#!/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: <NAME> <<EMAIL>>
#

'''
Pipek-Mezey localization
'''

import time
import numpy
import scipy.linalg
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 import __config__


def atomic_pops(mol, mo_coeff, method='meta_lowdin'):
    '''
    Kwargs:
        method : string
            one of mulliken, lowdin, meta_lowdin

    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.
    '''
    if getattr(mol, 'pbc_intor', None):  # whether mol object is a cell
        s = mol.pbc_intor('int1e_ovlp_sph', hermi=1)
    else:
        s = mol.intor_symmetric('int1e_ovlp')
    nmo = mo_coeff.shape[1]
    proj = numpy.empty((mol.natm,nmo,nmo))

    if method.lower() == '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.lower() in ('lowdin', 'meta_lowdin'):
        c = orth.restore_ao_character(mol, 'ANO')
        csc = reduce(lib.dot, (mo_coeff.conj().T, s, orth.orth_ao(mol, method, c, 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)
    else:
        raise KeyError('method = %s' % method)

    return proj


class PipekMezey(boys.Boys):

    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)  # should be 2 or 4

    def __init__(self, mol, mo_coeff=None):
        boys.Boys.__init__(self, mol, mo_coeff)
        self._keys = self._keys.union(['pop_method', 'exponent'])

    def dump_flags(self, verbose=None):
        boys.Boys.dump_flags(self, verbose)
        logger.info(self, 'pop_method = %s',self.pop_method)

    def gen_g_hop(self, u):
        mo_coeff = lib.dot(self.mo_coeff, u)
        pop = self.atomic_pops(self.mol, mo_coeff, self.pop_method)
        if self.exponent == 2:
            g0 = numpy.einsum('xii,xip->pi', pop, pop)
            g = -self.pack_uniq_var(g0-g0.conj().T) * 2
        elif self.exponent == 4:
            pop3 = numpy.einsum('xii->xi', pop)**3
            g0 = numpy.einsum('xi,xip->pi', pop3, pop)
            g = -self.pack_uniq_var(g0-g0.conj().T) * 4
        else:
            raise NotImplementedError('exponent %s' % self.exponent)

        h_diag = numpy.einsum('xii,xpp->pi', pop, pop) * 2
        g_diag = g0.diagonal()
        h_diag-= g_diag + g_diag.reshape(-1,1)
        h_diag+= numpy.einsum('xip,xip->pi', pop, pop) * 2
        h_diag+= numpy.einsum('xip,xpi->pi', pop, pop) * 2
        h_diag = -self.pack_uniq_var(h_diag) * 2

        g0 = g0 + g0.conj().T
        if self.exponent == 2:
            def h_op(x):
                x = self.unpack_uniq_var(x)
                norb = x.shape[0]
                hx = lib.dot(x.T, g0.T).conj()
                hx+= numpy.einsum('xip,xi->pi', pop, numpy.einsum('qi,xiq->xi', x, pop)) * 2
                hx-= numpy.einsum('xpp,xip->pi', pop,
                                  lib.dot(pop.reshape(-1,norb), x).reshape(-1,norb,norb)) * 2
                hx-= numpy.einsum('xip,xp->pi', pop, numpy.einsum('qp,xpq->xp', x, pop)) * 2
                return -self.pack_uniq_var(hx-hx.conj().T)
        else:
            def h_op(x):
                x = self.unpack_uniq_var(x)
                norb = x.shape[0]
                hx = lib.dot(x.T, g0.T).conj() * 2
                pop2 = numpy.einsum('xii->xi', pop)**2
                pop3 = numpy.einsum('xii->xi', pop)**3
                tmp = numpy.einsum('qi,xiq->xi', x, pop) * pop2
                hx+= numpy.einsum('xip,xi->pi', pop, tmp) * 12
                hx-= numpy.einsum('xp,xip->pi', pop3,
                                  lib.dot(pop.reshape(-1,norb), x).reshape(-1,norb,norb)) * 4
                tmp = numpy.einsum('qp,xpq->xp', x, pop) * pop2
                hx-= numpy.einsum('xip,xp->pi', pop, tmp) * 12
                return -self.pack_uniq_var(hx-hx.conj().T)

        return g, h_op, h_diag

    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)
        pop = self.atomic_pops(self.mol, mo_coeff, self.pop_method)
        if self.exponent == 2:
            g0 = numpy.einsum('xii,xip->pi', pop, pop)
            g = -self.pack_uniq_var(g0-g0.conj().T) * 2
        else:
            pop3 = numpy.einsum('xii->xi', pop)**3
            g0 = numpy.einsum('xi,xip->pi', pop3, pop)
            g = -self.pack_uniq_var(g0-g0.conj().T) * 4
        return g

    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)
        pop = self.atomic_pops(self.mol, mo_coeff, self.pop_method)
        if self.exponent == 2:
            return numpy.einsum('xii,xii->', pop, pop)
        else:
            pop2 = numpy.einsum('xii->xi', pop)**2
            return numpy.einsum('xi,xi', pop2, pop2)

    @lib.with_doc(atomic_pops.__doc__)
    def atomic_pops(self, mol, mo_coeff, method=None):
        if method is None:
            method = self.pop_method
        return atomic_pops(mol, mo_coeff, method)

PM = Pipek = PipekMezey

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.    0.5   -0.4
      '''
    mol.basis = 'ccpvdz'
    mol.build()
    mf = scf.RHF(mol).run()

    mo = PM(mol).kernel(mf.mo_coeff[:,5:9], verbose=4)
