#!/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.
#
# Authors: Zhi-Hao Cui <zhcui0408@gmail.com>
#
"""
Restricted DFT+U with kpoint sampling.
Based on KRHF routine.
Refs: PRB, 1998, 57, 1505.
"""
import itertools as it
import numpy as np
import scipy.linalg as la
from functools import reduce
from pyscf import lib
from pyscf.lib import logger
from pyscf import __config__
from pyscf.pbc.dft import krks
from pyscf.data.nist import HARTREE2EV
from pyscf import lo
from pyscf.lo import iao
from pyscf.pbc import gto as pgto
[docs]
def get_veff(ks, cell=None, dm=None, dm_last=0, vhf_last=0, hermi=1,
kpts=None, kpts_band=None):
"""
Coulomb + XC functional + Hubbard U terms.
.. note::
This is a replica of pyscf.dft.rks.get_veff with kpts added.
This function will change the ks object.
Args:
ks : an instance of :class:`RKS`
XC functional are controlled by ks.xc attribute. Attribute
ks.grids might be initialized.
dm : ndarray or list of ndarrays
A density matrix or a list of density matrices
Returns:
Veff : ``(nkpts, nao, nao)`` or ``(*, nkpts, nao, nao)`` ndarray
Veff = J + Vxc + V_U.
"""
if cell is None: cell = ks.cell
if dm is None: dm = ks.make_rdm1()
if kpts is None: kpts = ks.kpts
# J + V_xc
vxc = krks.get_veff(ks, cell, dm, dm_last=dm_last, vhf_last=vhf_last,
hermi=hermi, kpts=kpts, kpts_band=kpts_band)
vxc = _add_Vhubbard(vxc, ks, dm, kpts)
return vxc
def _add_Vhubbard(vxc, ks, dm, kpts):
'''Add Hubbard U to Vxc matrix inplace.
'''
C_ao_lo = ks.C_ao_lo
ovlp = ks.get_ovlp()
nkpts = len(kpts)
nlo = C_ao_lo.shape[-1]
rdm1_lo = np.zeros((nkpts, nlo, nlo), dtype=np.complex128)
for k in range(nkpts):
C_inv = np.dot(C_ao_lo[k].conj().T, ovlp[k])
rdm1_lo[k] = mdot(C_inv, dm[k], C_inv.conj().T)
is_ibz = hasattr(kpts, "kpts_ibz")
if is_ibz:
rdm1_lo_0 = kpts.dm_at_ref_cell(rdm1_lo)
alphas = ks.alpha
if not hasattr(alphas, '__len__'): # not a list or tuple
alphas = [alphas] * len(ks.U_idx)
E_U = 0.0
weight = getattr(kpts, "weights_ibz", np.repeat(1.0/nkpts, nkpts))
logger.info(ks, "-" * 79)
with np.printoptions(precision=5, suppress=True, linewidth=1000):
for idx, val, lab, alpha in zip(ks.U_idx, ks.U_val, ks.U_lab, alphas):
lab_string = " "
for l in lab:
lab_string += "%9s" %(l.split()[-1])
lab_sp = lab[0].split()
logger.info(ks, "local rdm1 of atom %s: ",
" ".join(lab_sp[:2]) + " " + lab_sp[2][:2])
U_mesh = np.ix_(idx, idx)
P_loc = 0.0
for k in range(nkpts):
S_k = ovlp[k]
C_k = C_ao_lo[k][:, idx]
P_k = rdm1_lo[k][U_mesh]
E_U += weight[k] * (val * 0.5) * (P_k.trace() - np.dot(P_k, P_k).trace() * 0.5)
vhub_loc = (np.eye(P_k.shape[-1]) - P_k) * (val * 0.5)
if alpha is not None:
# The alpha perturbation is only applied to the linear term of
# the local density.
E_U += weight[k] * alpha * P_k.trace()
vhub_loc += np.eye(P_k.shape[-1]) * alpha
SC = np.dot(S_k, C_k)
vhub_loc = SC.dot(vhub_loc).dot(SC.conj().T)
if vxc.dtype == np.float64:
vhub_loc = vhub_loc.real
vxc[k] += vhub_loc
if not is_ibz:
P_loc += P_k
if is_ibz:
P_loc = rdm1_lo_0[U_mesh].real
else:
P_loc = P_loc.real / nkpts
logger.info(ks, "%s\n%s", lab_string, P_loc)
logger.info(ks, "-" * 79)
if E_U.real < 0.0 and all(np.asarray(ks.U_val) > 0):
logger.warn(ks, "E_U (%g) is negative...", E_U.real)
vxc = lib.tag_array(vxc, E_U=E_U)
return vxc
[docs]
def energy_elec(ks, dm_kpts=None, h1e_kpts=None, vhf=None):
"""
Electronic energy for KRKSpU.
"""
if h1e_kpts is None: h1e_kpts = ks.get_hcore(ks.cell, ks.kpts)
if dm_kpts is None: dm_kpts = ks.make_rdm1()
if vhf is None or getattr(vhf, 'ecoul', None) is None:
vhf = ks.get_veff(ks.cell, dm_kpts)
weight = getattr(ks.kpts, "weights_ibz",
np.array([1.0/len(h1e_kpts),]*len(h1e_kpts)))
e1 = np.einsum('k,kij,kji', weight, h1e_kpts, dm_kpts)
tot_e = e1 + vhf.ecoul + vhf.exc + vhf.E_U
ks.scf_summary['e1'] = e1.real
ks.scf_summary['coul'] = vhf.ecoul.real
ks.scf_summary['exc'] = vhf.exc.real
ks.scf_summary['E_U'] = vhf.E_U.real
logger.debug(ks, 'E1 = %s Ecoul = %s Exc = %s EU = %s', e1, vhf.ecoul,
vhf.exc, vhf.E_U)
return tot_e.real, vhf.ecoul + vhf.exc + vhf.E_U
[docs]
def set_U(ks, U_idx, U_val):
"""
Regularize the U_idx and U_val to each atom,
and set ks.U_idx, ks.U_val, ks.U_lab.
"""
assert len(U_idx) == len(U_val)
ks.U_val = []
ks.U_idx = []
ks.U_lab = []
lo_labels = np.asarray(ks.cell.ao_labels())
for i, idx in enumerate(U_idx):
if isinstance(idx, str):
lab_idx = ks.cell.search_ao_label(idx)
labs = lo_labels[lab_idx]
labs = zip(lab_idx, labs)
for j, idxj in it.groupby(labs, key=lambda x: x[1].split()[0]):
ks.U_idx.append(list(list(zip(*idxj))[0]))
ks.U_val.append(U_val[i])
else:
ks.U_idx.append(idx)
ks.U_val.append(U_val[i])
ks.U_val = np.asarray(ks.U_val) / HARTREE2EV
for idx, val in zip(ks.U_idx, ks.U_val):
ks.U_lab.append(lo_labels[idx])
if len(ks.U_idx) == 0:
logger.warn(ks, "No sites specified for Hubbard U. "
"Please check if 'U_idx' is correctly specified")
[docs]
def make_minao_lo(ks, minao_ref):
"""
Construct minao local orbitals.
"""
cell = ks.cell
nao = cell.nao
kpts = getattr(ks.kpts, "kpts_ibz", ks.kpts)
nkpts = len(kpts)
ovlp = ks.get_ovlp()
C_ao_minao, labels = proj_ref_ao(cell, minao=minao_ref, kpts=kpts,
return_labels=True)
for k in range(nkpts):
C_ao_minao[k] = lo.vec_lowdin(C_ao_minao[k], ovlp[k])
C_ao_lo = np.zeros((nkpts, nao, nao), dtype=np.complex128)
for idx, lab in zip(ks.U_idx, ks.U_lab):
idx_minao = [i for i, l in enumerate(labels) if l in lab]
assert len(idx_minao) == len(idx)
C_ao_sub = C_ao_minao[:, :, idx_minao]
C_ao_lo[:, :, idx] = C_ao_sub
return C_ao_lo
[docs]
def proj_ref_ao(mol, minao='minao', kpts=None, return_labels=False):
"""
Get a set of reference AO spanned by the calculation basis.
Not orthogonalized.
Args:
return_labels: if True, return the labels as well.
"""
nkpts = len(kpts)
pmol = iao.reference_mol(mol, minao)
s1 = np.asarray(mol.pbc_intor('int1e_ovlp', hermi=1, kpts=kpts))
s2 = np.asarray(pmol.pbc_intor('int1e_ovlp', hermi=1, kpts=kpts))
s12 = np.asarray(pgto.cell.intor_cross('int1e_ovlp', mol, pmol, kpts=kpts))
#s21 = np.swapaxes(s12, -1, -2).conj()
C_ao_lo = np.zeros((nkpts, s1.shape[-1], s2.shape[-1]), dtype=np.complex128)
for k in range(nkpts):
s1cd_k = la.cho_factor(s1[k])
#s2cd_k = la.cho_factor(s2[k])
C_ao_lo[k] = la.cho_solve(s1cd_k, s12[k])
if return_labels:
labels = pmol.ao_labels()
return C_ao_lo, labels
else:
return C_ao_lo
[docs]
def mdot(*args):
'''
Compute the dot product of a list of arrays in a single function call.
'''
return reduce(np.dot, args)
[docs]
class KRKSpU(krks.KRKS):
"""
RKSpU class adapted for PBCs with k-point sampling.
"""
_keys = {"U_idx", "U_val", "C_ao_lo", "U_lab", 'alpha'}
get_veff = get_veff
energy_elec = energy_elec
to_hf = lib.invalid_method('to_hf')
def __init__(self, cell, kpts=np.zeros((1,3)), xc='LDA,VWN',
exxdiv=getattr(__config__, 'pbc_scf_SCF_exxdiv', 'ewald'),
U_idx=[], U_val=[], C_ao_lo='minao', minao_ref='MINAO', **kwargs):
"""
DFT+U args:
U_idx: can be
list of list: each sublist is a set of LO indices to add U.
list of string: each string is one kind of LO orbitals,
e.g. ['Ni 3d', '1 O 2pz'], in this case,
LO should be aranged as ao_labels order.
or a combination of these two.
U_val: a list of effective U [in eV], i.e. U-J in Dudarev's DFT+U.
each U corresponds to one kind of LO orbitals, should have
the same length as U_idx.
C_ao_lo: LO coefficients, can be
np.array, shape ((spin,), nkpts, nao, nlo),
string, in 'minao'.
minao_ref: reference for minao orbitals, default is 'MINAO'.
Attributes:
U_idx: same as the input.
U_val: effectiv U-J [in AU]
C_ao_loc: np.array
alpha: the perturbation [in AU] used to compute U in LR-cDFT.
Refs: Cococcioni and de Gironcoli, PRB 71, 035105 (2005)
"""
super(self.__class__, self).__init__(cell, kpts, xc=xc, exxdiv=exxdiv, **kwargs)
set_U(self, U_idx, U_val)
if isinstance(C_ao_lo, str):
if C_ao_lo.upper() == 'MINAO':
self.C_ao_lo = make_minao_lo(self, minao_ref)
else:
raise NotImplementedError
else:
self.C_ao_lo = np.asarray(C_ao_lo)
if self.C_ao_lo.ndim == 4:
self.C_ao_lo = self.C_ao_lo[0]
# The perturbation (eV) used to compute U in LR-cDFT.
self.alpha = None
[docs]
def dump_flags(self, verbose=None):
super().dump_flags(verbose)
log = logger.new_logger(self, verbose)
if log.verbose >= logger.INFO:
_print_U_info(self, log)
return self
[docs]
def nuc_grad_method(self):
raise NotImplementedError
def _print_U_info(mf, log):
alphas = mf.alpha
if not hasattr(alphas, '__len__'): # not a list or tuple
alphas = [alphas] * len(mf.U_idx)
log.info("-" * 79)
log.info('U indices and values: ')
for idx, val, lab, alpha in zip(mf.U_idx, mf.U_val, mf.U_lab, alphas):
log.info('%6s [%.6g eV] ==> %-100s', format_idx(idx),
val * HARTREE2EV, "".join(lab))
if alpha is not None:
log.info('alpha for LR-cDFT %s (eV)', alpha*HARTREE2EV)
log.info("-" * 79)
[docs]
def linear_response_u(mf_plus_u, alphalist=(0.02, 0.05, 0.08)):
'''
Refs:
[1] M. Cococcioni and S. de Gironcoli, Phys. Rev. B 71, 035105 (2005)
[2] H. J. Kulik, M. Cococcioni, D. A. Scherlis, and N. Marzari, Phys. Rev. Lett. 97, 103001 (2006)
[3] Heather J. Kulik, J. Chem. Phys. 142, 240901 (2015)
[4] https://hjkgrp.mit.edu/tutorials/2011-05-31-calculating-hubbard-u/
[5] https://hjkgrp.mit.edu/tutorials/2011-06-28-hubbard-u-multiple-sites/
Args:
alphalist :
alpha parameters (in eV) are the displacements for the linear
response calculations. For each alpha in this list, the DFT+U with
U=u0+alpha, U=u0-alpha are evaluated. u0 is the U value from the
reference mf_plus_u object, which will be treated as a standard DFT
functional.
'''
is_ibz = hasattr(mf_plus_u.kpts, "kpts_ibz")
if is_ibz:
raise NotImplementedError
assert isinstance(mf_plus_u, KRKSpU)
assert len(mf_plus_u.U_idx) > 0
if not mf_plus_u.converged:
mf_plus_u.run()
assert mf_plus_u.converged
# The bare density matrix without adding U
bare_dm = mf_plus_u.make_rdm1()
mf = mf_plus_u.copy()
log = logger.new_logger(mf)
alphalist = np.asarray(alphalist)
alphalist = np.append(-alphalist[::-1], alphalist)
nkpts = len(mf.kpts)
C_ao_lo = mf.C_ao_lo
ovlp = mf.get_ovlp()
C_inv = [[C_ao_lo[k][:,local_idx].conj().T.dot(ovlp[k]) for k in range(nkpts)]
for local_idx in mf.U_idx]
bare_occupancies = []
final_occupancies = []
for alpha in alphalist:
# All in atomic unit
mf.alpha = alpha / HARTREE2EV
mf.kernel(dm0=bare_dm)
local_occ = 0
for c in C_inv:
C_on_site = [c[k].dot(mf.mo_coeff[k]) for k in range(nkpts)]
rdm1_lo = mf.make_rdm1(C_on_site, mf.mo_occ)
local_occ += sum(x.trace().real for x in rdm1_lo)
local_occ /= nkpts
final_occupancies.append(local_occ)
# The first iteration of SCF
fock = mf.get_fock(dm=bare_dm)
e, mo = mf.eig(fock, ovlp)
local_occ = 0
for c in C_inv:
C_on_site = [c[k].dot(mo[k]) for k in range(nkpts)]
rdm1_lo = mf.make_rdm1(C_on_site, mf.mo_occ)
local_occ += sum(x.trace().real for x in rdm1_lo)
local_occ /= nkpts
bare_occupancies.append(local_occ)
log.info('alpha=%f bare_occ=%g final_occ=%g',
alpha, bare_occupancies[-1], final_occupancies[-1])
chi0, occ0 = np.polyfit(alphalist, bare_occupancies, deg=1)
chif, occf = np.polyfit(alphalist, final_occupancies, deg=1)
log.info('Line fitting chi0 = %f x + %f', chi0, occ0)
log.info('Line fitting chif = %f x + %f', chif, occf)
Uresp = 1./chi0 - 1./chif
log.note('Uresp = %f, chi0 = %f, chif = %f', Uresp, chi0, chif)
return Uresp
