, p, Wmn, Del_00, Del_P0, freqs, wts, qij, q_abs):
'''
Compute self-energy on real axis using contour deformation
'''
nocc = gw.nocc
nkpts = gw.nkpts
# This code does not support metals
homo = -99.
lumo = 99.
for k in range(nkpts):
if homo < gw._scf.mo_energy[k][nocc-1]:
homo = gw._scf.mo_energy[k][nocc-1]
if lumo > gw._scf.mo_energy[k][nocc]:
lumo = gw._scf.mo_energy[k][nocc]
ef = (homo+lumo)/2.
nmo = gw.nmo
sign = np.zeros((nkpts,nmo),dtype=np.int64)
for k in range(nkpts):
sign[k] = np.sign(ef-gw._scf.mo_energy[k])
sigmaI = get_sigmaI_diag(gw, ep, kp, p, Wmn, Del_00, Del_P0, sign, freqs, wts)
sigmaR = get_sigmaR_diag(gw, ep, kp, p, ef, freqs, qij, q_abs)
return sigmaI + sigmaR
[docs]
def get_rho_response(gw, omega, mo_energy, Lpq, kL, kidx):
'''
Compute density response function in auxiliary basis at freq iw
'''
nkpts, naux, nmo, nmo = Lpq.shape
nocc = gw.nocc
kpts = gw.kpts
kscaled = gw.mol.get_scaled_kpts(kpts)
kscaled -= kscaled[0]
# Compute Pi for kL
Pi = np.zeros((naux,naux),dtype=np.complex128)
for i, kpti in enumerate(kpts):
# Find ka that conserves with ki and kL (-ki+ka+kL=G)
a = kidx[i]
eia = mo_energy[i,:nocc,None] - mo_energy[a,None,nocc:]
eia = eia/(omega**2+eia*eia)
Pia = einsum('Pia,ia->Pia',Lpq[i][:,:nocc,nocc:],eia)
# Response from both spin-up and spin-down density
Pi += 4./nkpts * einsum('Pia,Qia->PQ',Pia,Lpq[i][:,:nocc,nocc:].conj())
return Pi
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def get_WmnI_diag(gw, orbs, kptlist, freqs, max_memory=8000):
'''
Compute GW correlation self-energy (diagonal elements)
in MO basis on imaginary axis
'''
mo_energy = np.array(gw._scf.mo_energy)
mo_coeff = np.array(gw._scf.mo_coeff)
nmo = gw.nmo
nkpts = gw.nkpts
kpts = gw.kpts
nklist = len(kptlist)
nw = len(freqs)
norbs = len(orbs)
mydf = gw.with_df
# possible kpts shift center
kscaled = gw.mol.get_scaled_kpts(kpts)
kscaled -= kscaled[0]
Del_00, Del_P0, qij, q_abs = None, None, None, None
if gw.fc:
# Set up q mesh for q->0 finite size correction
q_pts = np.array([1e-3,0,0]).reshape(1,3)
q_abs = gw.mol.get_abs_kpts(q_pts)
# Get qij = 1/sqrt(Omega) * < psi_{ik} | e^{iqr} | psi_{ak-q} > at q: (nkpts, nocc, nvir)
qij = get_qij(gw, q_abs[0], mo_coeff)
Wmn = np.zeros((nkpts,nklist,nmo,norbs,nw),dtype=np.complex128)
if gw.fc:
Del_P0 = np.zeros((nklist,norbs,nw),dtype=np.complex128)
Del_00 = np.zeros(nw,dtype=np.complex128)
for kL in range(nkpts):
# Lij: (ki, L, i, j) for looping every kL
Lij = []
# kidx: save kj that conserves with kL and ki (-ki+kj+kL=G)
# kidx_r: save ki that conserves with kL and kj (-ki+kj+kL=G)
kidx = np.zeros((nkpts),dtype=np.int64)
kidx_r = np.zeros((nkpts),dtype=np.int64)
for i, kpti in enumerate(kpts):
for j, kptj in enumerate(kpts):
# Find (ki,kj) that satisfies momentum conservation with kL
kconserv = -kscaled[i] + kscaled[j] + kscaled[kL]
is_kconserv = np.linalg.norm(np.round(kconserv) - kconserv) < 1e-12
if is_kconserv:
kidx[i] = j
kidx_r[j] = i
logger.debug(gw, "Read Lpq (kL: %s / %s, ki: %s, kj: %s)"%(kL+1, nkpts, i, j))
Lij_out = None
# Read (L|pq) and ao2mo transform to (L|ij)
Lpq = []
for LpqR, LpqI, sign \
in mydf.sr_loop([kpti, kptj], max_memory=0.1*gw._scf.max_memory, compact=False):
Lpq.append(LpqR+LpqI*1.0j)
# support uneqaul naux on different k points
Lpq = np.vstack(Lpq).reshape(-1,nmo**2)
tao = []
ao_loc = None
moij, ijslice = _conc_mos(mo_coeff[i], mo_coeff[j])[2:]
Lij_out = _ao2mo.r_e2(Lpq, moij, ijslice, tao, ao_loc, out=Lij_out)
Lij.append(Lij_out.reshape(-1,nmo,nmo))
Lij = np.asarray(Lij)
naux = Lij.shape[1]
if kL == 0:
for w in range(nw):
# body dielectric matrix eps_body
Pi = get_rho_response(gw, freqs[w], mo_energy, Lij, kL, kidx)
eps_body_inv = np.linalg.inv(np.eye(naux)-Pi)
if gw.fc:
# head dielectric matrix eps_00
Pi_00 = get_rho_response_head(gw, freqs[w], mo_energy, qij)
eps_00 = 1. - 4. * np.pi/np.linalg.norm(q_abs[0])**2 * Pi_00
# wings dielectric matrix eps_P0
Pi_P0 = get_rho_response_wing(gw, freqs[w], mo_energy, Lij, qij)
eps_P0 = -np.sqrt(4.*np.pi) / np.linalg.norm(q_abs[0]) * Pi_P0
# inverse dielectric matrix
eps_inv_00 = 1./(eps_00 - np.dot(np.dot(eps_P0.conj(),eps_body_inv),eps_P0))
eps_inv_P0 = -eps_inv_00 * np.dot(eps_body_inv, eps_P0)
# head correction
Del_00[w] = 2./np.pi * (6.*np.pi**2/gw.mol.vol/nkpts)**(1./3.) * (eps_inv_00 - 1.)
wings_const = np.sqrt(gw.mol.vol/4./np.pi**3) * (6.*np.pi**2/gw.mol.vol/nkpts)**(2./3.)
eps_inv_PQ = eps_body_inv
for k in range(nklist):
kn = kptlist[k]
# Find km that conserves with kn and kL (-km+kn+kL=G)
km = kidx_r[kn]
Qmn = einsum('Pmn,PQ->Qmn',Lij[km][:,:,orbs].conj(),eps_inv_PQ-np.eye(naux))
Wmn[km,k,:,:,w] = 1./nkpts * einsum('Qmn,Qmn->mn',Qmn,Lij[km][:,:,orbs])
if gw.fc:
# compute wing correction
Wn_P0 = einsum('Pnm,P->nm',Lij[kn],eps_inv_P0).diagonal()
Wn_P0 = Wn_P0.real * 2.
Del_P0[k,:,w] = wings_const * Wn_P0[orbs]
else:
for w in range(nw):
Pi = get_rho_response(gw, freqs[w], mo_energy, Lij, kL, kidx)
Pi_inv = np.linalg.inv(np.eye(naux)-Pi)-np.eye(naux)
for k in range(nklist):
kn = kptlist[k]
# Find km that conserves with kn and kL (-km+kn+kL=G)
km = kidx_r[kn]
Qmn = einsum('Pmn,PQ->Qmn',Lij[km][:,:,orbs].conj(),Pi_inv)
Wmn[km,k,:,:,w] = 1./nkpts * einsum('Qmn,Qmn->mn',Qmn,Lij[km][:,:,orbs])
return Wmn, Del_00, Del_P0, qij, q_abs
[docs]
def get_sigmaI_diag(gw, omega, kp, p, Wmn, Del_00, Del_P0, sign, freqs, wts):
'''
Compute self-energy by integrating on imaginary axis
'''
mo_energy = gw._scf.mo_energy
nkpts = gw.nkpts
sigma = 0j
for k in range(nkpts):
emo = omega - 1j*gw.eta*sign[k] - mo_energy[k]
g0 = wts[None,:]*emo[:,None] / ((emo**2)[:,None]+(freqs**2)[None,:])
sigma += -einsum('mw,mw',g0,Wmn[k])/np.pi
if gw.fc and k == kp:
sigma += -einsum('w,w->',Del_00,g0[p])/np.pi
sigma += -einsum('w,w->',Del_P0,g0[p])/np.pi
return sigma
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def get_rho_response_R(gw, omega, mo_energy, Lpq, kL, kidx):
'''
Compute density response function in auxiliary basis at freq iw
'''
nkpts, naux, nmo, nmo = Lpq.shape
nocc = gw.nocc
kpts = gw.kpts
kscaled = gw.mol.get_scaled_kpts(kpts)
kscaled -= kscaled[0]
# Compute Pi for kL
Pi = np.zeros((naux,naux),dtype=np.complex128)
for i, kpti in enumerate(kpts):
# Find ka that conserves with ki and kL (-ki+ka+kL=G)
a = kidx[i]
eia = mo_energy[i,:nocc,None] - mo_energy[a,None,nocc:]
eia = 1./(omega+eia+2j*gw.eta) + 1./(-omega+eia)
Pia = einsum('Pia,ia->Pia',Lpq[i][:,:nocc,nocc:],eia)
# Response from both spin-up and spin-down density
Pi += 2./nkpts * einsum('Pia,Qia->PQ',Pia,Lpq[i][:,:nocc,nocc:].conj())
return Pi
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def get_sigmaR_diag(gw, omega, kn, orbp, ef, freqs, qij, q_abs):
'''
Compute self-energy for poles inside coutour
(more and more expensive away from Fermi surface)
'''
mo_energy = np.array(gw._scf.mo_energy)
mo_coeff = np.array(gw._scf.mo_coeff)
nmo = gw.nmo
nkpts = gw.nkpts
kpts = gw.kpts
mydf = gw.with_df
# possible kpts shift center
kscaled = gw.mol.get_scaled_kpts(kpts)
kscaled -= kscaled[0]
idx = []
for k in range(nkpts):
if omega > ef:
fm = 1.0
idx.append(np.where((mo_energy[k]<omega) & (mo_energy[k]>ef))[0])
else:
fm = -1.0
idx.append(np.where((mo_energy[k]>omega) & (mo_energy[k]<ef))[0])
sigmaR = 0j
for kL in range(nkpts):
# Lij: (ki, L, i, j) for looping every kL
Lij = []
# kidx: save kj that conserves with kL and ki (-ki+kj+kL=G)
# kidx_r: save ki that conserves with kL and kj (-ki+kj+kL=G)
kidx = np.zeros((nkpts),dtype=np.int64)
kidx_r = np.zeros((nkpts),dtype=np.int64)
for i, kpti in enumerate(kpts):
for j, kptj in enumerate(kpts):
# Find (ki,kj) that satisfies momentum conservation with kL
kconserv = -kscaled[i] + kscaled[j] + kscaled[kL]
is_kconserv = np.linalg.norm(np.round(kconserv) - kconserv) < 1e-12
if is_kconserv:
kidx[i] = j
kidx_r[j] = i
km = kidx_r[kn]
if len(idx[km]) > 0:
for i, kpti in enumerate(kpts):
for j, kptj in enumerate(kpts):
# Find (ki,kj) that satisfies momentum conservation with kL
kconserv = -kscaled[i] + kscaled[j] + kscaled[kL]
is_kconserv = np.linalg.norm(np.round(kconserv) - kconserv) < 1e-12
if is_kconserv:
kidx[i] = j
kidx_r[j] = i
#logger.debug(gw, "Read Lpq (kL: %s / %s, ki: %s, kj: %s)"%(kL+1, nkpts, i, j))
Lij_out = None
# Read (L|pq) and ao2mo transform to (L|ij)
Lpq = []
for LpqR, LpqI, sign \
in mydf.sr_loop([kpti, kptj], max_memory=0.1*gw._scf.max_memory, compact=False):
Lpq.append(LpqR+LpqI*1.0j)
# support uneqaul naux on different k points
Lpq = np.vstack(Lpq).reshape(-1,nmo**2)
tao = []
ao_loc = None
moij, ijslice = _conc_mos(mo_coeff[i], mo_coeff[j])[2:]
Lij_out = _ao2mo.r_e2(Lpq, moij, ijslice, tao, ao_loc, out=Lij_out)
Lij.append(Lij_out.reshape(-1,nmo,nmo))
Lij = np.asarray(Lij)
naux = Lij.shape[1]
if kL == 0:
km = kidx_r[kn]
if len(idx[km]) > 0:
for m in idx[km]:
em = mo_energy[km][m] - omega
# body dielectric matrix eps_body
Pi = get_rho_response_R(gw, abs(em), mo_energy, Lij, kL, kidx)
eps_body_inv = np.linalg.inv(np.eye(naux)-Pi)
if gw.fc and m == orbp:
# head dielectric matrix eps_00
Pi_00 = get_rho_response_head_R(gw, abs(em), mo_energy, qij)
eps_00 = 1. - 4. * np.pi/np.linalg.norm(q_abs[0])**2 * Pi_00
# wings dielectric matrix eps_P0
Pi_P0 = get_rho_response_wing_R(gw, abs(em), mo_energy, Lij, qij)
eps_P0 = -np.sqrt(4.*np.pi) / np.linalg.norm(q_abs[0]) * Pi_P0
# inverse dielectric matrix
eps_inv_00 = 1./(eps_00 - np.dot(np.dot(eps_P0.conj(),eps_body_inv),eps_P0))
eps_inv_P0 = -eps_inv_00 * np.dot(eps_body_inv, eps_P0)
eps_inv_PQ = eps_body_inv
# body
Qmn = einsum('P,PQ->Q',Lij[km][:,m,orbp].conj(),eps_inv_PQ-np.eye(naux))
Wmn = 1./nkpts * einsum('Q,Q->',Qmn,Lij[km][:,m,orbp])
sigmaR += fm * Wmn
if gw.fc and m == orbp:
# head correction
Del_00 = 2./np.pi * (6.*np.pi**2/gw.mol.vol/nkpts)**(1./3.) * (eps_inv_00 - 1.)
sigmaR += fm * Del_00
# wings correction
wings_const = np.sqrt(gw.mol.vol/4./np.pi**3) * (6.*np.pi**2/gw.mol.vol/nkpts)**(2./3.)
Wn_P0 = einsum('P,P->',Lij[kn][:,m,orbp].conj(),eps_inv_P0)
Wn_P0 = Wn_P0.real * 2.
sigmaR += fm * wings_const * Wn_P0
else:
km = kidx_r[kn]
if len(idx[km]) > 0:
for m in idx[km]:
em = mo_energy[km][m] - omega
Pi = get_rho_response_R(gw, abs(em), mo_energy, Lij, kL, kidx)
Pi_inv = np.linalg.inv(np.eye(naux)-Pi)-np.eye(naux)
Qmn = einsum('P,PQ->Q',Lij[km][:,m,orbp].conj(),Pi_inv)
Wmn = 1./nkpts * einsum('Q,Q->',Qmn,Lij[km][:,m,orbp])
sigmaR += fm * Wmn
return sigmaR
[docs]
def get_rho_response_head_R(gw, omega, mo_energy, qij):
'''
Compute head (G=0, G'=0) density response function in auxiliary basis at freq w
'''
nkpts, nocc, nvir = qij.shape
nocc = gw.nocc
kpts = gw.kpts
# Compute Pi head
Pi_00 = 0j
for i, kpti in enumerate(kpts):
eia = mo_energy[i,:nocc,None] - mo_energy[i,None,nocc:]
eia = 1./(omega+eia+2j*gw.eta) + 1./(-omega+eia)
Pi_00 += 2./nkpts * einsum('ia,ia->',eia,qij[i].conj()*qij[i])
return Pi_00
[docs]
def get_rho_response_wing_R(gw, omega, mo_energy, Lpq, qij):
'''
Compute density response function in auxiliary basis at freq iw
'''
nkpts, naux, nmo, nmo = Lpq.shape
nocc = gw.nocc
kpts = gw.kpts
# Compute Pi for kL
Pi = np.zeros(naux,dtype=np.complex128)
for i, kpti in enumerate(kpts):
eia = mo_energy[i,:nocc,None] - mo_energy[i,None,nocc:]
eia = 1./(omega+eia+2j*gw.eta) + 1./(-omega+eia)
eia_q = eia * qij[i].conj()
Pi += 2./nkpts * einsum('Pia,ia->P',Lpq[i][:,:nocc,nocc:],eia_q)
return Pi
[docs]
def get_rho_response_head(gw, omega, mo_energy, qij):
'''
Compute head (G=0, G'=0) density response function in auxiliary basis at freq iw
'''
nkpts, nocc, nvir = qij.shape
nocc = gw.nocc
kpts = gw.kpts
# Compute Pi head
Pi_00 = 0j
for i, kpti in enumerate(kpts):
eia = mo_energy[i,:nocc,None] - mo_energy[i,None,nocc:]
eia = eia/(omega**2+eia*eia)
Pi_00 += 4./nkpts * einsum('ia,ia->',eia,qij[i].conj()*qij[i])
return Pi_00
[docs]
def get_rho_response_wing(gw, omega, mo_energy, Lpq, qij):
'''
Compute wing (G=P, G'=0) density response function in auxiliary basis at freq iw
'''
nkpts, naux, nmo, nmo = Lpq.shape
nocc = gw.nocc
kpts = gw.kpts
# Compute Pi wing
Pi = np.zeros(naux,dtype=np.complex128)
for i, kpti in enumerate(kpts):
eia = mo_energy[i,:nocc,None] - mo_energy[i,None,nocc:]
eia = eia/(omega**2+eia*eia)
eia_q = eia * qij[i].conj()
Pi += 4./nkpts * einsum('Pia,ia->P',Lpq[i][:,:nocc,nocc:],eia_q)
return Pi
[docs]
def get_qij(gw, q, mo_coeff, uniform_grids=False):
'''
Compute qij = 1/Omega * |< psi_{ik} | e^{iqr} | psi_{ak-q} >|^2 at q: (nkpts, nocc, nvir)
'''
nocc = gw.nocc
nmo = gw.nmo
nvir = nmo - nocc
kpts = gw.kpts
nkpts = len(kpts)
cell = gw.mol
mo_energy = gw._scf.mo_energy
if uniform_grids:
mydf = df.FFTDF(cell, kpts=kpts)
coords = cell.gen_uniform_grids(mydf.mesh)
else:
coords, weights = dft.gen_grid.get_becke_grids(cell,level=5)
ngrid = len(coords)
qij = np.zeros((nkpts,nocc,nvir),dtype=np.complex128)
for i, kpti in enumerate(kpts):
ao_p = dft.numint.eval_ao(cell, coords, kpt=kpti, deriv=1)
ao = ao_p[0]
ao_grad = ao_p[1:4]
if uniform_grids:
ao_ao_grad = einsum('mg,xgn->xmn',ao.T.conj(),ao_grad) * cell.vol / ngrid
else:
ao_ao_grad = einsum('g,mg,xgn->xmn',weights,ao.T.conj(),ao_grad)
q_ao_ao_grad = -1j * einsum('x,xmn->mn',q,ao_ao_grad)
q_mo_mo_grad = np.dot(np.dot(mo_coeff[i][:,:nocc].T.conj(), q_ao_ao_grad), mo_coeff[i][:,nocc:])
enm = 1./(mo_energy[i][nocc:,None] - mo_energy[i][None,:nocc])
dens = enm.T * q_mo_mo_grad
qij[i] = dens / np.sqrt(cell.vol)
return qij
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 qaudrature 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]
[docs]
class KRGWCD(lib.StreamObject):
linearized = getattr(__config__, 'gw_gw_GW_linearized', False)
eta = getattr(__config__, 'gw_gw_GW_eta', 1e-3)
fc = getattr(__config__, 'gw_gw_GW_fc', True)
_keys = set([
'linearized', 'eta', 'fc', 'frozen', 'mol', 'with_df',
'kpts', 'nkpts', 'mo_energy', 'mo_coeff', 'mo_occ', 'sigma',
])
def __init__(self, mf, frozen=0):
self.mol = mf.mol
self._scf = mf
self.verbose = self.mol.verbose
self.stdout = self.mol.stdout
self.max_memory = mf.max_memory
#TODO: implement frozen orbs
if frozen > 0:
raise NotImplementedError
self.frozen = frozen
# DF-KGW must use GDF integrals
if getattr(mf, 'with_df', None):
self.with_df = mf.with_df
else:
raise NotImplementedError
##################################################
# don't modify the following attributes, they are not input options
self._nocc = None
self._nmo = None
self.kpts = mf.kpts
self.nkpts = len(self.kpts)
# 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
[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
nkpts = self.nkpts
log.info('GW nocc = %d, nvir = %d, nkpts = %d', nocc, nvir, nkpts)
if self.frozen is not None:
log.info('frozen orbitals %s', str(self.frozen))
logger.info(self, 'use perturbative linearized QP eqn = %s', self.linearized)
logger.info(self, 'GW finite size corrections = %s', self.fc)
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, orbs=None, kptlist=None, nw=100):
"""
Input:
kptlist: self-energy k-points
orbs: self-energy orbs
nw: grid number
Output:
mo_energy: GW quasiparticle energy
"""
if mo_coeff is None:
mo_coeff = np.array(self._scf.mo_coeff)
if mo_energy is None:
mo_energy = np.array(self._scf.mo_energy)
nmo = self.nmo
naux = self.with_df.get_naoaux()
nkpts = self.nkpts
mem_incore = (2*nkpts*nmo**2*naux) * 16/1e6
mem_now = lib.current_memory()[0]
if (mem_incore + mem_now > 0.99*self.max_memory):
logger.warn(self, 'Memory may not be enough!')
raise NotImplementedError
cput0 = (logger.process_clock(), logger.perf_counter())
self.dump_flags()
self.converged, self.mo_energy, self.mo_coeff = \
kernel(self, mo_energy, mo_coeff, orbs=orbs,
kptlist=kptlist, nw=nw, 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