#!/usr/bin/env python
# Copyright 2014-2019 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: Qiming Sun <osirpt.sun@gmail.com>
#
'''
Pipek-Mezey localization
ref. JCTC 10, 642 (2014); DOI:10.1021/ct401016x
'''
import numpy
from functools import reduce
from pyscf import lib
from pyscf.lib import logger
from pyscf.lo import orth
from pyscf.lo import boys
from pyscf.lo import pipek_jacobi
from pyscf import __config__
[docs]
def atomic_pops(mol, mo_coeff, method='meta_lowdin', mf=None, s=None):
'''
Kwargs:
method : string
The atomic population projection scheme. It can be mulliken,
lowdin, meta_lowdin, iao, or becke
Returns:
A 3-index tensor [A,i,j] indicates the population of any orbital-pair
density |i><j| for each species (atom in this case). This tensor is
used to construct the population and gradients etc.
You can customize the PM localization wrt other population metric,
such as the charge of a site, the charge of a fragment (a group of
atoms) by overwriting this tensor. See also the example
pyscf/examples/loc_orb/40-hubbard_model_PM_localization.py for the PM
localization of site-based population for hubbard model.
'''
method = method.lower().replace('_', '-')
nmo = mo_coeff.shape[1]
proj = numpy.empty((mol.natm,nmo,nmo))
if s is None:
if getattr(mol, 'pbc_intor', None): # whether mol object is a cell
s = mol.pbc_intor('int1e_ovlp', hermi=1)
else:
s = mol.intor_symmetric('int1e_ovlp')
if method == 'becke':
from pyscf.dft import gen_grid
if not (getattr(mf, 'grids', None) and getattr(mf, '_numint', None)):
# Call DFT to initialize grids and numint objects
mf = mol.RKS()
grids = mf.grids
ni = mf._numint
if not isinstance(grids, gen_grid.Grids):
raise NotImplementedError('PM becke scheme for PBC systems')
# The atom-wise Becke grids (without concatenated to a vector of grids)
coords, weights = grids.get_partition(mol, concat=False)
for i in range(mol.natm):
ao = ni.eval_ao(mol, coords[i], deriv=0)
aow = numpy.einsum('pi,p->pi', ao, weights[i])
charge_matrix = lib.dot(aow.conj().T, ao)
proj[i] = reduce(lib.dot, (mo_coeff.conj().T, charge_matrix, mo_coeff))
elif method == 'mulliken':
for i, (b0, b1, p0, p1) in enumerate(mol.offset_nr_by_atom()):
csc = reduce(numpy.dot, (mo_coeff[p0:p1].conj().T, s[p0:p1], mo_coeff))
proj[i] = (csc + csc.conj().T) * .5
elif method in ('lowdin', 'meta-lowdin'):
csc = reduce(lib.dot, (mo_coeff.conj().T, s, orth.orth_ao(mol, method, 'ANO', s=s)))
for i, (b0, b1, p0, p1) in enumerate(mol.offset_nr_by_atom()):
proj[i] = numpy.dot(csc[:,p0:p1], csc[:,p0:p1].conj().T)
elif method in ('iao', 'ibo'):
from pyscf.lo import iao
assert mf is not None
# FIXME: How to handle UHF/UKS object?
orb_occ = mf.mo_coeff[:,mf.mo_occ>0]
iao_coeff = iao.iao(mol, orb_occ)
#
# IAO is generally not orthogonalized. For simplicity, we take Lowdin
# orthogonalization here. Other orthogonalization can be used. Results
# should be very closed to the Lowdin-orth orbitals
#
# PM with Mulliken population of non-orth IAOs can be found in
# ibo.PipekMezey function
#
iao_coeff = orth.vec_lowdin(iao_coeff, s)
csc = reduce(lib.dot, (mo_coeff.conj().T, s, iao_coeff))
iao_mol = iao.reference_mol(mol)
for i, (b0, b1, p0, p1) in enumerate(iao_mol.offset_nr_by_atom()):
proj[i] = numpy.dot(csc[:,p0:p1], csc[:,p0:p1].conj().T)
else:
raise KeyError('method = %s' % method)
return proj
[docs]
class PipekMezey(boys.OrbitalLocalizer):
'''The Pipek-Mezey localization optimizer that maximizes the orbital
population
Args:
mol : Mole object
Kwargs:
mo_coeff : size (N,N) np.array
The orbital space to localize for PM localization.
When initializing the localization optimizer ``bopt = PM(mo_coeff)``,
Note these orbitals ``mo_coeff`` may or may not be used as initial
guess, depending on the attribute ``.init_guess`` . If ``.init_guess``
is set to None, the ``mo_coeff`` will be used as initial guess. If
``.init_guess`` is 'atomic', a few atomic orbitals will be
constructed inside the space of the input orbitals and the atomic
orbitals will be used as initial guess.
Note when calling .kernel(orb) method with a set of orbitals as
argument, the orbitals will be used as initial guess regardless of
the value of the attributes .mo_coeff and .init_guess.
Attributes for PM class:
verbose : int
Print level. Default value equals to :class:`Mole.verbose`.
max_memory : float or int
Allowed memory in MB. Default value equals to :class:`Mole.max_memory`.
conv_tol : float
Converge threshold. Default 1e-6
conv_tol_grad : float
Converge threshold for orbital rotation gradients. Default 1e-3
max_cycle : int
The max. number of macro iterations. Default 100
max_iters : int
The max. number of iterations in each macro iteration. Default 20
max_stepsize : float
The step size for orbital rotation. Small step (0.005 - 0.05) is preferred.
Default 0.03.
init_guess : str or None
Initial guess for optimization. If set to None, orbitals defined
by the attribute .mo_coeff will be used as initial guess. If set
to 'atomic', atomic orbitals will be used as initial guess.
Default 'atomic'
pop_method : str
How the orbital population is calculated, see JCTC 10, 642
(2014) for discussion. Options are:
- 'meta-lowdin' (default) as defined in JCTC 10, 3784 (2014)
- 'mulliken' original Pipek-Mezey scheme, JCP 90, 4916 (1989)
- 'lowdin' Lowdin charges, JCTC 10, 642 (2014)
- 'iao' or 'ibo' intrinsic atomic orbitals, JCTC 9, 4384 (2013)
- 'becke' Becke charges, JCTC 10, 642 (2014)
The IAO and Becke charges do not depend explicitly on the
basis set, and have a complete basis set limit [JCTC 10,
642 (2014)].
exponent : int
The power to define norm. It can be 2 or 4. Default 2.
Saved results
mo_coeff : ndarray
Localized orbitals
'''
pop_method = getattr(__config__, 'lo_pipek_PM_pop_method', 'meta_lowdin')
conv_tol = getattr(__config__, 'lo_pipek_PM_conv_tol', 1e-6)
exponent = getattr(__config__, 'lo_pipek_PM_exponent', 2) # any integer >= 2
_keys = {'pop_method', 'conv_tol', 'exponent'}
def __init__(self, mol, mo_coeff=None, mf=None, pop_method=None):
boys.OrbitalLocalizer.__init__(self, mol, mo_coeff)
self._scf = mf
if pop_method is not None:
self.pop_method = pop_method
[docs]
def dump_flags(self, verbose=None):
boys.OrbitalLocalizer.dump_flags(self, verbose)
logger.info(self, 'pop_method = %s',self.pop_method)
[docs]
def gen_g_hop(self, u):
exponent = self.exponent
mo_coeff = lib.dot(self.mo_coeff, u)
projR = self.atomic_pops(self.mol, mo_coeff, self.pop_method).real
pop = lib.einsum('xii->xi', projR)
popexp1 = pop**(exponent-1)
popexp2 = pop**(exponent-2)
# gradient
g = lib.einsum('xj,xij->ij', popexp1, projR)
g = -2 * exponent * self.pack_uniq_var(g - g.T)
# hessian diagonal
g1 = lib.einsum('xi,xi->i', popexp1, pop)
g2 = lib.einsum('xi,xj->ij', popexp1, pop)
h_diag = -2 * exponent * (g1[:,None] - g2)
g1 = lib.einsum('xi,xij->ij', popexp2, projR**2.)
h_diag += 4 * exponent * (exponent-1) * g1
h_diag = -self.pack_uniq_var(h_diag + h_diag.T)
# hessian vector product
G = lib.einsum('xi,xij->ij', popexp1, projR)
G += G.T
def h_op(x):
xR = self.unpack_uniq_var(x).real
# contributions from (nabla proj) x (nabla proj)
j0 = popexp2 * lib.einsum('xik,ki->xi', projR, xR)
j1 = lib.einsum('xi,xij->ij', j0, projR)
hx = -4 * exponent * (exponent-1) * j1
# contributions from nabla^2 proj: symmetric terms
j1 = numpy.dot(G,xR)
hx += -exponent * j1
# contributions from nabla^2 proj: asymmetric terms
j1 = lib.einsum('xj,xik,kj->ij', popexp1, projR, xR)
hx += 2 * exponent * j1
hx -= hx.T
return -self.pack_uniq_var(hx)
return g, h_op, h_diag
[docs]
def get_grad(self, u=None):
if u is None: u = numpy.eye(self.mo_coeff.shape[1])
exponent = self.exponent
mo_coeff = lib.dot(self.mo_coeff, u)
projR = self.atomic_pops(self.mol, mo_coeff, self.pop_method).real
popexp1 = lib.einsum('xii->xi', projR)**(exponent-1)
g = lib.einsum('xj,xij->ij', popexp1, projR)
g = -2 * exponent * self.pack_uniq_var(g - g.T)
return g
[docs]
def cost_function(self, u=None):
if u is None: u = numpy.eye(self.mo_coeff.shape[1])
mo_coeff = lib.dot(self.mo_coeff, u)
projR = self.atomic_pops(self.mol, mo_coeff, self.pop_method).real
return (lib.einsum('xii->xi', projR)**self.exponent).sum()
[docs]
@lib.with_doc(atomic_pops.__doc__)
def atomic_pops(self, mol, mo_coeff, method=None, s=None):
if method is None:
method = self.pop_method
if method.lower() in ('iao', 'ibo') and self._scf is None:
logger.error(self, 'PM with IAO scheme should include an scf '
'object when creating PM object.\n PM(mol, mf=scf_object)')
raise ValueError('PM attribute method is not valid')
return atomic_pops(mol, mo_coeff, method, self._scf, s=s)
[docs]
def stability_jacobi(self):
return pipek_jacobi.PipekMezey_stability_jacobi(self)
PM = Pipek = PipekMezey
if __name__ == '__main__':
from pyscf import gto, scf
mol = gto.Mole()
mol.atom = '''
O 0.00000 0.00000 0.11779
H 0.00000 0.75545 -0.47116
H 0.00000 -0.75545 -0.47116
'''
mol.basis = 'ccpvdz'
mol.build()
mf = scf.RHF(mol).run()
mlo = PM(mol)
mlo.verbose = 4
mlo.exponent = 2 # integer >= 2
mo0 = mf.mo_coeff[:,mf.mo_occ>1e-6]
mo = mlo.kernel(mo0)
isstable, mo1 = mlo.stability_jacobi()
if not isstable:
mo = mlo.kernel(mo1)
isstable, mo1 = mlo.stability_jacobi()
assert( isstable )
