oep.h

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/*
 * oep.h
 *
 *  Created on: Nov 6, 2019
 *      Author: fbischoff
 */
 
#ifndef SRC_APPS_CHEM_OEP_H_
#define SRC_APPS_CHEM_OEP_H_
 
 
 
#include<madness/chem/nemo.h>
#include<madness/chem/SCFOperators.h>
#include<madness/chem/projector.h>
#include<madness/chem/MolecularOrbitals.h>
 
namespace madness {
 
/// Class to compute terms of the potential
struct divide_add_interpolate {
 
    double thresh_high=1.e-5;
    double thresh_low=1.e-7;
    double eps_regularize=1.e-8;
    double log_high, log_low;
    bool square_denominator=false;
 
    divide_add_interpolate(double hi, double lo, double eps_regularize) :
        thresh_high(hi), thresh_low(lo), eps_regularize(eps_regularize),
            log_high(log10(thresh_high)), log_low(log10(thresh_low)) {}
 
    std::size_t get_result_size() const {
        return 1;
    }
 
    /// @param[in]  t   vector containing: oep, refdensity, longrange
    /// @return     (num1/denom1 - num2/denom2) * mask + (1-mask)*longrange
    std::vector<Tensor<double> > operator()(const Key<3> & key,
            const std::vector<Tensor<double> >& t) const {
 
        const Tensor<double>& refdens=t[0];
        const Tensor<double>& num1=t[1];
        const Tensor<double>& denom1=t[2];
        const Tensor<double>& num2=t[3];
        const Tensor<double>& denom2=t[4];
        const Tensor<double>& longrange=t[5];
 
        // set 1 if above munge interval (dens > thresh_high) or 0 if below munge interval (dens < thresh_low)
        // interpolate inbetween with logarithmic form of (r - lo)/(hi - lo) because this yields near linear behavior
        Tensor<double> U = copy(refdens); // copy refdens in oder to have the same dimension in the tensor
        U.fill(1.0); // start with a fuction that is 1.0 everywhere
        ITERATOR(
            U,
            double r = refdens(IND);
            double result=num1(IND)/(denom1(IND)+eps_regularize)
                    - num2(IND)/(denom2(IND)+eps_regularize);
            if (square_denominator) result=num1(IND)/(denom1(IND)*denom1(IND)+eps_regularize)
                    - num2(IND)/(denom2(IND)*denom2(IND)+eps_regularize);
            if (r > thresh_high) {
                U(IND) = result;
            } else if (r < thresh_low) {
                U(IND) = longrange(IND);
            } else {
                // mask = 1 if refdens>hi, 0 if refdens < lo, interpolate in between
                double mask=(log10(refdens(IND)) - log_low)/(log_high - log_low);
                U(IND)=mask*result + (1.0-mask)*longrange(IND);
            }
        );
        return std::vector<Tensor<double> > (1,U);
    }
};
 
/// TODO:
///  - change algorithm to handle localized orbitals
///  - do not recompute the OEP potentials that don't depend on the orbital energies/Fock matrix elements
///  - do not recompute the HF contributions to the OEP potentials
///  - think about the long-range part of the Slater potential (or medium-range)
class OEP_Parameters : public QCCalculationParametersBase {
public:
    OEP_Parameters() {
        initialize<std::vector<std::string> >("model",{"dcep"},"comment on this: oaep ocep dcep mrks");
        initialize<unsigned int>("maxiter",150,"maximum number of iterations in OEP algorithm");
        initialize<bool>("restart",false,"restart from previous OEP calculation");
        initialize<bool>("no_compute",false,"read from previous OEP calculation, no computation");
        initialize<double>("levelshift",0.0,"shift occupied orbital energies in the BSH operator");
//      initialize<double>("conv_threshold",1.e-5,"comment on this");
        initialize<double>("density_threshold_high",1.e-6,"comment on this");
        initialize<double>("density_threshold_low",1.e-8,"comment on this");
        initialize<double>("density_threshold_inv",1.e-9,"comment on this");
        initialize<std::vector<double> >("kain_param",{1.0e-8, 3.0},"comment on this");
 
//      std::vector<bool> oep_model = {false, false, false, false};
        initialize<unsigned int>("saving_amount",0,"choose level 0, 1, 2 or 3 for saving functions");
        initialize<unsigned int>("save_iter_orbs",0,"if > 0 save all orbitals every ... iterations (needs a lot of storage!");
        initialize<unsigned int>("save_iter_density",0,"if > 0 save KS density every ... iterations");
        initialize<unsigned int>("save_iter_IKS",0,"if > 0 save IKS every ... iterations");
        initialize<unsigned int>("save_iter_kin_tot_KS",0,"if > 0 save kin_tot_KS every ... iterations");
        initialize<unsigned int>("save_iter_kin_P_KS",0,"if > 0 save kin_P_KS every ... iterations");
        initialize<unsigned int>("save_iter_corrections",0,"if > 0 save OEP correction(s) every ... iterations");
        initialize<unsigned int>("save_iter_effective_potential",0,"if > 0 save effective potential every ... iterations");
    }
 
    OEP_Parameters(World& world, const commandlineparser& parser) : OEP_Parameters() {
        read_input_and_commandline_options(world, parser, "oep");
    }
 
    OEP_Parameters(const OEP_Parameters& other) = default;
 
 
    void set_derived_values(const Nemo::NemoCalculationParameters& nparam) {
        set_derived_value("density_threshold_high",10.0*nparam.econv());
        set_derived_value("density_threshold_low",0.01*get<double>("density_threshold_high"));
        if (dens_thresh_hi()<dens_thresh_lo()) {
            MADNESS_EXCEPTION("confused thresholds for the long-range transition",1);
        }
        set_derived_value("density_threshold_inv",0.1*get<double>("density_threshold_low"));
 
        if (levelshift()>0.0) {
            print("levelshift > 0.0 in oep parameters\n\n");
            MADNESS_EXCEPTION("levelshift > 0.0 in oep parameters",1);
        }
    }
 
    // convenience functions
    std::vector<std::string> model() const {return get<std::vector<std::string> >("model");}
 
    bool restart() const {return get<bool>("restart");}
    bool no_compute() const {return get<bool>("no_compute");}
    unsigned int maxiter() const {return get<unsigned int>("maxiter");}
    double levelshift() const {return get<double>("levelshift");}
//  double conv_thresh() const {return get<double>("conv_threshold");}
    double dens_thresh_hi() const {return get<double>("density_threshold_high");}
    double dens_thresh_lo() const {return get<double>("density_threshold_low");}
    double dens_thresh_inv() const{return get<double>("density_threshold_inv");}
    unsigned int saving_amount() const {return get<unsigned int>("saving_amount");}
    unsigned int save_iter_corrections        () const {return get<unsigned int>("save_iter_corrections");}
 
    std::vector<double> kain_param() const {return get<std::vector<double> >("kain_param");}
 
};
 
 
class OEP : public Nemo {
 
private:
 
    /// parameters for this OEP calculation
    OEP_Parameters oep_param;
 
    /// the wave function reference that determines the local potential
    std::shared_ptr<Nemo> reference;
 
    /// the final local potential
    real_function_3d Vfinal;
 
public:
 
    OEP(World& world, const commandlineparser& parser)
            : Nemo(world, parser),
              oep_param(world, parser) {
 
        // add tight convergence criteria
        std::vector<std::string> convergence_crit=param.get<std::vector<std::string> >("convergence_criteria");
        if (std::find(convergence_crit.begin(),convergence_crit.end(),"each_energy")==convergence_crit.end()) {
            convergence_crit.push_back("each_energy");
        }
        param.set_derived_value("convergence_criteria",convergence_crit);
        calc->param.set_derived_value("convergence_criteria",convergence_crit);
 
        // set reference
        set_reference(std::make_shared<Nemo>(world,parser));
        reference->param.set_derived_value("convergence_criteria",convergence_crit);
        reference->get_calc()->param.set_derived_value("convergence_criteria",convergence_crit);
 
        oep_param.set_derived_values(param);
    }
 
    std::string name() const {return "oep";}
 
    static void help() {
        print_header2("help page for oep");
        print("The oep code computes local exchange potentials based on a Hartree-Fock calculation from nemo");
        print("oep --print_parameters\n");
        print("You can perform a simple calculation by running\n");
        print("oep --geometry=h2o.xyz\n");
        print("provided you have an xyz file in your directory.");
 
    }
 
    static void print_parameters() {
        OEP_Parameters param;
        print("default parameters for the oep program are");
        param.print("oep", "end");
        print("\n\nthe molecular geometry must be specified in a separate block:");
        Molecule::print_parameters();
    }
 
    void set_reference(const std::shared_ptr<Nemo> reference1) {
        reference=reference1;
    }
 
    std::shared_ptr<Nemo> get_reference() const {
        return reference;
    }
 
    void print_parameters(std::vector<std::string> what) const {
        for (auto w : what) {
            if (w=="oep") oep_param.print("oep");
            else if (w=="reference") reference->param.print("dft");
            else if (w=="oep_calc") param.print("oep_calc");
            else {MADNESS_EXCEPTION(std::string("unknown parameter set to print "+w).c_str(),1);}
        }
    }
 
    real_function_3d get_final_potential() const {
        return Vfinal;
    }
 
    virtual double value() {
        return value(calc->molecule.get_all_coords());
    }
 
    /// update the json file with calculation input and output
    void output_calc_info_schema(const double& energy) const;
 
 
    virtual double value(const Tensor<double>& x) {
        reference->value();
        set_protocol(param.econv());
        calc->copy_data(world,*(reference->get_calc()));
        double energy=0.0;
        Tensor<double> fock;
        bool load_mos=(oep_param.restart() or oep_param.no_compute());
        if (load_mos) load_restartdata(fock);
 
        if (not oep_param.no_compute())  energy=solve(reference->get_calc()->get_amo());
        output_calc_info_schema(energy);
        return energy;
    };
 
    /// Iterative energy calculation for approximate OEP with EXACT EXCHANGE functional
    /// for other functionals, slater potential must be modified
    /// HF orbitals and eigenvalues are used as the guess here
    /// note that KS_nemo is a reference and changes oep->get_calc()->amo orbitals
    /// same for orbital energies (eigenvalues) KS_eigvals which is oep->get_calc()->aeps
    /// converged if norm, total energy difference and orbital energy differences (if not OAEP) are converged
    double solve(const vecfuncT& HF_nemo);
 
    void analyze();
 
    double iterate(const std::string model, const vecfuncT& HF_nemo, const tensorT& HF_eigvals,
            vecfuncT& KS_nemo, tensorT& KS_Fock, real_function_3d& Voep,
            const real_function_3d Vs) const;
 
    virtual std::shared_ptr<Fock<double,3>> make_fock_operator() const;
 
//    MolecularOrbitals<double,3> to_MO() const {
//      std::vector<std::string> str_irreps;
//      vecfuncT aaa=symmetry_projector(calc->amo,R_square,str_irreps);
//      return MolecularOrbitals<double,3>(aaa,this->get_calc()->aeps,str_irreps,this->get_calc()->aocc,this->get_calc()->aset);
//    }
 
    void save_restartdata(const Tensor<double>& fock) const;
 
    void load_restartdata(Tensor<double>& fock);
 
    std::tuple<Tensor<double>, vecfuncT> recompute_HF(const vecfuncT& HF_nemo) const;
 
    /// The following function tests all essential parts of the OEP program qualitatively and some also quantitatively
    bool selftest();
 
    bool need_ocep_correction(const std::string& model) const {
        return (model=="ocep") or (model=="dcep") or (model=="mrks");
    }
 
    bool need_dcep_correction(const std::string& model) const {
        return (model=="dcep");
    }
 
    bool need_mrks_correction(const std::string& model) const {
        return (model=="mrks");
    }
 
    /// print orbital energies in reverse order with optional shift
    void print_orbens(const tensorT orbens) const {
        for (long i = orbens.size() - 1; i >= 0; i--) {
            printf(" e%2.2lu = %12.8f Eh\n", i, orbens(i));
        }
    }
 
    double compute_and_print_final_energies(const std::string model, const real_function_3d& Voep,
            const vecfuncT& KS_nemo, const tensorT& KS_Fock,
            const vecfuncT& HF_nemo, const tensorT& HF_Fock) const;
 
    /// compute density from orbitals with ragularization (Bischoff, 2014_1, equation (19))
    real_function_3d compute_density(const vecfuncT& nemo) const {
        real_function_3d density = 2.0*R_square*dot(world, nemo, nemo); // 2 because closed shell
        return density;
    }
 
    /// compute Delta rho as an indicator for the result's quality
    double compute_delta_rho(const real_function_3d rho_HF, const real_function_3d rho_KS) const {
        // from Ospadov_2017, equation (26)
        real_function_3d rho_diff = abs(rho_KS - rho_HF);
        double Drho = rho_diff.trace();
        return Drho;
    }
 
     /// compute Slater potential (Kohut, 2014, equation (15))
    real_function_3d compute_slater_potential(const vecfuncT& nemo) const {
 
        Exchange<double,3> K(world,reference->get_calc()->param.lo());
         K.set_bra_and_ket(R_square * nemo, nemo);
        const vecfuncT Knemo = K(nemo);
        // 2.0*R_square in numerator and density (rho) cancel out upon division
        real_function_3d numerator = -1.0*dot(world, nemo, Knemo);
        real_function_3d rho = dot(world, nemo, nemo);
 
        // long-range asymptotic behavior for Slater potential is \int 1/|r-r'| * |phi_HOMO|^2 dr'
        // in order to compute this lra, use Coulomb potential with only HOMO density (= |phi_HOMO|^2)
//        Coulomb J(world);
//        J.reset_poisson_operator_ptr(param.lo(),param.econv());
//        real_function_3d lra = -1.0*J.compute_potential(R_square*square(nemo[homo_ind]));
        Coulomb<double,3> J(world,this);
        real_function_3d lra=-1.0/(param.nalpha()+param.nbeta())*J.compute_potential(this);
//        print("compute long-range part of the Slater potential from the full molecular density");
        if (oep_param.saving_amount() >= 3) save(lra, "lra_slater");
 
        real_function_3d zero=real_factory_3d(world);
        real_function_3d one=real_factory_3d(world).functor([](const coord_3d& r){return 1.0;});
        std::vector<real_function_3d> args={R_square*rho,numerator,rho,zero,one,lra};
        refine_to_common_level(world,args);
 
        divide_add_interpolate op(oep_param.dens_thresh_hi(), oep_param.dens_thresh_lo(),
                oep_param.dens_thresh_inv());
        real_function_3d VSlater=multi_to_multi_op_values(op,args)[0];
        return VSlater;
    }
 
 
    /// compute the total kinetic energy density of equation (6) from Kohut
 
    /// return without the NCF and factor 2 for closed shell !
    real_function_3d compute_total_kinetic_density(const vecfuncT& nemo) const {
 
        // compute tau = 1/2 * sum |\nabla phi_i|^2
        // = 1/2 * sum {(\nabla R)^2 * nemo_i^2 + 2 * R * nemo_i * (\nabla R) * (\nabla nemo_i)) + R^2 * (\nabla nemo_i)^2}
        // = 1/2 * R^2 * sum {U1dot * nemo_i^2 + 2 * nemo_i * U1 * (\nabla nemo_i)) + (\nabla nemo_i)^2}
 
        // get \nabla R and (\nabla R)^2 via and U1 = -1/R \nabla R and U1dot = (1/R \nabla R)^2
        const vecfuncT U1 = this->ncf->U1vec();
        NuclearCorrelationFactor::U1_dot_U1_functor u1_dot_u1(ncf.get());
        const real_function_3d U1dot = real_factory_3d(world).functor(u1_dot_u1).truncate_on_project();
 
        real_function_3d u1u1term=U1dot*dot(world,nemo,nemo);
        real_function_3d u1dnterm=real_factory_3d(world).compressed();
        real_function_3d dndnterm=real_factory_3d(world).compressed();
 
        for (int idim=0; idim<3; ++idim) {
            real_derivative_3d D(world,idim);
            if(param.dft_deriv() == "bspline") D.set_bspline1();
            vecfuncT nemo_copy=copy(world,nemo);
            refine(world,nemo_copy);
            std::vector<real_function_3d> dnemo=apply(world,D,nemo_copy);
            u1dnterm+=U1[idim]*dot(world,nemo_copy,dnemo);
            refine(world,dnemo);
            dndnterm+=dot(world,dnemo,dnemo);
        }
 
        real_function_3d tau=0.5*(u1u1term - 2.0*u1dnterm + dndnterm);
 
        return tau;
 
    }
 
    /// compute the Pauli kinetic energy density divided by the density tau_P/rho with equation (16) from Ospadov, 2017
    real_function_3d compute_Pauli_kinetic_density(const vecfuncT& nemo) const {
 
        // compute the numerator tau_P
 
        // get \nabla nemo
        std::vector<vecfuncT> grad_nemo(nemo.size());
        for (size_t i = 0; i < nemo.size(); i++) {
            vecfuncT nemo_copy=copy(world,nemo);
            refine(world,nemo_copy);
 
            if(param.dft_deriv() == "bspline") grad_nemo[i] = grad_bspline_one(nemo_copy[i]);  // gradient using b-spline
            else grad_nemo[i] = grad(nemo_copy[i]);  // default gradient using abgv
        }
 
        vecfuncT grad_nemo_term;
        for (size_t i = 0; i < nemo.size(); i++) {
            for (size_t j = i + 1; j < nemo.size(); j++) {
                vecfuncT tmp = nemo[i]*grad_nemo[j] - nemo[j]*grad_nemo[i];
                grad_nemo_term.push_back(dot(world, tmp, tmp));
            }
        }
        real_function_3d numerator = sum(world, grad_nemo_term); // numerator = tau_P * 2 * rho / R^4
 
        return numerator;
 
    }
 
    /// compute correction of the given model
 
    /// shift of the diagonal elements of the fock matrix results in a global shift
    /// of the potential
    /// \f[
    /// \frac{\bar \epsilon}{\rho} = \frac{1}{\rho}\sum_{ij}\phi_i(F_ij+\delta_ij s)\phi_j
    ///        = \frac{1}{\rho} ( \sum_{ij}\phi_i F_ij \phi_j + s\sum_i\phi_i\phi_i )
    ///        = s + \frac{1}{\rho} \sum_{ij}\phi_i F_ij \phi_j
    /// \f]
    real_function_3d compute_ocep_correction(const real_function_3d& ocep_numerator_HF,
            const vecfuncT& nemoHF, const vecfuncT& nemoKS,
            const tensorT& fockHF, const tensorT& fockKS) const {
 
        if (fockKS.normf()<1.e-10) {
            real_function_3d zero=real_factory_3d(world);
            return zero;
        }
 
        // compute HOMO energies and the long-range asymptotics
        auto [eval, evec] = syev(fockKS);
        double homoKS = -eval.max();
 
        auto [eval1, evec1] = syev(fockHF);
        double homoHF = -eval1.max();
 
        double longrange=homoHF-homoKS;
 
        tensorT fock1=copy(fockKS);
        for (int i=0; i<fock1.dim(0); ++i) fock1(i,i)-=longrange;
 
        // 2.0*R_square in numerator and density (rho) cancel out upon division
//      real_function_3d numeratorHF=-1.0*compute_energy_weighted_density_local(nemoHF,fockHF);
//      real_function_3d numeratorHF=-1.0*compute_energy_weighted_density(nemoHF,eigvalsHF);
        real_function_3d numeratorKS=-1.0*compute_energy_weighted_density_local(nemoKS,fock1);
//      real_function_3d numeratorKS=-1.0*compute_energy_weighted_density(nemoKS,eval);
 
        // 2.0*R_square in numerator and density (rho) cancel out upon division
        real_function_3d densityKS = dot(world, nemoKS, nemoKS);
        real_function_3d densityHF = dot(world, nemoHF, nemoHF);
 
        real_function_3d lra_func=real_factory_3d(world);
        std::vector<real_function_3d> args={densityKS,ocep_numerator_HF,densityHF,
                numeratorKS,densityKS,lra_func};
        refine_to_common_level(world,args);
 
        divide_add_interpolate op(oep_param.dens_thresh_hi(), oep_param.dens_thresh_lo(),
                oep_param.dens_thresh_inv());
        real_function_3d correction=multi_to_multi_op_values(op,args)[0];
 
 
        return correction;
 
    }
 
    /// return without the NCF and factor 2 for closed shell !
 
    /// follow Eq. (4) of Ryabinkin2014
    real_function_3d compute_energy_weighted_density_local(const vecfuncT& nemo,
            const tensorT& fock) const {
        return dot(world,nemo,transform(world,nemo,fock));
    }
 
    /// compute correction of the given model
    real_function_3d compute_dcep_correction(const real_function_3d& dcep_numerator_HF,
            const vecfuncT& nemoHF, const vecfuncT& nemoKS) const {
 
        // 2.0*R_square in numerator and density (rho) cancel out upon division
//      real_function_3d numeratorHF=compute_total_kinetic_density(nemoHF);
        real_function_3d numeratorKS=compute_total_kinetic_density(nemoKS);
 
        // 2.0*R_square in numerator and density (rho) cancel out upon division
        real_function_3d densityKS = dot(world, nemoKS, nemoKS);
        real_function_3d densityHF = dot(world, nemoHF, nemoHF);
 
        real_function_3d lra_func=real_factory_3d(world).functor([](const coord_3d& r)
                {return 0.0;});
 
        std::vector<real_function_3d> args={densityKS,dcep_numerator_HF,densityHF,
                numeratorKS,densityKS,lra_func};
        refine_to_common_level(world,args);
 
        divide_add_interpolate op(oep_param.dens_thresh_hi(), oep_param.dens_thresh_lo(),
                oep_param.dens_thresh_inv());
        real_function_3d correction=multi_to_multi_op_values(op,args)[0];
 
        return correction;
    }
 
    /// compute correction of the given model
    real_function_3d compute_mrks_correction(const real_function_3d& mrks_numerator_HF,
            const vecfuncT& nemoHF, const vecfuncT& nemoKS) const {
 
        // 2.0*R_square in numerator and density (rho) cancel out upon division
//      real_function_3d numeratorHF=compute_Pauli_kinetic_density(nemoHF);
        real_function_3d numeratorKS=compute_Pauli_kinetic_density(nemoKS);
 
        // 2.0*R_square in numerator and density (rho) cancel out upon division
        real_function_3d densityKS = dot(world, nemoKS, nemoKS);
        real_function_3d densityHF = dot(world, nemoHF, nemoHF);
 
        // longrange correction is zero
        real_function_3d lra_func=real_factory_3d(world).functor([](const coord_3d& r) {return 0.0;});
 
        real_function_3d denssq=square(densityKS);
        std::vector<real_function_3d> args={densityKS,mrks_numerator_HF,densityHF,
                numeratorKS,densityKS,lra_func};
        refine_to_common_level(world,args);
 
        divide_add_interpolate op(oep_param.dens_thresh_hi(), oep_param.dens_thresh_lo(),
                oep_param.dens_thresh_inv());
        op.square_denominator=true;
        real_function_3d correction=0.5*multi_to_multi_op_values(op,args)[0];
 
        return correction;
    }
 
    /// compute all potentials from given nemos except kinetic energy
    void compute_nemo_potentials(const vecfuncT& nemo, vecfuncT& Jnemo, vecfuncT& Unemo) const {
 
        // compute Coulomb part
        compute_coulomb_potential(nemo, Jnemo);
 
        // compute nuclear potential part
        Nuclear<double,3> Unuc(world, this->ncf);
        Unemo = Unuc(nemo);
 
    }
 
    /// compute Coulomb potential
    void compute_coulomb_potential(const vecfuncT& nemo, vecfuncT& Jnemo) const {
 
        Coulomb<double,3> J(world, this);
        real_function_3d density=this->compute_density(nemo);
        J.potential()=J.compute_potential(density);
        Jnemo = J(nemo);
        truncate(world, Jnemo);
 
    }
 
    /// compute exchange potential (needed for Econv)
    void compute_exchange_potential(const vecfuncT& nemo, vecfuncT& Knemo) const {
 
        Exchange<double,3> K(world,this->get_calc()->param.lo());
        K.set_bra_and_ket(R_square * nemo, nemo);
        Knemo = K(nemo);
        truncate(world, Knemo);
 
    }
 
    /// compute Evir using Levy-Perdew virial relation (Kohut_2014, (43) or Ospadov_2017, (25))
    double compute_exchange_energy_vir(const vecfuncT& nemo, const real_function_3d Vx) const {
 
        // make vector of functions r = (x, y, z)
        auto monomial_x = [] (const coord_3d& r) {return r[0];};
        auto monomial_y = [] (const coord_3d& r) {return r[1];};
        auto monomial_z = [] (const coord_3d& r) {return r[2];};
        vecfuncT r(3);
        r[0]=real_factory_3d(world).functor(monomial_x);
        r[1]=real_factory_3d(world).functor(monomial_y);
        r[2]=real_factory_3d(world).functor(monomial_z);
 
        real_function_3d rhonemo = 2.0*dot(world, nemo, nemo); // 2 because closed shell
 
        real_function_3d rhoterm = 3*rhonemo + dot(world, r, -2.0*ncf->U1vec()*rhonemo + grad(rhonemo,true));
        double Ex = inner(Vx, R_square*rhoterm);
        return Ex;
 
    }
 
    /// compute exchange energy using the expectation value of the exchange operator
    double compute_exchange_energy_conv(const vecfuncT phi, const vecfuncT Kphi) const {
 
        double Ex = -1.0*inner(world, phi, Kphi).sum();
        return Ex;
 
    }
 
    /// compute all energy contributions except exchange and sum up for total energy
    /// the exchange energy must be computed priorly with the compute_exchange_energy_... methods
    std::vector<double> compute_energy(const vecfuncT& nemo, const double E_X) const {
 
        // compute kinetic energy
        double E_kin = compute_kinetic_energy(nemo);
 
        real_function_3d density=compute_density(nemo);
 
        // compute external potential (nuclear attraction)
        real_function_3d Vext = calc->potentialmanager->vnuclear();
        Coulomb<double,3> J(world,this);
        real_function_3d Jpotential=J.compute_potential(density);
 
        // compute remaining energies: nuclear attraction, Coulomb, nuclear repulsion
        // computed as expectation values (see Szabo, Ostlund (3.81))
        const double E_ext = inner(Vext,density); // 2 because closed shell
        const double E_J = 0.5*inner(density,Jpotential);
        const double E_nuc = calc->molecule.nuclear_repulsion_energy();
        double energy = E_kin + E_ext + E_J + E_X + E_nuc;
 
        if (world.rank() == 0) {
            printf("\n                      kinetic energy %15.8f Eh\n", E_kin);
            printf("  electron-nuclear attraction energy %15.8f Eh\n", E_ext);
            printf("                      Coulomb energy %15.8f Eh\n", E_J);
            printf("                     exchange energy %15.8f Eh\n", E_X);
            printf("    nuclear-nuclear repulsion energy %15.8f Eh\n", E_nuc);
            printf("                        total energy %15.8f Eh\n\n", energy);
        }
        return {energy,E_kin,E_ext,E_J,E_X};
 
    }
 
    /// compute diagonal elements of Fock matrix
    Tensor<double> compute_fock_diagonal_elements(const Tensor<double>& KS_eigvals,
            const vecfuncT& phi, const vecfuncT& Kphi, const real_function_3d& Vx) const {
        return KS_eigvals - inner(world, phi, Kphi) - inner(world, phi, Vx*phi);
    }
 
    /// cumpute E^(0) = \sum_i \epsilon_i^KS
    double compute_E_zeroth(const tensorT eigvals) const {
        double E_0 = 0.0;
        for (long i = 0; i < eigvals.size(); i++) {
            E_0 += eigvals(i);
        }
        E_0 *= 2.0; // closed shell: every orbital energy must be counted twice
        return E_0;
    }
 
    /// compute E^(1) = 1/2*\sum_ij <ij||ij> - \sum_i <i|J + Vx|i> = \sum_i <i|- 0.5*J - 0.5*K - Vx|i>
    double compute_E_first(const vecfuncT phi, const vecfuncT Jphi, const vecfuncT Kphi, const real_function_3d Vx) const {
 
        //compute expectation values:
        const double E_J = inner(world, phi, Jphi).sum();
        const double E_K = inner(world, phi, Kphi).sum();
        const double E_Vx = inner(world, phi, Vx*phi).sum();
        printf("  E_J   =  %15.8f Eh\n", E_J);
        printf("  E_K   =  %15.8f Eh\n", E_K);
        printf("  E_Vx  =  %15.8f Eh\n", E_Vx);
        const double E_nuc = calc->molecule.nuclear_repulsion_energy();
 
        double E_1 = -1.0*(E_J + E_K + 2.0*E_Vx) + E_nuc;
        return E_1;
 
    }
 
};
 
 
 
} /* namespace madness */
 
#endif /* SRC_APPS_CHEM_OEP_H_ */