#!/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>
#
'''
Foster-Boys localization
'''
import numpy
from functools import reduce
from pyscf import lib
from pyscf.lib import logger
from pyscf.soscf import ciah
from pyscf.lo import orth, cholesky_mos
from pyscf import __config__
[docs]
def kernel(localizer, mo_coeff=None, callback=None, verbose=None):
from pyscf.tools import mo_mapping
if mo_coeff is not None:
localizer.mo_coeff = numpy.asarray(mo_coeff, order='C')
if localizer.mo_coeff.shape[1] <= 1:
return localizer.mo_coeff
if localizer.verbose >= logger.WARN:
localizer.check_sanity()
localizer.dump_flags()
cput0 = (logger.process_clock(), logger.perf_counter())
log = logger.new_logger(localizer, verbose=verbose)
if localizer.conv_tol_grad is None:
conv_tol_grad = numpy.sqrt(localizer.conv_tol*.1)
log.info('Set conv_tol_grad to %g', conv_tol_grad)
else:
conv_tol_grad = localizer.conv_tol_grad
if mo_coeff is None:
if getattr(localizer, 'mol', None) and localizer.mol.natm == 0:
# For customized Hamiltonian
u0 = localizer.get_init_guess('random')
else:
u0 = localizer.get_init_guess(localizer.init_guess)
else:
u0 = localizer.get_init_guess(None)
rotaiter = ciah.rotate_orb_cc(localizer, u0, conv_tol_grad, verbose=log)
u, g_orb, stat = next(rotaiter)
cput1 = log.timer('initializing CIAH', *cput0)
tot_kf = stat.tot_kf
tot_hop = stat.tot_hop
conv = False
e_last = 0
for imacro in range(localizer.max_cycle):
norm_gorb = numpy.linalg.norm(g_orb)
u0 = lib.dot(u0, u)
e = localizer.cost_function(u0)
e_last, de = e, e-e_last
log.info('macro= %d f(x)= %.14g delta_f= %g |g|= %g %d KF %d Hx',
imacro+1, e, de, norm_gorb, stat.tot_kf+1, stat.tot_hop)
cput1 = log.timer('cycle= %d'%(imacro+1), *cput1)
if (norm_gorb < conv_tol_grad and abs(de) < localizer.conv_tol
and stat.tot_hop < localizer.ah_max_cycle):
conv = True
if callable(callback):
callback(locals())
if conv:
break
u, g_orb, stat = rotaiter.send(u0)
tot_kf += stat.tot_kf
tot_hop += stat.tot_hop
rotaiter.close()
log.info('macro X = %d f(x)= %.14g |g|= %g %d intor %d KF %d Hx',
imacro+1, e, norm_gorb,
(imacro+1)*2, tot_kf+imacro+1, tot_hop)
# Sort the localized orbitals, to make each localized orbitals as close as
# possible to the corresponding input orbitals
sorted_idx = mo_mapping.mo_1to1map(u0)
localizer.mo_coeff = lib.dot(localizer.mo_coeff, u0[:,sorted_idx])
return localizer.mo_coeff
[docs]
def dipole_integral(mol, mo_coeff, charge_center=None):
# The gauge origin has no effects for maximization |<r>|^2
# Set to charge center for physical significance of <r>
if charge_center is None:
charge_center = (numpy.einsum('z,zx->x', mol.atom_charges(), mol.atom_coords())
/ mol.atom_charges().sum())
with mol.with_common_origin(charge_center):
dip = numpy.asarray([reduce(lib.dot, (mo_coeff.conj().T, x, mo_coeff))
for x in mol.intor_symmetric('int1e_r', comp=3)])
return dip
[docs]
def atomic_init_guess(mol, mo_coeff):
s = mol.intor_symmetric('int1e_ovlp')
c = orth.orth_ao(mol, s=s)
mo = reduce(numpy.dot, (c.conj().T, s, mo_coeff))
# Find the AOs which have largest overlap to MOs
idx = numpy.argsort(numpy.einsum('pi,pi->p', mo.conj(), mo))
nmo = mo.shape[1]
idx = sorted(idx[-nmo:])
# Rotate mo_coeff, make it as close as possible to AOs
u, w, vh = numpy.linalg.svd(mo[idx])
return lib.dot(u, vh).conj().T
[docs]
class OrbitalLocalizer(lib.StreamObject, ciah.CIAHOptimizerMixin):
conv_tol = getattr(__config__, 'lo_boys_Boys_conv_tol', 1e-6)
conv_tol_grad = getattr(__config__, 'lo_boys_Boys_conv_tol_grad', None)
max_cycle = getattr(__config__, 'lo_boys_Boys_max_cycle', 100)
max_iters = getattr(__config__, 'lo_boys_Boys_max_iters', 20)
max_stepsize = getattr(__config__, 'lo_boys_Boys_max_stepsize', .05)
ah_trust_region = getattr(__config__, 'lo_boys_Boys_ah_trust_region', 3)
ah_start_tol = getattr(__config__, 'lo_boys_Boys_ah_start_tol', 1e9)
ah_max_cycle = getattr(__config__, 'lo_boys_Boys_ah_max_cycle', 40)
init_guess = getattr(__config__, 'lo_boys_Boys_init_guess', 'atomic')
_keys = {
'conv_tol', 'conv_tol_grad', 'max_cycle', 'max_iters',
'max_stepsize', 'ah_trust_region', 'ah_start_tol',
'ah_max_cycle', 'init_guess', 'mol', 'mo_coeff',
}
def __init__(self, mol, mo_coeff=None):
self.mol = mol
self.stdout = mol.stdout
self.verbose = mol.verbose
self.mo_coeff = mo_coeff
[docs]
def dump_flags(self, verbose=None):
log = logger.new_logger(self, verbose)
log.info('\n')
log.info('******** %s ********', self.__class__)
log.info('conv_tol = %s' , self.conv_tol )
log.info('conv_tol_grad = %s' , self.conv_tol_grad )
log.info('max_cycle = %s' , self.max_cycle )
log.info('max_stepsize = %s' , self.max_stepsize )
log.info('max_iters = %s' , self.max_iters )
log.info('kf_interval = %s' , self.kf_interval )
log.info('kf_trust_region = %s', self.kf_trust_region)
log.info('ah_start_tol = %s' , self.ah_start_tol )
log.info('ah_start_cycle = %s' , self.ah_start_cycle )
log.info('ah_level_shift = %s' , self.ah_level_shift )
log.info('ah_conv_tol = %s' , self.ah_conv_tol )
log.info('ah_lindep = %s' , self.ah_lindep )
log.info('ah_max_cycle = %s' , self.ah_max_cycle )
log.info('ah_trust_region = %s', self.ah_trust_region)
log.info('init_guess = %s' , self.init_guess )
[docs]
def get_init_guess(self, key='atomic'):
'''Generate initial guess for localization.
Kwargs:
key : str or bool
If key is 'atomic', initial guess is based on the projected
atomic orbitals. False
'''
nmo = self.mo_coeff.shape[1]
if isinstance(key, str) and key.lower() == 'atomic':
u0 = atomic_init_guess(self.mol, self.mo_coeff)
elif isinstance(key, str) and key.lower().startswith('cho'):
mo_init = cholesky_mos(self.mo_coeff)
S = self.mol.intor_symmetric('int1e_ovlp')
u0 = numpy.linalg.multi_dot([self.mo_coeff.T, S, mo_init])
else:
u0 = numpy.eye(nmo)
if (isinstance(key, str) and key.lower().startswith('rand')
or numpy.linalg.norm(self.get_grad(u0)) < 1e-5):
# Add noise to kick initial guess out of saddle point
dr = numpy.cos(numpy.arange((nmo-1)*nmo//2)) * 1e-3
u0 = self.extract_rotation(dr)
return u0
[docs]
def gen_g_hop(self, u):
raise NotImplementedError
[docs]
def get_grad(self, u=None):
raise NotImplementedError
[docs]
def cost_function(self, u=None):
raise NotImplementedError
kernel = kernel
[docs]
class Boys(OrbitalLocalizer):
r'''
Base class oflocalization optimizer that maximizes the orbital dipole
\sum_i | <i| r |i> |^2
Args:
mol : Mole object
Kwargs:
mo_coeff : size (N,N) np.array
The orbital space to localize for Boys localization.
When initializing the localization optimizer ``bopt = Boys(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 Boys 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. If set
to 'cholesky', then cholesky orbitals will be used as the initial guess.
Default is 'atomic'.
Saved results
mo_coeff : ndarray
Localized orbitals
'''
[docs]
def gen_g_hop(self, u):
mo_coeff = lib.dot(self.mo_coeff, u)
dip = dipole_integral(self.mol, mo_coeff)
g0 = numpy.einsum('xii,xip->pi', dip, dip)
g = -self.pack_uniq_var(g0-g0.conj().T) * 2
h_diag = numpy.einsum('xii,xpp->pi', dip, dip) * 2
h_diag-= g0.diagonal() + g0.diagonal().reshape(-1,1)
h_diag+= numpy.einsum('xip,xip->pi', dip, dip) * 2
h_diag+= numpy.einsum('xip,xpi->pi', dip, dip) * 2
h_diag = -self.pack_uniq_var(h_diag) * 2
#:nmo = mo_coeff.shape[1]
#:h = numpy.einsum('xjj,xjq,pk->pjqk', dip, dip, numpy.eye(nmo))
#:h+= numpy.einsum('xqq,xjq,pk->pjqk', dip, dip, numpy.eye(nmo))
#:h+= numpy.einsum('xjq,xjp,jk->pjqk', dip, dip, numpy.eye(nmo))
#:h+= numpy.einsum('xjp,xkp,pq->pjqk', dip, dip, numpy.eye(nmo))
#:h-= numpy.einsum('xjj,xkp,jq->pjqk', dip, dip, numpy.eye(nmo))
#:h-= numpy.einsum('xpp,xjq,pk->pjqk', dip, dip, numpy.eye(nmo))
#:h-= numpy.einsum('xjp,xpq,pk->pjqk', dip, dip, numpy.eye(nmo))*2
#:h = h - h.transpose(0,1,3,2)
#:h = h - h.transpose(1,0,2,3)
#:h = h + h.transpose(2,3,0,1)
#:h *= -.5
#:idx = numpy.tril_indices(nmo, -1)
#:h = h[idx][:,idx[0],idx[1]]
g0 = g0 + g0.conj().T
def h_op(x):
x = self.unpack_uniq_var(x)
norb = x.shape[0]
#:hx = numpy.einsum('qp,xjj,xjq->pj', x, dip, dip)
#:hx+= numpy.einsum('qp,xqq,xjq->pj', x, dip, dip)
#:hx+= numpy.einsum('jk,xkk,xkp->pj', x, dip, dip)
#:hx+= numpy.einsum('jk,xpp,xkp->pj', x, dip, dip)
#:hx+= numpy.einsum('qj,xjq,xjp->pj', x, dip, dip)
#:hx+= numpy.einsum('pk,xjp,xkp->pj', x, dip, dip)
#:hx-= numpy.einsum('qp,xpp,xjq->pj', x, dip, dip) * 2
#:hx-= numpy.einsum('qp,xjp,xpq->pj', x, dip, dip) * 2
#:hx+= numpy.einsum('qj,xjp,xjq->pj', x, dip, dip)
#:hx+= numpy.einsum('pk,xkp,xjp->pj', x, dip, dip)
#:hx-= numpy.einsum('jk,xjj,xkp->pj', x, dip, dip) * 2
#:hx-= numpy.einsum('jk,xkj,xjp->pj', x, dip, dip) * 2
#:return -self.pack_uniq_var(hx)
#:hx = numpy.einsum('iq,qp->pi', g0, x)
hx = lib.dot(x.T, g0.T).conj()
#:hx+= numpy.einsum('qi,xiq,xip->pi', x, dip, dip) * 2
hx+= numpy.einsum('xip,xi->pi', dip, numpy.einsum('qi,xiq->xi', x, dip)) * 2
#:hx-= numpy.einsum('qp,xpp,xiq->pi', x, dip, dip) * 2
hx-= numpy.einsum('xpp,xip->pi', dip,
lib.dot(dip.reshape(-1,norb), x).reshape(3,norb,norb)) * 2
#:hx-= numpy.einsum('qp,xip,xpq->pi', x, dip, dip) * 2
hx-= numpy.einsum('xip,xp->pi', dip, numpy.einsum('qp,xpq->xp', x, dip)) * 2
return -self.pack_uniq_var(hx-hx.conj().T)
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])
mo_coeff = lib.dot(self.mo_coeff, u)
dip = dipole_integral(self.mol, mo_coeff)
g0 = numpy.einsum('xii,xip->pi', dip, dip)
g = -self.pack_uniq_var(g0-g0.conj().T) * 2
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)
charge_center = (numpy.einsum('z,zx->x', self.mol.atom_charges(), self.mol.atom_coords())
/ self.mol.atom_charges().sum())
dip = dipole_integral(self.mol, mo_coeff, charge_center)
with self.mol.with_common_origin(charge_center):
r2 = self.mol.intor_symmetric('int1e_r2')
r2 = numpy.einsum('pi,pi->', mo_coeff, lib.dot(r2, mo_coeff))
val = r2 - numpy.einsum('xii,xii->', dip, dip)
return val
FB = BF = Boys
if __name__ == '__main__':
from pyscf import gto, scf
mol = gto.Mole()
mol.atom = '''
O 0. 0. 0.2
H 0. -0.5 -0.4
H 0. 0.7 -0.2
'''
mol.basis = 'ccpvdz'
mol.build()
mf = scf.RHF(mol).run()
mo = mf.mo_coeff[:,:3]
loc = Boys(mol, mo)
u0 = numpy.eye(3)
dx = 1e-5
g_num = []
hdiag_num = []
h_op, hdiag = loc.gen_g_hop(u0)[1:]
for i in range(3):
dr = numpy.zeros(3)
dr[i] = dx
u = loc.extract_rotation(dr)
cf1 =-loc.cost_function(u0)
cf2 =-loc.cost_function(u0.dot(u))
cg1 = loc.get_grad(u0)
cg2 = loc.get_grad(u0.dot(u))
g_num.append((cf2-cf1)/dx)
print('hx', abs(cg2-cg1-h_op(dr)).sum())
hdiag_num.append(h_op(dr/dx)[i])
print('g', numpy.array(g_num), loc.get_grad(u0)*2)
print('hdiag', numpy.array(hdiag_num), hdiag)
mo = Boys(mol).kernel(mf.mo_coeff[:,5:9], verbose=4)