#!/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 copy
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 = super(ks.__class__, ks).get_veff(cell, dm, dm_last=dm_last,
vhf_last=vhf_last, hermi=hermi, kpts=kpts,
kpts_band=kpts_band)
# V_U
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)
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 in zip(ks.U_idx, ks.U_val, ks.U_lab):
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]
SC = np.dot(S_k, C_k)
vxc[k] += mdot(SC, (np.eye(P_k.shape[-1]) - P_k)
* (val * 0.5), SC.conj().T).astype(vxc[k].dtype,copy=False)
E_U += weight[k] * (val * 0.5) * (P_k.trace() - np.dot(P_k, P_k).trace() * 0.5)
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 (%s) 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(copy.deepcopy(idx))
ks.U_val.append(U_val[i])
ks.U_val = np.asarray(ks.U_val) / HARTREE2EV
logger.info(ks, "-" * 79)
logger.debug(ks, 'U indices and values: ')
for idx, val in zip(ks.U_idx, ks.U_val):
ks.U_lab.append(lo_labels[idx])
logger.debug(ks, '%6s [%.6g eV] ==> %-100s', format_idx(idx),
val * HARTREE2EV, "".join(lo_labels[idx]))
logger.info(ks, "-" * 79)
[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])
labels = np.asarray(labels)
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 = set(["U_idx", "U_val", "C_ao_lo", "U_lab"])
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'):
"""
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'.
"""
super(self.__class__, self).__init__(cell, kpts, xc=xc, exxdiv=exxdiv)
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]
[docs]
def nuc_grad_method(self):
raise NotImplementedError
if __name__ == '__main__':
from pyscf.pbc import gto
np.set_printoptions(3, linewidth=1000, suppress=True)
cell = gto.Cell()
cell.unit = 'A'
cell.atom = 'C 0., 0., 0.; C 0.8917, 0.8917, 0.8917'
cell.a = '''0. 1.7834 1.7834
1.7834 0. 1.7834
1.7834 1.7834 0. '''
cell.basis = 'gth-dzvp'
cell.pseudo = 'gth-pade'
cell.verbose = 7
cell.build()
kmesh = [2, 2, 2]
kpts = cell.make_kpts(kmesh, wrap_around=True)
#U_idx = ["2p", "2s"]
#U_val = [5.0, 2.0]
U_idx = ["1 C 2p"]
U_val = [5.0]
mf = KRKSpU(cell, kpts, U_idx=U_idx, U_val=U_val, C_ao_lo='minao',
minao_ref='gth-szv')
mf.conv_tol = 1e-10
print (mf.U_idx)
print (mf.U_val)
print (mf.C_ao_lo.shape)
print (mf.kernel())