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lsda_gga_functionals.cc
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lsda_gga_functionals.cc
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/*
* @BEGIN LICENSE
*
* mydft by Psi4 Developer, a plugin to:
*
* Psi4: an open-source quantum chemistry software package
*
* Copyright (c) 2007-2016 The Psi4 Developers.
*
* The copyrights for code used from other parties are included in
* the corresponding files.
*
* This program is free software; you can redistribute it and/or modify
* it under the terms of the GNU General Public License as published by
* the Free Software Foundation; either version 2 of the License, or
* (at your option) any later version.
*
* This program is distributed in the hope that it will be useful,
* but WITHOUT ANY WARRANTY; without even the implied warranty of
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
* GNU General Public License for more details.
*
* You should have received a copy of the GNU General Public License along
* with this program; if not, write to the Free Software Foundation, Inc.,
* 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA.
*
* @END LICENSE
*/
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include <algorithm>
#include <vector>
#include <utility>
#include <tuple>
#include "psi4/libpsi4util/libpsi4util.h"
// for dft
#include "psi4/libfock/v.h"
#include "psi4/libfunctional/superfunctional.h"
// for grid
#include "psi4/libfock/points.h"
#include "psi4/libfock/cubature.h"
#include "psi4/psi4-dec.h"
#include "psi4/liboptions/liboptions.h"
#include "psi4/libpsio/psio.hpp"
#include "psi4/libmints/wavefunction.h"
#include "psi4/libmints/mintshelper.h"
#include "psi4/libmints/matrix.h"
#include "psi4/libmints/vector.h"
#include "psi4/libmints/basisset.h"
#include "psi4/libmints/molecule.h"
#include "psi4/lib3index/dftensor.h"
#include "psi4/libqt/qt.h"
// jk object
#include "psi4/libfock/jk.h"
// for dft
#include "psi4/libfock/v.h"
#include "psi4/libfunctional/superfunctional.h"
#include "mcpdft_solver.h"
namespace psi{ namespace mcpdft {
//###########################################################
//# The exchange and correlation functional implementations #
//###########################################################
double MCPDFTSolver::Gfunction(double r, double A, double a1, double b1, double b2, double b3, double b4, double p) {
double G = -2.0 * A * (1.0 + a1 * r) * log( 1.0 + pow( 2.0 * A * ( b1 * sqrt(r) + b2 * r + b3 * pow(r,3.0/2.0) + b4 * pow(r, p+1.0 ) ) ,-1.0 ) );
return G;
}
//+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
//+++++++++++++++++++ Exchange functionals ++++++++++++++++++
// LDA/LSDA, B86_MGC, B88, PBE +
//+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
double MCPDFTSolver::EX_LDA(std::shared_ptr<Vector> rho_a, std::shared_ptr<Vector> rho_b){
const double alpha = (2.0/3.0); // Slater value
const double Cx = (9.0/8.0) * alpha * pow(3.0/M_PI,1.0/3.0);
double * rho_ap = rho_a->pointer();
double * rho_bp = rho_b->pointer();
double exc = 0.0;
for (int p = 0; p < phi_points_; p++) {
double rho = rho_ap[p] + rho_bp[p];
exc += -Cx * pow( rho, 4.0/3.0) * grid_w_->pointer()[p];
}
return exc;
}
double MCPDFTSolver::EX_LSDA(std::shared_ptr<Vector> RHO_A, std::shared_ptr<Vector> RHO_B){
const double alpha = (2.0/3.0); // Slater value
const double Cx = (9.0/8.0) * alpha * pow(3.0/M_PI,1.0/3.0);
double * rho_ap = RHO_A->pointer();
double * rho_bp = RHO_B->pointer();
double exc = 0.0;
for (int p = 0; p < phi_points_; p++) {
double exa = pow(2.0,1.0/3.0) * Cx * pow( rho_ap[p], 4.0/3.0) ;
double exb = pow(2.0,1.0/3.0) * Cx * pow( rho_bp[p], 4.0/3.0) ;
double ex_LSDA = exa + exb;
exc += - ex_LSDA * grid_w_->pointer()[p];
}
return exc;
}
double MCPDFTSolver::EX_LSDA(std::shared_ptr<Vector> RHO_A, std::shared_ptr<Vector> RHO_B, std::shared_ptr<Vector> ZETA){
const double alpha = (2.0/3.0); // Slater value
const double Cx = (9.0/8.0) * alpha * pow(3.0/M_PI,1.0/3.0);
double * rho_ap = RHO_A->pointer();
double * rho_bp = RHO_B->pointer();
double * zeta_p = ZETA->pointer();
double exc = 0.0;
for (int p = 0; p < phi_points_; p++) {
// build f(zeta) weight factor where f(0) = 0 and f(1) = 1
double fZet = ( pow( (1.0 + zeta_p[p]) ,4.0/3.0 ) + pow( (1.0 - zeta_p[p]) ,4.0/3.0) - 2.0) / ( 2.0 * pow(2,1.0/3.0) - 2.0 );
double ex0 = Cx * pow( (rho_ap[p] + rho_bp[p]), 1.0/3.0) ;
double ex1 = pow(2.0,1.0/3.0) * ex0;
double ex_LSDA = ex0 + (ex1 - ex0) * fZet;
exc += - ex_LSDA * (rho_ap[p] + rho_bp[p]) * grid_w_->pointer()[p];
}
return exc;
}
double MCPDFTSolver::EX_B86_MGC(){
const double Cx = 0.73855876638202240586;
const double beta = 0.00375;
const double c = pow(2.0,1.0/3.0) * Cx;
double * rho_ap = rho_a_->pointer();
double * rho_bp = rho_b_->pointer();
double * sigma_aap = sigma_aa_->pointer();
double * sigma_bbp = sigma_bb_->pointer();
double exc = 0.0;
for (int p = 0; p < phi_points_; p++) {
double rhoa = rho_ap[p];
double rhob = rho_bp[p];
double rhoa_43 = pow( rhoa, 4.0/3.0);
double rhob_43 = pow( rhob, 4.0/3.0);
double Xa = sqrt(sigma_aap[p]) / rhoa_43;
double Xb = sqrt(sigma_bbp[p]) / rhob_43;
double Xa_2 = Xa * Xa;
double Xb_2 = Xb * Xb;
exc += ( -c * rhoa_43 - (beta * Xa_2 * rhoa_43) / pow(1.0 + 0.007 * Xa_2,4.0/5.0) ) * grid_w_->pointer()[p];
exc += ( -c * rhob_43 - (beta * Xb_2 * rhob_43) / pow(1.0 + 0.007 * Xb_2,4.0/5.0) ) * grid_w_->pointer()[p];
}
return exc;
}
double MCPDFTSolver::EX_B88(std::shared_ptr<Vector> RHO_A, std::shared_ptr<Vector> RHO_B,
std::shared_ptr<Vector> SIGMA_AA, std::shared_ptr<Vector> SIGMA_BB){
const double Cx = 0.73855876638202240586;
const double beta = 0.0042;
const double c = pow(2.0,1.0/3.0) * Cx;
double * rho_ap = RHO_A->pointer();
double * rho_bp = RHO_B->pointer();
double * sigma_aap = SIGMA_AA->pointer();
double * sigma_bbp = SIGMA_BB->pointer();
double exc = 0.0;
for (int p = 0; p < phi_points_; p++) {
double rhoa = rho_ap[p];
double rhob = rho_bp[p];
double rhoa_43 = pow( rhoa, 4.0/3.0);
double rhob_43 = pow( rhob, 4.0/3.0);
double Xa = sqrt(sigma_aap[p]) / rhoa_43;
double Xb = sqrt(sigma_bbp[p]) / rhob_43;
double Xa_2 = Xa * Xa;
double Xb_2 = Xb * Xb;
exc += -rhoa_43 * ( c + (beta * Xa_2) / (1.0 + 6.0 * beta * Xa * asinh(Xa)) ) * grid_w_->pointer()[p];
exc += -rhob_43 * ( c + (beta * Xb_2) / (1.0 + 6.0 * beta * Xb * asinh(Xb)) ) * grid_w_->pointer()[p];
}
return exc;
}
double MCPDFTSolver::EX_B88_I(std::shared_ptr<Vector> RHO_A, std::shared_ptr<Vector> RHO_B,
std::shared_ptr<Vector> SIGMA_AA, std::shared_ptr<Vector> SIGMA_BB){
double tol = 1.0e-20;
const double Cx = 0.73855876638202240586;
const double beta = 0.0042;
const double c = pow(2.0,1.0/3.0) * Cx;
double * rho_ap = RHO_A->pointer();
double * rho_bp = RHO_B->pointer();
double * sigma_aap = SIGMA_AA->pointer();
double * sigma_bbp = SIGMA_BB->pointer();
double exc = 0.0;
for (int p = 0; p < phi_points_; p++) {
double rhoa = rho_ap[p];
double rhob = rho_bp[p];
double rho = rhoa + rhob;
double sigmaaa = sigma_aap[p];
double sigmabb = sigma_bbp[p];
double rhoa_43 = 0.0;
double rhob_43 = 0.0;
double Xa = 0.0;
double Xb = 0.0;
double Xa_2 = 0.0;
double Xb_2 = 0.0;
if ( rho > tol ) {
if ( rhoa < tol ){
rhoa = 0.0;
sigmabb = std::max(0.0,sigma_bbp[p]);
rhob_43 = pow( rhob, 4.0/3.0);
Xb = sqrt(sigma_bbp[p]) / rhob_43;
Xb_2 = Xb * Xb;
}else if ( rhob < tol ){
rhob = 0.0;
sigmaaa = std::max(0.0,sigma_aap[p]);
rhoa_43 = pow( rhoa, 4.0/3.0);
Xa = sqrt(sigma_aap[p]) / rhoa_43;
Xa_2 = Xa * Xa;
}else{
sigmaaa = std::max(0.0,sigma_aap[p]);
sigmabb = std::max(0.0,sigma_bbp[p]);
rhoa_43 = pow( rhoa, 4.0/3.0);
rhob_43 = pow( rhob, 4.0/3.0);
Xa = sqrt(sigma_aap[p]) / rhoa_43;
Xb = sqrt(sigma_bbp[p]) / rhob_43;
Xa_2 = Xa * Xa;
Xb_2 = Xb * Xb;
}
exc += -rhoa_43 * ( c + (beta * Xa_2) / (1.0 + 6.0 * beta * Xa * asinh(Xa)) ) * grid_w_->pointer()[p];
exc += -rhob_43 * ( c + (beta * Xb_2) / (1.0 + 6.0 * beta * Xb * asinh(Xb)) ) * grid_w_->pointer()[p];
}else{
exc += 0.0;
}
}
return exc;
}
double MCPDFTSolver::EX_PBE(std::shared_ptr<Vector> RHO_A, std::shared_ptr<Vector> RHO_B,
std::shared_ptr<Vector> SIGMA_AA, std::shared_ptr<Vector> SIGMA_BB) {
const double delta = 0.06672455060314922;
const double MU = (1.0/3.0) * delta * M_PI * M_PI;
const double KAPPA = 0.804;
double * rho_ap = RHO_A->pointer();
double * rho_bp = RHO_B->pointer();
double * sigma_aap = SIGMA_AA->pointer();
double * sigma_bbp = SIGMA_BB->pointer();
double exc = 0.0;
for (int p = 0; p < phi_points_; p++) {
double rhoa = rho_ap[p];
double rhob = rho_bp[p];
double rhoa_43 = pow( rhoa, 4.0/3.0);
double rhob_43 = pow( rhob, 4.0/3.0);
double Xa = sqrt(sigma_aap[p]) / rhoa_43;
double Xb = sqrt(sigma_bbp[p]) / rhob_43;
double Sa = (Xa * pow(6.0, 2.0/3.0)) / (12.0 * pow(M_PI, 2.0/3.0));
double Sb = (Xb * pow(6.0, 2.0/3.0)) / (12.0 * pow(M_PI, 2.0/3.0));
double Fsa = 1.0 + KAPPA - KAPPA * pow( (1.0 + (MU * pow(Sa,2.0)) / KAPPA ), -1.0 );
double Fsb = 1.0 + KAPPA - KAPPA * pow( (1.0 + (MU * pow(Sb,2.0)) / KAPPA ), -1.0 );
auto E = [](double rhos, double Fss) -> double{
double temp = -0.75 * pow(3.0, 1.0/3.0) * pow(M_PI, 2.0/3.0) * pow(rhos,4.0/3.0) * Fss / M_PI;
return temp;
};
double EX_GGAa = 0.5 * E(2.0*rhoa,Fsa);
double EX_GGAb = 0.5 * E(2.0*rhob,Fsb);
exc += ( EX_GGAa + EX_GGAb ) * grid_w_->pointer()[p];
}
return exc;
}
double MCPDFTSolver::EX_PBE_I(std::shared_ptr<Vector> RHO_A, std::shared_ptr<Vector> RHO_B,
std::shared_ptr<Vector> SIGMA_AA, std::shared_ptr<Vector> SIGMA_BB) {
const double delta = 0.06672455060314922;
const double MU = (1.0/3.0) * delta * M_PI * M_PI;
const double KAPPA = ( (options_.get_str("MCPDFT_FUNCTIONAL") == "REVPBE") ? 1.245 : 0.804 );
double tol = 1.0e-20;
double * rho_ap = RHO_A->pointer();
double * rho_bp = RHO_B->pointer();
double * sigma_aap = SIGMA_AA->pointer();
double * sigma_bbp = SIGMA_BB->pointer();
auto kF = [](double RHO) -> double {
double dum = pow(3.0 * M_PI * M_PI * RHO ,1.0/3.0);
return dum;
};
auto eX = [=](double RHO) -> double {
double temp = -(3.0 * kF(RHO)) / (4.0 * M_PI);
return temp;
};
auto FX = [=](double SIGMA) -> double {
double temp = 1.0 + KAPPA - KAPPA * pow( (1.0 + (MU * pow(SIGMA,2.0)) / KAPPA ), -1.0 );
return temp;
};
auto S = [=](double RHO, double SIGMA) -> double {
double temp = sqrt(SIGMA) / (2.0 * kF(RHO) * RHO);
return temp;
};
double exc = 0.0;
for (int p = 0; p < phi_points_; p++) {
double rhoa = rho_ap[p];
double rhob = rho_bp[p];
double rho = rhoa + rhob;
double sigmaaa = sigma_aap[p];
double sigmabb = sigma_bbp[p];
double sigma = 0.0;
if ( rho > tol ) {
if ( rhoa < tol ){
rho = rhob;
sigmabb = std::max(0.0,sigmabb);
sigma = sigmabb;
double zk = eX(2.0 * rho) * FX(S(2.0 * rho, 4.0 * sigma));
exc += rho * zk * grid_w_->pointer()[p];
}else if ( rhob < tol ){
rho = rhoa;
sigmaaa = std::max(0.0,sigmaaa);
sigma = sigmaaa;
double zk = eX(2.0 * rho) * FX(S(2.0 * rho, 4.0 * sigma));
exc += rho * zk * grid_w_->pointer()[p];
}else {
double zka = rhoa * eX(2.0 * rhoa) * FX(S(2.0 * rhoa, 4.0 * sigmaaa));
double zkb = rhob * eX(2.0 * rhob) * FX(S(2.0 * rhob, 4.0 * sigmabb));
double zk = zka + zkb;
exc += zk * grid_w_->pointer()[p];
}
}else{
//double zk = 0.0;
exc += 0.0;
}
}
return exc;
}
//
//
// const double alpha = (2.0/3.0); // Slater value
// const double Cx = (9.0/8.0) * alpha * pow(3.0/M_PI,1.0/3.0);
// const double MU = 0.2195149727645171;
// const double KAPPA = 0.804;
//
// double * rho_ap = rho_a_->pointer();
// double * rho_bp = rho_b_->pointer();
//
// double * rho_a_xp = rho_a_x_->pointer();
// double * rho_b_xp = rho_b_x_->pointer();
//
// double * rho_a_yp = rho_a_y_->pointer();
// double * rho_b_yp = rho_b_y_->pointer();
//
// double * rho_a_zp = rho_a_z_->pointer();
// double * rho_b_zp = rho_b_z_->pointer();
//
// double * zeta_p = zeta_->pointer();
// double * sigma_aap = sigma_aa_->pointer();
// double * sigma_abp = sigma_ab_->pointer();
// double * sigma_bbp = sigma_bb_->pointer();
//
// double exc = 0.0;
// for (int p = 0; p < phi_points_; p++) {
//
// double sig = sigma_aap[p] + 2.0 * sigma_abp[p] + sigma_bbp[p];
// // double sig = sigma_aap[p];
// double rhoa = rho_ap[p];
// double rhob = rho_bp[p];
// double rho = rhoa + rhob;
// // local fermi wave vector
// double kf = pow( ( 3.0 * pow(M_PI,2.0) * rho), 1.0/3.0);
// // double kfa = pow( ( 3.0 * pow(M_PI,2.0) * rhoa ) , 1.0/3.0);
// // double kfb = pow( ( 3.0 * pow(M_PI,2.0) * rhob ) , 1.0/3.0);
//
// // double EXa = -(3.0 * kfa) / (4.0 * M_PI);
// // double EXb = -(3.0 * kfb) / (4.0 * M_PI);
// // double absDelRho = sqrt( ( pow( (rho_a_xp[p] + rho_b_xp[p]) ,2.0) + pow( (rho_a_yp[p] + rho_b_yp[p]) ,2.0) + pow( (rho_a_zp[p] + rho_b_zp[p]) ,2.0) ) );
// double absDelRho = sqrt( sig );
//
// // double Sa = sqrt(4.0 * grada_2) / (2.0 * kfa * rhoa);
// // double Sb = sqrt(4.0 * gradb_2) / (2.0 * kfb * rhob);
//
// double s = absDelRho / ( 2.0 * kf * rho);
// // double sa = absDelRho / ( 2.0 * kfa *2.0* rho_ap[p] );
// // double sb = absDelRho / ( 2.0 * kfb *2.0* rho_bp[p] );
//
// double Fs = pow( (1.0 + 1.296 * pow(s,2.0) + 14.0 * pow(s,4.0) + 0.2 * pow(s,6.0) ) , 1.0/15.0 ) ;
// // double Fsa = pow( (1.0 + 1.296 * pow(sa,2.0) + 14.0 * pow(sa,4.0) + 0.2 * pow(sa,6.0) ) , 1.0/15.0 );
// // double Fsb = pow( (1.0 + 1.296 * pow(sb,2.0) + 14.0 * pow(sb,4.0) + 0.2 * pow(sb,6.0) ) , 1.0/15.0 );
// // double Fs = 1.0 + KAPPA - KAPPA * pow( (1.0 + (MU * pow(s,2.0)) / KAPPA ), -1.0 );
// // double Fsa = 1.0 + KAPPA - KAPPA * pow( (1.0 + (MU * pow(Sa,2.0)) / KAPPA ), -1.0 );
// // double Fsb = 1.0 + KAPPA - KAPPA * pow( (1.0 + (MU * pow(Sb,2.0)) / KAPPA ), -1.0 );
//
// // double EX_GGAa = rhoa * EXa * Fsa;
// // double EX_GGAb = rhob * EXb * Fsb;
// exc += -Cx * pow( rho, 4.0/3.0) * Fs * grid_w_->pointer()[p];
// // exc += -pow(2.0,1.0/3.0) * Cx * ( pow( rho_ap[p], 4.0/3.0) + pow( rho_bp[p], 4.0/3.0) ) * Fs * grid_w_->pointer()[p];
// // exc += -pow(2.0,1.0/3.0) * Cx * ( pow( rho_ap[p], 4.0/3.0) * Fsa + pow( rho_bp[p], 4.0/3.0) * Fsb ) * grid_w_->pointer()[p];
// // exc += -0.5 * ( -3.0 *2.0* kfa/(4*M_PI) * rho_ap[p] * Fsa - 3.0 *2.0* kfb / (4*M_PI) * rho_bp[p] * Fsb ) * grid_w_->pointer()[p];
// // exc += 0.5 * ( EX_GGAa + EX_GGAb ) * grid_w_->pointer()[p];
// }
// return exc;
// }
//+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
//+++++++++++++++++ Correlation Functionals +++++++++++++++++
// B88_OP, VWN3, LYP, PW92, PBE
//+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
// Note: This correlation functional depends on B86MGC exchange functional
// with empirical atomic parameters t and u defined below.
// From T. Tsuneda, J. Chem. Phys. 110, 10664 (1999).
double MCPDFTSolver::EC_B88_OP(std::shared_ptr<Vector> RHO_A, std::shared_ptr<Vector> RHO_B, std::shared_ptr<Vector> SIGMA_AA, std::shared_ptr<Vector> SIGMA_BB){
const double beta = 0.0042;
double * rho_ap = RHO_A->pointer();
double * rho_bp = RHO_B->pointer();
double * sigma_aap = SIGMA_AA->pointer();
double * sigma_bbp = SIGMA_BB->pointer();
double exc = 0.0;
for (int p = 0; p < phi_points_; p++) {
double rhoa = rho_ap[p];
double rhob = rho_bp[p];
double rhoa_43 = pow( rhoa, 4.0/3.0);
double rhob_43 = pow( rhob, 4.0/3.0);
double rhoa_13 = pow( rhoa, 1.0/3.0);
double rhob_13 = pow( rhob, 1.0/3.0);
double sigmaaa = sigma_aap[p];
double sigmabb = sigma_bbp[p];
double Xa = sqrt(sigmaaa) / rhoa_43;
double Xb = sqrt(sigmabb) / rhob_43;
double Xa_2 = Xa * Xa;
double Xb_2 = Xb * Xb;
auto Ks = [=](double Xs, double Xs_2) -> double{
double temp3 = 3.0 * pow(3.0/(4.0 * M_PI),1.0/3.0) + 2.0 * (beta * Xs_2) / (1.0 + 6.0 * beta * Xs * asinh(Xs));
return temp3;
};
double Ka = Ks(Xa, Xa_2);
double Kb = Ks(Xb, Xb_2);
double BETA_ab = 2.3670 * (rhoa_13 * rhob_13 * Ka * Kb) / (rhoa_13 * Ka + rhob_13 * Kb);
exc += -rhoa * rhob * ( (1.5214 * BETA_ab + 0.5764) / ( pow(BETA_ab,4.0) + 1.1284 * pow(BETA_ab,3.0) + 0.3183 * pow(BETA_ab,2.0) ) ) * grid_w_->pointer()[p];
}
return exc;
}
// double MCPDFTSolver::EC_B88(){
//
// const double Cx = 0.73855876638202240586;
// const double c = pow(2.0,1.0/3.0) * Cx;
// const double t = 0.63;
// const double u = 0.96;
// const double beta = 0.00375;
// // const double beta = 0.0042;
// const double lambda = 0.007;
//
// double * rho_ap = rho_a_->pointer();
// double * rho_bp = rho_b_->pointer();
// double * sigma_aap = sigma_aa_->pointer();
// double * sigma_bbp = sigma_bb_->pointer();
// double * tau_ap = tau_a_->pointer();
// double * tau_bp = tau_b_->pointer();
//
// double exc = 0.0;
// for (int p = 0; p < phi_points_; p++) {
//
// double rhoa = rho_ap[p];
// double rhob = rho_bp[p];
// double rho = rhoa + rhob;
// double rhoa_43 = pow( rhoa, 4.0/3.0);
// double rhob_43 = pow( rhob, 4.0/3.0);
// double rhoa_13 = pow( rhoa, 1.0/3.0);
// double rhob_13 = pow( rhob, 1.0/3.0);
// double taua = tau_ap[p];
// double taub = tau_bp[p];
// double sigmaaa = sigma_aap[p];
// double sigmabb = sigma_bbp[p];
// double Xa = sqrt(sigmaaa) / rhoa_43;
// double Xb = sqrt(sigmabb) / rhob_43;
// double Xa_2 = Xa * Xa;
// double Xb_2 = Xb * Xb;
//
// auto xyfunc = [=](double rhos_13, double Xs_2) -> double{
//
// double temp = 0.5 * pow(c * rhos_13 + (beta * Xs_2 * rhos_13) / pow(1.0 + lambda * Xs_2 ,4.0/5.0 ) ,-1.0);
// return temp;
// };
//
// auto rfunc = [=](double rhos, double rhos_43, double Xs_2) -> double{
//
// double temp1 = 0.5 * rhos * pow( c * rhos_43 + (beta * Xs_2 * rhos_43)/ pow(1.0 + lambda * Xs_2 ,4.0/5.0) ,-1.0);
// return temp1;
// };
//
// auto dfunc = [=](double rhos, double taus, double sigmass) -> double {
//
// double temp2 = taus - 0.25 * sigmass / rhos;
// return temp2;
// };
//
// double x = xyfunc(rhoa_13,Xa_2);
// double y = xyfunc(rhob_13,Xb_2);
// double q = t * (x + y);
// double q_2 = q * q;
// double r_a = rfunc(rhoa, rhoa_43, Xa_2);
// double r_b = rfunc(rhob, rhob_43, Xb_2);
// double z_a = 2.0 * u * r_a;
// double z_b = 2.0 * u * r_b;
// double d_a = dfunc(rhoa, taua, sigmaaa);
// double d_b = dfunc(rhob, taub, sigmabb);
//
// double f = -0.8 * rhoa * rhob * q_2 * (1.0 - log(1.0 + q) / q);
// double g_a = -0.01 * rhoa * d_a * pow(z_a,4.0) * ( 1.0 - 2.0 * log(1.0 + 0.5 * z_a) / z_a );
// double g_b = -0.01 * rhob * d_b * pow(z_b,4.0) * ( 1.0 - 2.0 * log(1.0 + 0.5 * z_b) / z_b );
//
// auto Ks = [=](double Xs, double Xs_2) -> double{
//
// double temp3 = 3.0 * pow(3.0/(4.0 * M_PI),1.0/3.0) + 2.0 * (beta * Xs_2) / (1.0 + 6.0 * beta * Xs * asinh(Xs));
// return temp3;
// };
//
// double Ka = Ks(Xa, Xa_2);
// double Kb = Ks(Xb, Xb_2);
//
// exc += ( f + g_a + g_b ) * grid_w_->pointer()[p];
// double BETA_ab = 2.3670 * (rhoa_13 * rhob_13 * Ka * Kb) / (rhoa_13 * Ka + rhob_13 * Kb);
// // exc += -rhoa * rhob * ( (1.5214 * BETA_ab + 0.5764) / ( pow(BETA_ab,4.0) + 1.1284 * pow(BETA_ab,3.0) + 0.3183 * pow(BETA_ab,2.0) ) ) * grid_w_->pointer()[p];
//
// }
// return exc;
// }
double MCPDFTSolver::EC_VWN3_RPA(std::shared_ptr<Vector> RHO_A, std::shared_ptr<Vector> RHO_B, std::shared_ptr<Vector> ZETA, std::shared_ptr<Vector> RS){
const double k1 = 0.0310907;
const double k2 = 0.01554535;
const double l1 = -0.409286;
const double l2 = -0.743294;
const double m1 = 13.0720;
const double m2 = 20.1231;
const double n1 = 42.7198;
const double n2 = 101.578;
double * rho_ap = RHO_A->pointer();
double * rho_bp = RHO_B->pointer();
double * zeta_p = ZETA->pointer();
double * rs_p = RS->pointer();
double exc = 0.0;
for (int p = 0; p < phi_points_; p++) {
double rhoa = rho_ap[p];
double rhob = rho_bp[p];
double rho = rhoa + rhob;
double zeta = zeta_p[p];
double rs = rs_p[p];
double x = sqrt(rs);
double y = (9.0/8.0) * pow(1.0 + zeta, 4.0/3.0) + (9.0/8.0) * pow(1.0 - zeta, 4.0/3.0) - (9.0/4.0);
double z = (4.0 * y) / (9.0 * pow(2.0,1.0/3.0) - 9.0);
auto X = [](double i, double c, double d) -> double{
double temp = pow(i,2.0) + c * i + d;
return temp;
};
auto Q = [](double c, double d) -> double{
double temp1 = sqrt( 4 * d - pow(c,2.0) );
return temp1;
};
auto q = [=](double A, double p, double c, double d) -> double{
double dum1 = A * ( log( pow(x,2.0) / X(x,c,d) ) + 2.0 * c * atan( Q(c,d)/(2.0*x + c) ) * pow(Q(c,d),-1.0)
- c * p * ( log( pow(x-p,2.0) / X(x,c,d) ) + 2.0 * (c + 2.0 * p) * atan( Q(c,d)/(2.0*x + c) ) * pow(Q(c,d),-1.0) ) * pow(X(p,c,d),-1.0) );
return dum1;
};
double Lambda = q(k1, l1, m1, n1);
double lambda = q(k2, l2, m2, n2);
double e = Lambda + z * (lambda - Lambda);
exc += e * rho * grid_w_->pointer()[p];
}
return exc;
}
// Conv. (III)
double MCPDFTSolver::EC_VWN3_RPA_III(std::shared_ptr<Vector> RHO_A, std::shared_ptr<Vector> RHO_B){ //, std::shared_ptr<Vector> ZETTA){
double tol = 1.0e-20;
const double ecp1 = 0.03109070000;
const double ecp2 = -0.409286;
const double ecp3 = 13.0720;
const double ecp4 = 42.7198;
const double ecf1 = 0.01554535000;
const double ecf2 = -0.743294;
const double ecf3 = 20.1231;
const double ecf4 = 101.578;
const double d2Fz = 1.7099209341613656173;
double * rho_ap = RHO_A->pointer();
double * rho_bp = RHO_B->pointer();
auto x = [](double RHO) -> double {
double rs = pow( 3.0 / ( 4.0 * M_PI * RHO ) , 1.0/3.0 );
double dum = sqrt(rs);
return dum;
};
auto Fz = [](double ZETA) -> double {
double dum = (pow((1.0 + ZETA) ,4.0/3.0) + pow((1.0 - ZETA) ,4.0/3.0) - 2.0) / (2.0 * pow(2.0,1.0/3.0) - 2.0);
return dum;
};
auto X = [](double i, double c, double d) -> double{
double temp = pow(i,2.0) + c * i + d;
return temp;
};
auto Q = [](double c, double d) -> double{
double temp1 = sqrt( 4 * d - pow(c,2.0) );
return temp1;
};
auto q = [=](double RHO, double A, double p, double c, double d) -> double{
double dum1 = A * ( log( pow(x(RHO),2.0) / X(x(RHO),c,d) ) + 2.0 * c * atan( Q(c,d)/(2.0*x(RHO) + c) ) * pow(Q(c,d),-1.0)
- c * p * ( log( pow(x(RHO)-p,2.0) / X(x(RHO),c,d) ) + 2.0 * (c + 2.0 * p) * atan( Q(c,d)/(2.0*x(RHO) + c) )
* pow(Q(c,d),-1.0) ) * pow(X(p,c,d),-1.0) );
return dum1;
};
auto EcP = [=](double RHO) -> double {
double dumm = q(RHO,ecp1,ecp2,ecp3,ecp4);
return dumm;
};
auto EcF = [=](double RHO) -> double {
double dum = q(RHO,ecf1,ecf2,ecf3,ecf4);
return dum;
};
double exc = 0.0;
for (int p = 0; p < phi_points_; p++) {
double rhoa = rho_ap[p];
double rhob = rho_bp[p];
double rho = rhoa + rhob;
double zeta = 0.0;
if ( rho > tol ) {
if ( rhoa < tol ){
rho = rhob;
zeta = 1.0;
}else if ( rhob < tol ){
rho = rhoa;
zeta = 1.0;
}else {/* if (!(rhoa < tol) && !(rhob < tol) ) */
zeta = (rhoa - rhob) / rho;
}
double zk = EcP(rho) + Fz(zeta) * (EcF(rho) - EcP(rho));
exc += rho * zk * grid_w_->pointer()[p];
}else{
double zk = 0.0;
exc += 0.0;
}
}
return exc;
}
// Conv. (I)
double MCPDFTSolver::EC_LYP_I(std::shared_ptr<Vector> RHO_A, std::shared_ptr<Vector> RHO_B,
std::shared_ptr<Vector> SIGMA_AA, std::shared_ptr<Vector> SIGMA_AB, std::shared_ptr<Vector> SIGMA_BB) {
double tol = 1.0e-20;
const double A = 0.04918;
const double B = 0.132;
const double C = 0.2533;
const double Dd = 0.349;
double * rho_ap = RHO_A->pointer();
double * rho_bp = RHO_B->pointer();
double * sigma_aap = SIGMA_AA->pointer();
double * sigma_abp = SIGMA_AB->pointer();
double * sigma_bbp = SIGMA_BB->pointer();
double exc = 0.0;
for (int p = 0; p < phi_points_; p++) {
double rhoa = rho_ap[p];
double rhob = rho_bp[p];
double rho = rhoa + rhob;
double sigmaaa = sigma_aap[p];
double sigmaab = sigma_abp[p];
double sigmabb = sigma_bbp[p];
double sigma = sigmaaa + sigmabb + 2.0 * sigmaab;
if ( rho > tol ) {
if ( rhoa < tol ){
double zk = 0.0;
}else if ( rhob < tol ){
double zk = 0.0;
}else{
sigmabb = std::max(0.0,sigmabb);
sigmaaa = std::max(0.0,sigmaaa);
double rho_2 = pow(rho,2.0);
double rho_a_2 = pow(rhoa,2.0);
double rho_b_2 = pow(rhob,2.0);
double rho_m13 = pow(rho,-1.0/3.0);
double rho_a_83 = pow(rhoa,8.0/3.0);
double rho_b_83 = pow(rhob,8.0/3.0);
double f = pow( 1.0 + Dd * rho_m13, -1.0);
double omega = exp(-C * rho_m13) * f * pow(rho,-11.0/3.0);
double delta = (C + Dd * f) * rho_m13;
double zk = -4.0 * A * rhoa * rhob * f / rho - A * B * omega * (rhoa * rhob * (36.462398978764777098 * (rho_a_83 + rho_b_83)
+ (2.6111111111111111111 - 0.38888888888888888889 * delta) * sigma - (2.5 - 0.055555555555555555556 * delta) * (sigmaaa + sigmabb)
- 0.11111111111111111111 * (delta - 11.0) * ((rhoa * sigmaaa + rhob * sigmabb) / rho ) ) - 0.66666666666666666667 * rho_2 * sigma
+ (0.66666666666666666667 * rho_2 - rho_a_2) * sigmabb + (0.66666666666666666667 * rho_2 - rho_b_2) * sigmaaa);
exc += zk * grid_w_->pointer()[p];
}
}else{
double zk = 0.0;
exc += zk * grid_w_->pointer()[p];
}
}
return exc;
}
double MCPDFTSolver::EC_PBE(std::shared_ptr<Vector> RHO_A, std::shared_ptr<Vector> RHO_B,
std::shared_ptr<Vector> SIGMA_AA, std::shared_ptr<Vector> SIGMA_AB, std::shared_ptr<Vector> SIGMA_BB){
double tol = 1.0e-20;
double * rho_ap = RHO_A->pointer();
double * rho_bp = RHO_B->pointer();
double * sigma_aap = SIGMA_AA->pointer();
double * sigma_abp = SIGMA_AB->pointer();
double * sigma_bbp = SIGMA_BB->pointer();
double exc = 0.0;
for (int p = 0; p < phi_points_; p++) {
double rhoa = rho_ap[p];
double rhob = rho_bp[p];
double sigmaaa = sigma_aap[p];
double sigmaab = sigma_abp[p];
double sigmabb = sigma_bbp[p];
double rho = rhoa + rhob;
double sigma = sigmaaa + sigmabb + 2.0 * sigmaab;
if ( rho > tol ) {
if ( rhoa < tol ){
rho = rhob;
sigmabb = std::max(0.0,sigmabb);
sigma = sigmabb;
double t2 = 1.0 / rhob;
double t3 = pow(t2 ,1.0/3.0);
double t6 = pow(t2 ,1.0/6.0);
double t9 = sqrt(t2);
double t11 = t3 * t3;
double t17 = log(1.0 + 0.3216395899738507e2 / (0.1112037486309468e2 * t6 + 0.3844746237447211e1 * t3 + 0.1644733775567609e1
* t9 + 0.2405871291288192 * t11));
double t18 = (1.0 + 0.1274696188700087 * t3) * t17;
double t20 = pow(rhob ,2.0);
double t21 = pow(rhob ,1.0/3.0);
double t23 = 1.0 / t21 / t20;
double t26 = exp(0.2000000587336264e1 * t18);
double t27 = t26 - 1.0;
double t31 = 0.2162211495206379 / t27 * sigmabb * t23;
double t33 = pow(t27 ,2.0);
double t35 = pow(sigmabb ,2.0);
double t37 = pow(t20 ,2.0);
double t38 = pow(t21 ,2.0);
double t49 = log(1.0 + 0.2162211495206379 * sigmabb * t23 * (1.0 + t31) /(1.0 + t31 + 0.4675158550002605e-1 / t33 * t35 / t38 / t37));
double zk = rhob * (-0.310907e-1 * t18 + 0.1554534543482745e-1 * t49);
exc += rho * zk * grid_w_->pointer()[p];
}else if ( rhob < tol ){
rho = rhoa;
sigmaaa = std::max(0.0,sigmaaa);
sigma = sigmaaa;
double t2 = 1.0 / rhoa;
double t3 = pow(t2 ,1.0/3.0);
double t6 = pow(t2 ,1.0/6.0);
double t9 = sqrt(t2);
double t11 = t3 * t3;
double t17 = log(1.0 + 0.3216395899738507e2 / (0.1112037486309468e2 * t6 + 0.3844746237447211e1 * t3 + 0.1644733775567609e1
* t9 + 0.2405871291288192 * t11));
double t18 = (1.0 + 0.1274696188700087 * t3) * t17;
double t20 = pow(rhoa ,2.0);
double t21 = pow(rhoa ,1.0/3.0);
double t23 = 1.0 / t21 / t20;
double t26 = exp(0.2000000587336264e1 * t18);
double t27 = t26 - 1.0;
double t31 = 0.2162211495206379 / t27 * sigmaaa * t23;
double t33 = pow(t27 ,2.0);
double t35 = pow(sigmaaa ,2.0);
double t37 = pow(t20 ,2.0);
double t38 = pow(t21 ,2.0);
double t49 = log(1.0 + 0.2162211495206379 * sigmaaa * t23 * (1.0 + t31) /(1.0 + t31 + 0.4675158550002605e-1 / t33 * t35 / t38 / t37));
double zk = rhoa * (-0.310907e-1 * t18 + 0.1554534543482745e-1 * t49);
exc += rho * zk * grid_w_->pointer()[p];
}else{
double t4 = 1/rho;
double t5 = pow( t4, 1.0/3.0);
double t7 = 1.0 + 0.1325688999052018 * t5;
double t8 = pow(t4, 1.0/6.0);
double t11 = sqrt(t4);
double t13 = pow(t5 ,2.0);
double t15 = 0.598255043577108e1 * t8 + 0.2225569421150687e1 * t5 + 0.8004286349993634 * t11 + 0.1897004325747559 * t13;
double t18 = 1.0 + 0.1608197949869254e2 / t15;
double t19 = log(t18);
double t21 = 0.621814e-1 * t7 * t19;
double t23 = 1.0 + 0.6901399211255825e-1 * t5;
double t28 = 0.8157414703487641e1 * t8 + 0.2247591863577616e1 * t5 + 0.4300972471276643 * t11 + 0.1911512595127338 * t13;
double t31 = 1.0 + 0.2960874997779344e2 / t28;
double t32 = log(t31);
double t33 = t23 * t32;
double t35 = rhoa - 1.0 * rhob;
double t36 = t35 * t4; // zeta
double t37 = 1.0 + t36;
double t38 = pow(t37 ,1.0/3.0);
double t41 = 1.0 - t36;
double t42 = pow(t41 ,1.0/3.0);
double t44 = t38 * t37 + t42 * t41 - 2.0;
double t45 = pow(t35 ,2.0);
double t46 = pow(t45 ,2.0);
double t47 = pow(rho ,2.0);
double t48 = pow(t47 ,2.0);
double t49 = 1.0 / t48;
double t50 = t46 * t49;
double t52 = 1.0 - t50;
double t55 = 0.37995525e-1 * t33 * t44 * t52;
double t57 = 1.0 + 0.1274696188700087 * t5;
double t62 = 0.1112037486309468e2 * t8 + 0.3844746237447211e1 * t5 + 0.1644733775567609e1 * t11 + 0.2405871291288192 * t13;
double t65 = 1.0 + 0.3216395899738507e2 / t62;
double t66 = log(t65);
double t69 = -0.310907e-1 * t57 * t66 + t21;
double t70 = t69 * t44;
double t72 = 0.1923661050931536e1 * t70 * t50;
double t73 = pow(t38 ,2.0);
double t75 = pow(t42 ,2.0);
double t77 = 0.5 * t73 + 0.5 * t75;
double t78 = pow(t77 ,2.0);
double t79 = t78 * t77;
double t80 = 1.0 / t78;
double t81 = sigma * t80;
double t82 = pow(rho ,1.0/3.0);
double t84 = 1.0 / t82 / t47;
double t85 = -t21 + t55 + t72;
double t86 = 1.0 / t79;
double t89 = exp(-0.3216396844291482e2 * t85 * t86);
double t90 = t89 - 1.0;
double t91 = 1.0 / t90;
double t92 = t91 * sigma;
double t93 = t80 * t84;
double t95 = 0.1362107888567592 * t92 * t93;
double t96 = 1.0 + t95;
double t98 = pow(t90 ,2.0);
double t99 = 1.0 / t98;
double t100 = pow(sigma ,2.0);
double t101 = t99 * t100;
double t102 = pow(t78 ,2.0);
double t103 = 1.0 / t102;
double t104 = pow(t82 ,2.0);
double t106 = 1.0 / t104 / t48;
double t107 = t103 * t106;
double t110 = 1.0 + t95 + 0.1855337900098064e-1 * t101 * t107;
double t111 = 1.0 / t110;
double t115 = 1.0 + 0.1362107888567592 * t81 * t84 * t96 * t111;
double t116 = log(t115);
double t118 = 0.310906908696549e-1 * t79 * t116;
double zk = -t21 + t55 + t72 + t118;
exc += rho * zk * grid_w_->pointer()[p];
}
}else{
double zk = 0.0;
exc += rho * zk * grid_w_->pointer()[p];
}
}
return exc;
}
// From J.P. Perdew, and Y. Wang, Phys. Rev. B 45, 13244 (1992)
double MCPDFTSolver::EC_PW92_I(std::shared_ptr<Vector> RHO_A, std::shared_ptr<Vector> RHO_B){
const double pa = 1.0;
const double Aa = 0.0168869;