#!/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.
#
# Author: Tianyu Zhu <zhutianyu1991@gmail.com>
#

'''
Spin-restricted G0W0 approximation with analytic continuation
This implementation has N^4 scaling, and is faster than GW-CD (N^4)
and analytic GW (N^6) methods.
GW-AC is recommended for valence states only, and is inaccuarate for core states.

Method:
    See T. Zhu and G.K.-L. Chan, arxiv:2007.03148 (2020) for details
    Compute Sigma on imaginary frequency with density fitting,
    then analytically continued to real frequency

Useful References:
    J. Chem. Theory Comput. 12, 3623-3635 (2016)
    New J. Phys. 14, 053020 (2012)
'''

from functools import reduce
import numpy
import numpy as np
import h5py
from scipy.optimize import newton, least_squares

from pyscf import lib
from pyscf.lib import logger
from pyscf.ao2mo import _ao2mo
from pyscf import df, scf
from pyscf.mp.mp2 import get_nocc, get_nmo, get_frozen_mask
from pyscf import __config__

einsum = lib.einsum

def kernel(gw, mo_energy, mo_coeff, Lpq=None, orbs=None,
           nw=None, vhf_df=False, verbose=logger.NOTE):
    '''
    GW-corrected quasiparticle orbital energies
    Returns:
        A list :  converged, mo_energy, mo_coeff
    '''
    mf = gw._scf
    if gw.frozen is None:
        frozen = 0
    else:
        frozen = gw.frozen

    # only support frozen core
    assert (isinstance(frozen, int))
    assert (frozen < gw.nocc)

    if Lpq is None:
        Lpq = gw.ao2mo(mo_coeff)
    if orbs is None:
        orbs = range(gw.nmo)
    else:
        orbs = [x - frozen for x in orbs]
        if orbs[0] < 0:
            logger.warn(gw, 'GW orbs must be larger than frozen core!')
            raise RuntimeError

    # v_xc
    v_mf = mf.get_veff() - mf.get_j()
    v_mf = reduce(numpy.dot, (mo_coeff.T, v_mf, mo_coeff))

    nocc = gw.nocc
    nmo = gw.nmo

    # v_hf from DFT/HF density
    if vhf_df and frozen == 0:
        # density fitting for vk
        vk = -einsum('Lni,Lim->nm',Lpq[:,:,:nocc],Lpq[:,:nocc,:])
    else:
        # exact vk without density fitting
        dm = mf.make_rdm1()
        rhf = scf.RHF(gw.mol)
        vk = rhf.get_veff(gw.mol,dm) - rhf.get_j(gw.mol,dm)
        vk = reduce(numpy.dot, (mo_coeff.T, vk, mo_coeff))

    # Grids for integration on imaginary axis
    freqs,wts = _get_scaled_legendre_roots(nw)

    # Compute self-energy on imaginary axis i*[0,iw_cutoff]
    sigmaI,omega = get_sigma_diag(gw, orbs, Lpq, freqs, wts, iw_cutoff=5.)

    # Analytic continuation
    if gw.ac == 'twopole':
        coeff = AC_twopole_diag(sigmaI, omega, orbs, nocc)
    elif gw.ac == 'pade':
        coeff,omega_fit = AC_pade_thiele_diag(sigmaI, omega)

    conv = True
    mf_mo_energy = mo_energy.copy()
    ef = (mo_energy[nocc-1] + mo_energy[nocc])/2.
    mo_energy = np.zeros_like(gw._scf.mo_energy)
    for p in orbs:
        if gw.linearized:
            # linearized G0W0
            de = 1e-6
            ep = mf_mo_energy[p]
            #TODO: analytic sigma derivative
            if gw.ac == 'twopole':
                sigmaR = two_pole(ep-ef, coeff[:,p-orbs[0]]).real
                dsigma = two_pole(ep-ef+de, coeff[:,p-orbs[0]]).real - sigmaR.real
            elif gw.ac == 'pade':
                sigmaR = pade_thiele(ep-ef, omega_fit[p-orbs[0]], coeff[:,p-orbs[0]]).real
                dsigma = pade_thiele(ep-ef+de, omega_fit[p-orbs[0]], coeff[:,p-orbs[0]]).real - sigmaR.real
            zn = 1.0/(1.0-dsigma/de)
            e = ep + zn*(sigmaR.real + vk[p,p] - v_mf[p,p])
            mo_energy[p+frozen] = e
        else:
            # self-consistently solve QP equation
            def quasiparticle(omega):
                if gw.ac == 'twopole':
                    sigmaR = two_pole(omega-ef, coeff[:,p-orbs[0]]).real
                elif gw.ac == 'pade':
                    sigmaR = pade_thiele(omega-ef, omega_fit[p-orbs[0]], coeff[:,p-orbs[0]]).real
                return omega - mf_mo_energy[p] - (sigmaR.real + vk[p,p] - v_mf[p,p])
            try:
                e = newton(quasiparticle, mf_mo_energy[p], tol=1e-6, maxiter=100)
                mo_energy[p+frozen] = e
            except RuntimeError:
                conv = False

    if gw.verbose >= logger.DEBUG:
        numpy.set_printoptions(threshold=nmo)
        logger.debug(gw, '  GW mo_energy =\n%s', mo_energy)
        numpy.set_printoptions(threshold=1000)

    return conv, mo_energy, mo_coeff

def get_rho_response(omega, mo_energy, Lpq):
    '''
    Compute density response function in auxiliary basis at freq iw
    '''
    naux, nocc, nvir = Lpq.shape
    eia = mo_energy[:nocc,None] - mo_energy[None,nocc:]
    eia = eia/(omega**2+eia*eia)
    Pia = einsum('Pia,ia->Pia',Lpq,eia)
    # Response from both spin-up and spin-down density
    Pi = 4. * einsum('Pia,Qia->PQ',Pia,Lpq)

    return Pi

def get_sigma_diag(gw, orbs, Lpq, freqs, wts, iw_cutoff=None):
    '''
    Compute GW correlation self-energy (diagonal elements)
    in MO basis on imaginary axis
    '''
    mo_energy = _mo_energy_without_core(gw, gw._scf.mo_energy)
    nocc = gw.nocc
    nw = len(freqs)
    naux = Lpq.shape[0]
    norbs = len(orbs)

    # TODO: Treatment of degeneracy
    if (mo_energy[nocc] - mo_energy[nocc-1]) < 1e-3:
        logger.warn(gw, 'GW not well-defined for degeneracy!')
    ef = (mo_energy[nocc-1] + mo_energy[nocc])/2.

    # Integration on numerical grids
    if iw_cutoff is not None:
        nw_sigma = sum(iw < iw_cutoff for iw in freqs) + 1
    else:
        nw_sigma = nw + 1

    # Compute occ for -iw and vir for iw separately
    # to avoid branch cuts in analytic continuation
    omega_occ = np.zeros((nw_sigma),dtype=np.complex128)
    omega_vir = np.zeros((nw_sigma),dtype=np.complex128)
    omega_occ[1:] = -1j*freqs[:(nw_sigma-1)]
    omega_vir[1:] = 1j*freqs[:(nw_sigma-1)]
    orbs_occ = [i for i in orbs if i < nocc]
    norbs_occ = len(orbs_occ)

    emo_occ = omega_occ[None,:] + ef - mo_energy[:,None]
    emo_vir = omega_vir[None,:] + ef - mo_energy[:,None]

    sigma = np.zeros((norbs,nw_sigma),dtype=np.complex128)
    omega = np.zeros((norbs,nw_sigma),dtype=np.complex128)
    for p in range(norbs):
        orbp = orbs[p]
        if orbp < nocc:
            omega[p] = omega_occ.copy()
        else:
            omega[p] = omega_vir.copy()

    for w in range(nw):
        Pi = get_rho_response(freqs[w], mo_energy, Lpq[:,:nocc,nocc:])
        Pi_inv = np.linalg.inv(np.eye(naux)-Pi)-np.eye(naux)
        g0_occ = wts[w] * emo_occ / (emo_occ**2+freqs[w]**2)
        g0_vir = wts[w] * emo_vir / (emo_vir**2+freqs[w]**2)
        Qnm = einsum('Pnm,PQ->Qnm',Lpq[:,orbs,:],Pi_inv)
        Wmn = einsum('Qnm,Qmn->mn',Qnm,Lpq[:,:,orbs])
        sigma[:norbs_occ] += -einsum('mn,mw->nw',Wmn[:,:norbs_occ],g0_occ)/np.pi
        sigma[norbs_occ:] += -einsum('mn,mw->nw',Wmn[:,norbs_occ:],g0_vir)/np.pi

    return sigma, omega

def _get_scaled_legendre_roots(nw):
    """
    Scale nw Legendre roots, which lie in the
    interval [-1, 1], so that they lie in [0, inf)
    Ref: www.cond-mat.de/events/correl19/manuscripts/ren.pdf

    Returns:
        freqs : 1D ndarray
        wts : 1D ndarray
    """
    freqs, wts = np.polynomial.legendre.leggauss(nw)
    x0 = 0.5
    freqs_new = x0*(1.+freqs)/(1.-freqs)
    wts = wts*2.*x0/(1.-freqs)**2
    return freqs_new, wts

def _get_clenshaw_curtis_roots(nw):
    """
    Clenshaw-Curtis qaudrature on [0,inf)
    Ref: J. Chem. Phys. 132, 234114 (2010)
    Returns:
        freqs : 1D ndarray
        wts : 1D ndarray
    """
    freqs = np.zeros(nw)
    wts = np.zeros(nw)
    a = 0.2
    for w in range(nw):
        t = (w+1.0)/nw * np.pi/2.
        freqs[w] = a / np.tan(t)
        if w != nw-1:
            wts[w] = a*np.pi/2./nw/(np.sin(t)**2)
        else:
            wts[w] = a*np.pi/4./nw/(np.sin(t)**2)
    return freqs[::-1], wts[::-1]

def two_pole_fit(coeff, omega, sigma):
    cf = coeff[:5] + 1j*coeff[5:]
    f = cf[0] + cf[1]/(omega+cf[3]) + cf[2]/(omega+cf[4]) - sigma
    f[0] = f[0]/0.01
    return np.array([f.real,f.imag]).reshape(-1)

def two_pole(freqs, coeff):
    cf = coeff[:5] + 1j*coeff[5:]
    return cf[0] + cf[1]/(freqs+cf[3]) + cf[2]/(freqs+cf[4])

def AC_twopole_diag(sigma, omega, orbs, nocc):
    """
    Analytic continuation to real axis using a two-pole model
    Returns:
        coeff: 2D array (ncoeff, norbs)
    """
    norbs, nw = sigma.shape
    coeff = np.zeros((10,norbs))
    for p in range(norbs):
        # target = np.array([sigma[p].real,sigma[p].imag]).reshape(-1)
        if orbs[p] < nocc:
            x0 = np.array([0, 1, 1, 1, -1, 0, 0, 0, -1.0, -0.5])
        else:
            x0 = np.array([0, 1, 1, 1, -1, 0, 0, 0, 1.0, 0.5])
        #TODO: analytic gradient
        xopt = least_squares(two_pole_fit, x0, jac='3-point', method='trf', xtol=1e-10,
                             gtol = 1e-10, max_nfev=1000, verbose=0, args=(omega[p], sigma[p]))
        if xopt.success is False:
            print('WARN: 2P-Fit Orb %d not converged, cost function %e'%(p,xopt.cost))
        coeff[:,p] = xopt.x.copy()
    return coeff

def thiele(fn,zn):
    nfit = len(zn)
    g = np.zeros((nfit,nfit),dtype=np.complex128)
    g[:,0] = fn.copy()
    for i in range(1,nfit):
        g[i:,i] = (g[i-1,i-1]-g[i:,i-1])/((zn[i:]-zn[i-1])*g[i:,i-1])
    a = g.diagonal()
    return a

def pade_thiele(freqs,zn,coeff):
    nfit = len(coeff)
    X = coeff[-1]*(freqs-zn[-2])
    for i in range(nfit-1):
        idx = nfit-i-1
        X = coeff[idx]*(freqs-zn[idx-1])/(1.+X)
    X = coeff[0]/(1.+X)
    return X

def AC_pade_thiele_diag(sigma, omega):
    """
    Analytic continuation to real axis using a Pade approximation
    from Thiele's reciprocal difference method
    Reference: J. Low Temp. Phys. 29, 179 (1977)
    Returns:
        coeff: 2D array (ncoeff, norbs)
        omega: 2D array (norbs, npade)
    """
    idx = range(1,40,6)
    sigma1 = sigma[:,idx].copy()
    sigma2 = sigma[:,(idx[-1]+4)::4].copy()
    sigma = np.hstack((sigma1,sigma2))
    omega1 = omega[:,idx].copy()
    omega2 = omega[:,(idx[-1]+4)::4].copy()
    omega = np.hstack((omega1,omega2))
    norbs, nw = sigma.shape
    npade = nw // 2
    coeff = np.zeros((npade*2,norbs),dtype=np.complex128)
    for p in range(norbs):
        coeff[:,p] = thiele(sigma[p,:npade*2], omega[p,:npade*2])

    return coeff, omega[:,:npade*2]

def _mo_energy_without_core(gw, mo_energy):
    return mo_energy[get_frozen_mask(gw)]

def _mo_without_core(gw, mo):
    return mo[:,get_frozen_mask(gw)]

class GWAC(lib.StreamObject):

    linearized = getattr(__config__, 'gw_gw_GW_linearized', False)
    # Analytic continuation: pade or twopole
    ac = getattr(__config__, 'gw_gw_GW_ac', 'pade')

    def __init__(self, mf, frozen=None):
        self.mol = mf.mol
        self._scf = mf
        self.verbose = self.mol.verbose
        self.stdout = self.mol.stdout
        self.max_memory = mf.max_memory
        self.frozen = frozen

        # DF-GW must use density fitting integrals
        if getattr(mf, 'with_df', None):
            self.with_df = mf.with_df
        else:
            self.with_df = df.DF(mf.mol)
            self.with_df.auxbasis = df.make_auxbasis(mf.mol, mp2fit=True)
        self._keys.update(['with_df'])

##################################################
# don't modify the following attributes, they are not input options
        self._nocc = None
        self._nmo = None
        # self.mo_energy: GW quasiparticle energy, not scf mo_energy
        self.mo_energy = None
        self.mo_coeff = mf.mo_coeff
        self.mo_occ = mf.mo_occ
        self.sigma = None

        keys = set(('linearized','ac'))
        self._keys = set(self.__dict__.keys()).union(keys)

    def dump_flags(self):
        log = logger.Logger(self.stdout, self.verbose)
        log.info('')
        log.info('******** %s ********', self.__class__)
        log.info('method = %s', self.__class__.__name__)
        nocc = self.nocc
        nvir = self.nmo - nocc
        log.info('GW nocc = %d, nvir = %d', nocc, nvir)
        if self.frozen is not None:
            log.info('frozen = %s', self.frozen)
        logger.info(self, 'use perturbative linearized QP eqn = %s', self.linearized)
        logger.info(self, 'analytic continuation method = %s', self.ac)
        return self

    @property
    def nocc(self):
        return self.get_nocc()
    @nocc.setter
    def nocc(self, n):
        self._nocc = n

    @property
    def nmo(self):
        return self.get_nmo()
    @nmo.setter
    def nmo(self, n):
        self._nmo = n

    get_nocc = get_nocc
    get_nmo = get_nmo
    get_frozen_mask = get_frozen_mask

    def kernel(self, mo_energy=None, mo_coeff=None, Lpq=None, orbs=None, nw=100, vhf_df=False):
        """
        Input:
            orbs: self-energy orbs
            nw: grid number
            vhf_df: whether using density fitting for HF exchange
        Output:
            mo_energy: GW quasiparticle energy
        """
        if mo_coeff is None:
            mo_coeff = _mo_without_core(self, self._scf.mo_coeff)
        if mo_energy is None:
            mo_energy = _mo_energy_without_core(self, self._scf.mo_energy)

        cput0 = (logger.process_clock(), logger.perf_counter())
        self.dump_flags()
        self.converged, self.mo_energy, self.mo_coeff = \
                kernel(self, mo_energy, mo_coeff,
                       Lpq=Lpq, orbs=orbs, nw=nw, vhf_df=vhf_df, verbose=self.verbose)

        logger.warn(self, 'GW QP energies may not be sorted from min to max')
        logger.timer(self, 'GW', *cput0)
        return self.mo_energy

    def ao2mo(self, mo_coeff=None):
        if mo_coeff is None:
            mo_coeff = self.mo_coeff
        nmo = self.nmo
        naux = self.with_df.get_naoaux()
        mem_incore = (2*nmo**2*naux) * 8/1e6
        mem_now = lib.current_memory()[0]

        mo = numpy.asarray(mo_coeff, order='F')
        ijslice = (0, nmo, 0, nmo)
        Lpq = None
        if (mem_incore + mem_now < 0.99*self.max_memory) or self.mol.incore_anyway:
            Lpq = _ao2mo.nr_e2(self.with_df._cderi, mo, ijslice, aosym='s2', out=Lpq)
            return Lpq.reshape(naux,nmo,nmo)
        else:
            logger.warn(self, 'Memory may not be enough!')
            raise NotImplementedError


if __name__ == '__main__':
    from pyscf import gto, dft
    mol = gto.Mole()
    mol.verbose = 4
    mol.atom = [
        [8 , (0. , 0.     , 0.)],
        [1 , (0. , -0.7571 , 0.5861)],
        [1 , (0. , 0.7571 , 0.5861)]]
    mol.basis = 'def2-svp'
    mol.build()

    mf = dft.RKS(mol)
    mf.xc = 'pbe'
    mf.kernel()

    nocc = mol.nelectron//2
    nmo = mf.mo_energy.size
    nvir = nmo-nocc

    gw = GWAC(mf)
    gw.frozen = 0
    gw.linearized = False
    gw.ac = 'pade'
    gw.kernel(orbs=range(nocc-3,nocc+3))
    print(gw.mo_energy)
    assert (abs(gw.mo_energy[nocc-1]- -0.412849230989) < 1e-5)
    assert (abs(gw.mo_energy[nocc] -0.165745160102) < 1e-5)
