gfit.h

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/*
 This file is part of MADNESS.
 
 Copyright (C) 2007,2010 Oak Ridge National Laboratory
 
 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., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
 
 For more information please contact:
 
 Robert J. Harrison
 Oak Ridge National Laboratory
 One Bethel Valley Road
 P.O. Box 2008, MS-6367
 
 email: harrisonrj@ornl.gov
 tel:   865-241-3937
 fax:   865-572-0680
 
 
 $Id: key.h 2907 2012-06-14 10:15:05Z 3ru6ruWu $
 */
 
#ifndef MADNESS_MRA_GFIT_H__INCLUDED
#define MADNESS_MRA_GFIT_H__INCLUDED
 
/// \file gfit.h
/// \brief fit isotropic functions to a set of Gaussians with controlled precision
 
//#include <iostream>
 
#include <cmath>
#include "../constants.h"
#include "../tensor/basetensor.h"
#include "../tensor/slice.h"
#include "../tensor/tensor.h"
#include "../tensor/tensor_lapack.h"
#include "../world/madness_exception.h"
#include "../world/print.h"
#include <madness/mra/operatorinfo.h>
 
 
namespace madness {
 
template<typename T, std::size_t NDIM>
class GFit {
 
public:
 
    /// default ctor does nothing
    GFit() = default;
 
    GFit(OperatorInfo info) {
        double mu = info.mu;
        double lo = info.lo;
        double hi = info.hi;
        MADNESS_CHECK_THROW(hi>0,"hi must be positive in gfit: U need to set it manually in operator.h");
        double eps = info.thresh;
        bool debug=info.debug;
        OpType type = info.type;
 
 
        if (type==OT_G12) {*this=CoulombFit(lo,hi,eps,debug);
        } else if (type==OT_SLATER) {*this=SlaterFit(mu,lo,hi,eps,debug);
        } else if (type==OT_GAUSS)  {*this=GaussFit(mu,lo,hi,eps,debug);
        } else if (type==OT_F12)    {*this=F12Fit(mu,lo,hi,eps,debug);
        } else if (type==OT_FG12)   {*this=FGFit(mu,lo,hi,eps,debug);
        } else if (type==OT_F212)   {*this=F12sqFit(mu,lo,hi,eps,debug);
        } else if (type==OT_F2G12)  {*this=F2GFit(mu,lo,hi,eps,debug);
        } else if (type==OT_BSH)    {*this=BSHFit(mu,lo,hi,eps,debug);
        } else {
            print("Operator type not implemented: ",type);
            MADNESS_EXCEPTION("Operator type not implemented: ",1);
        }
 
    }
 
    /// copy constructor
    GFit(const GFit& other) = default;
 
    /// assignment operator
    GFit& operator=(const GFit& other) {
        coeffs_ = other.coeffs_;
        exponents_ = other.exponents_;
        return *this;
    }
 
    /// return a fit for the Coulomb function
 
    ///  f(r) = 1/r
    static GFit CoulombFit(double lo, double hi, double eps, bool prnt=false) {
        GFit fit=BSHFit(0.0,lo,hi,eps/(4.0*constants::pi),prnt);
        fit.coeffs_.scale(4.0*constants::pi);   // BSHFit scales by 1/(4 pi), undo here
        return fit;
    }
 
    /// return a fit for the bound-state Helmholtz function
 
    /// the BSH function is defined by
    ///  f(r) = 1/ (4 pi) exp(-\mu r)/r
    /// @param[in]  mu  the exponent of the BSH
    /// @param[in]  lo  the smallest length scale that needs to be precisely represented
    /// @param[in]  hi  the largest length scale that needs to be precisely represented
    /// @param[in]  eps the precision threshold
    /// @parma[in]  prnt    print level
    static GFit BSHFit(double mu, double lo, double hi, double eps, bool prnt=false) {
        GFit fit;
        bool fix_interval=false;
        if (NDIM==3) bsh_fit(mu,lo,hi,eps,fit.coeffs_,fit.exponents_,prnt,fix_interval);
        else bsh_fit_ndim(NDIM,mu,lo,hi,eps,fit.coeffs_,fit.exponents_,prnt);
 
        if (prnt) {
            print("bsh fit");
            auto exact = [&mu](const double r) -> double { return 1.0/(4.0 * constants::pi) * exp(-mu * r)/r; };
            fit.print_accuracy(exact, lo, hi);
        }
        return fit;
    }
 
    /// return a fit for the Slater function
 
    /// the Slater function is defined by
    ///  f(r) = exp(-\gamma r)
    /// @param[in]  gamma   the exponent of the Slater function
    /// @param[in]  lo  the smallest length scale that needs to be precisely represented
    /// @param[in]  hi  the largest length scale that needs to be precisely represented
    /// @param[in]  eps the precision threshold
    /// @parma[in]  prnt    print level
    static GFit SlaterFit(double gamma, double lo, double hi, double eps, bool prnt=false) {
        GFit fit;
        slater_fit(gamma,lo,hi,eps,fit.coeffs_,fit.exponents_,false);
        if (prnt) {
            print("Slater fit");
            auto exact = [&gamma](const double r) -> double { return exp(-gamma * r); };
            fit.print_accuracy(exact, lo, hi);
        }
        return fit;
    }
 
    /// return a (trivial) fit for a single Gauss function
 
    /// the Gauss function is defined by
    ///  f(r) = exp(-\gamma r^2)
    /// @param[in]  gamma   the exponent of the Gauss function
    /// @param[in]  lo  the smallest length scale that needs to be precisely represented
    /// @param[in]  hi  the largest length scale that needs to be precisely represented
    /// @param[in]  eps the precision threshold
    /// @parma[in]  prnt    print level
    static GFit GaussFit(double gamma, double lo, double hi, double eps, bool prnt=false) {
        GFit fit;
        fit.coeffs_=Tensor<T>(1);
        fit.exponents_=Tensor<double>(1);
        fit.coeffs_=1.0;
        fit.exponents_=gamma;
        if (prnt) {
            print("Gauss fit");
            auto exact = [&gamma](const double r) -> double { return exp(-gamma * r*r); };
            fit.print_accuracy(exact, lo, hi);
        }
        return fit;
    }
 
    /// return a fit for the F12 correlation factor
 
    /// the Slater function is defined by
    ///  f(r) = 1/(2 gamma) * (1 - exp(-\gamma r))
    /// @param[in]  gamma   the exponent of the Slater function
    /// @param[in]  lo  the smallest length scale that needs to be precisely represented
    /// @param[in]  hi  the largest length scale that needs to be precisely represented
    /// @param[in]  eps the precision threshold
    /// @parma[in]  prnt    print level
    static GFit F12Fit(double gamma, double lo, double hi, double eps, bool prnt=false) {
        GFit fit;
        f12_fit(gamma,lo*0.1,hi,eps*0.01,fit.coeffs_,fit.exponents_,false);
        fit.coeffs_*=(0.5/gamma);
        if (prnt) {
            print("f12 fit");
            auto exact=[&gamma](const double r) -> double {return 0.5/gamma*(1.0-exp(-gamma*r));};
            fit.print_accuracy(exact,lo,hi);
        }
        return fit;
    }
 
    /// return a fit for the F12^2 correlation factor
 
    /// the Slater function square is defined by
    ///  f(r) = [ 1/(2 gamma) * (1 - exp(-\gamma r)) ] ^2
    /// @param[in]  gamma   the exponent of the Slater function
    /// @param[in]  lo  the smallest length scale that needs to be precisely represented
    /// @param[in]  hi  the largest length scale that needs to be precisely represented
    /// @param[in]  eps the precision threshold
    /// @parma[in]  prnt    print level
    static GFit F12sqFit(double gamma, double lo, double hi, double eps, bool prnt=false) {
        GFit fit;
        f12sq_fit(gamma,lo*0.1,hi,eps*0.01,fit.coeffs_,fit.exponents_,false);
        fit.coeffs_*=(0.25/(gamma*gamma));
        if (prnt) {
            print("f12sq fit");
            auto exact=[&gamma](const double r) -> double {return std::pow(0.5/gamma*(1.0-exp(-gamma*r)),2.0);};
            fit.print_accuracy(exact,lo,hi);
        }
        return fit;
    }
 
 
    /// return a fit for the FG function
 
    /// fg = 1/(2 mu) * (1 - exp(-gamma r12))  / r12
    ///    = 1/(2 mu) *( 1/r12 - exp(-gamma r12)/r12)
    ///    = 1/(2 mu) * (coulomb - bsh)
    /// @param[in]  gamma   the exponent of the Slater function
    /// @param[in]  lo  the smallest length scale that needs to be precisely represented
    /// @param[in]  hi  the largest length scale that needs to be precisely represented
    /// @param[in]  eps the precision threshold
    /// @parma[in]  prnt    print level
    static GFit FGFit(double gamma, double lo, double hi, double eps, bool prnt=false) {
        GFit bshfit,coulombfit;
        eps*=0.1;
        lo*=0.1;
//        bool restrict_interval=false;
        bool fix_interval=true;
        bsh_fit(gamma,lo,hi,eps,bshfit.coeffs_,bshfit.exponents_,false,fix_interval);
        bsh_fit(0.0,lo,hi,eps,coulombfit.coeffs_,coulombfit.exponents_,false,fix_interval);
        // check the exponents are identical
        auto diffexponents=(coulombfit.exponents() - bshfit.exponents());
        MADNESS_CHECK(diffexponents.normf()/coulombfit.exponents().size()<1.e-12);
        auto diffcoefficients=(coulombfit.coeffs() - bshfit.coeffs());
        GFit fgfit;
        fgfit.exponents_=bshfit.exponents_;
        fgfit.coeffs_=4.0*constants::pi*0.5/gamma*diffcoefficients;
        GFit<double,3>::prune_small_coefficients(eps,lo,hi,fgfit.coeffs_,fgfit.exponents_);
 
        if (prnt) {
            print("fg fit");
            auto exact=[&gamma](const double r) -> double {return 0.5/gamma*(1.0-exp(-gamma*r))/r;};
            fgfit.print_accuracy(exact,lo,hi);
        }
        return fgfit;
    }
 
    /// return a fit for the F2G function
 
    /// f2g = (1/(2 mu) * (1 - exp(-gamma r12)))^2  / r12
    ///    = 1/(4 mu^2) * [ 1/r12 - 2 exp(-gamma r12)/r12) + exp(-2 gamma r12)/r12 ]
    /// @param[in]  gamma   the exponent of the Slater function
    /// @param[in]  lo  the smallest length scale that needs to be precisely represented
    /// @param[in]  hi  the largest length scale that needs to be precisely represented
    /// @param[in]  eps the precision threshold
    /// @parma[in]  prnt    print level
    static GFit F2GFit(double gamma, double lo, double hi, double eps, bool prnt=false) {
        GFit bshfit,coulombfit,bsh2fit;
        eps*=0.1;
        lo*=0.1;
//        bool restrict_interval=false;
        bool fix_interval=true;
        bsh_fit(gamma,lo,hi,eps,bshfit.coeffs_,bshfit.exponents_,false,fix_interval);
        bsh_fit(2.0*gamma,lo,hi,eps,bsh2fit.coeffs_,bsh2fit.exponents_,false,fix_interval);
        bsh_fit(0.0,lo,hi,eps,coulombfit.coeffs_,coulombfit.exponents_,false,fix_interval);
 
        // check the exponents are identical
        auto diffexponents=(coulombfit.exponents() - bshfit.exponents());
        MADNESS_CHECK(diffexponents.normf()/coulombfit.exponents().size()<1.e-12);
        auto diffexponents1=(coulombfit.exponents() - bsh2fit.exponents());
        MADNESS_CHECK(diffexponents1.normf()/coulombfit.exponents().size()<1.e-12);
 
        auto coefficients=(coulombfit.coeffs() - 2.0* bshfit.coeffs() + bsh2fit.coeffs());
        GFit f2gfit;
        f2gfit.exponents_=bshfit.exponents_;
        // additional factor 4 pi due to implementation of bsh_fit
        double fourpi=4.0*constants::pi;
        double fourmu2=4.0*gamma*gamma;
        f2gfit.coeffs_=fourpi/fourmu2*coefficients;
        GFit<double,3>::prune_small_coefficients(eps,lo,hi,f2gfit.coeffs_,f2gfit.exponents_);
 
        if (prnt) {
            print("fg fit");
            auto exact=[&gamma](const double r) -> double {
                return 0.25/(gamma*gamma)*(1.0-2.0*exp(-gamma*r)+exp(-2.0*gamma*r))/r;
            };
            f2gfit.print_accuracy(exact,lo,hi);
        }
        return f2gfit;
    }
 
 
    /// return a fit for a general isotropic function
 
    /// note that the error is controlled over a uniform grid, the boundaries
    /// will be poorly represented in general. Following Beylkin 2005
    static GFit GeneralFit() {
        MADNESS_EXCEPTION("General GFit still to be implemented",1);
        return GFit();
    }
 
    /// return the coefficients of the fit
    Tensor<T> coeffs() const {return coeffs_;}
 
    /// return the exponents of the fit
    Tensor<T> exponents() const {return exponents_;}
 
    void static prune_small_coefficients(const double eps, const double lo, const double hi,
                                  Tensor<double>& coeff, Tensor<double>& expnt) {
        double mid = lo + (hi-lo)*0.5;
        long npt=coeff.size();
        long i;
        for (i=npt-1; i>0; --i) {
            double cnew = coeff[i]*exp(-(expnt[i]-expnt[i-1])*mid*mid);
            double errlo = coeff[i]*exp(-expnt[i]*lo*lo) -
                           cnew*exp(-expnt[i-1]*lo*lo);
            double errhi = coeff[i]*exp(-expnt[i]*hi*hi) -
                           cnew*exp(-expnt[i-1]*hi*hi);
            if (std::max(std::abs(errlo),std::abs(errhi)) > 0.03*eps) break;
            npt--;
            coeff[i-1] = coeff[i-1] + cnew;
        }
        coeff = coeff(Slice(0,npt-1));
        expnt = expnt(Slice(0,npt-1));
    }
 
    void truncate_periodic_expansion(Tensor<double>& c, Tensor<double>& e,
            double L, bool discardG0) const {
        double tcut = 0.25/L/L;
 
        if (discardG0) {
            // Relies on expnts being in decreasing order
            for (int i=0; i<e.dim(0); ++i) {
                if (e(i) < tcut) {
                    c = c(Slice(0,i));
                    e = e(Slice(0,i));
                    break;
                }
            }
        } else {
//          // Relies on expnts being in decreasing order
//          int icut = -1;
//          for (int i=0; i<e.dim(0); ++i) {
//              if (e(i) < tcut) {
//                  icut = i;
//                  break;
//              }
//          }
//          if (icut > 0) {
//              for (int i=icut+1; i<e.dim(0); ++i) {
//                  c(icut) += c(i);
//              }
//              c = c(Slice(0,icut));
//              e = e(Slice(0,icut));
//          }
        }
    }
 
private:
 
    /// ctor taking an isotropic function
 
    /// the function will be represented with a uniform error on a uniform grid
    /// @param[in]  f   a 1d-function that implements T operator()
    template<typename funcT>
    GFit(const funcT& f) {
 
    }
 
    /// print coefficients and exponents, and values and errors
 
    /// @param[in]  op  the exact function, e.g. 1/r, exp(-mu r), etc
    /// @param[in]  lo  lower bound for the range r
    /// @param[in]  hi  higher bound for the range r
    template<typename opT>
    void print_accuracy(opT op, const double lo, const double hi) const {
 
        std::cout << "weights and roots" << std::endl;
        for (int i=0; i<coeffs_.size(); ++i)
            std::cout << i << " " << coeffs_[i] << " " << exponents_[i] << std::endl;
 
        long npt = 300;
        std::cout << "       x         value   abserr   relerr" << std::endl;
        std::cout << "  ------------  ------- -------- -------- " << std::endl;
        double step = exp(log(hi/lo)/(npt+1));
        for (int i=0; i<=npt; ++i) {
            double r = lo*(pow(step,i+0.5));
//            double exact = exp(-mu*r)/r/4.0/constants::pi;
            double exact = op(r);
            double test = 0.0;
            for (int j=0; j<coeffs_.dim(0); ++j)
                test += coeffs_[j]*exp(-r*r*exponents_[j]);
            double err = 0.0;
            if (exact) err = (exact-test)/exact;
            printf("  %.6e %8.1e %8.1e %8.1e\n",r, exact, exact-test, err);
        }
    }
 
    /// the coefficients of the expansion f(x) = \sum_m coeffs[m] exp(-exponents[m] * x^2)
    Tensor<T> coeffs_;
 
    /// the exponents of the expansion f(x) = \sum_m coeffs[m] exp(-exponents[m] * x^2)
    Tensor<T> exponents_;
 
    /// fit the function exp(-mu r)/r
 
    /// formulas taken from
    /// G. Beylkin and L. Monzon, On approximation of functions by exponential sums,
    /// Appl Comput Harmon A, vol. 19, no. 1, pp. 17-48, Jul. 2005.
    /// and
    /// R. J. Harrison, G. I. Fann, T. Yanai, and G. Beylkin,
    /// Multiresolution Quantum Chemistry in Multiwavelet Bases,
    /// Lecture Notes in Computer Science, vol. 2660, p. 707, 2003.
    static void bsh_fit(double mu, double lo, double hi, double eps,
            Tensor<double>& pcoeff, Tensor<double>& pexpnt, bool prnt, bool fix_interval) {
 
        if (mu < 0.0) throw "cannot handle negative mu in bsh_fit";
//        bool restrict_interval=(mu>0) and use_mu_for_restricting_interval;
 
 
        if ((mu > 0) and (not fix_interval)) {
//        if (restrict_interval) {
            // Restrict hi according to the exponential decay
            double r = -log(4*constants::pi*0.01*eps);
            r = -log(r * 4*constants::pi*0.01*eps);
            if (hi > r) hi = r;
        }
 
        double TT;
        double slo, shi;
 
        if (eps >= 1e-2) TT = 5;
        else if (eps >= 1e-4) TT = 10;
        else if (eps >= 1e-6) TT = 14;
        else if (eps >= 1e-8) TT = 18;
        else if (eps >= 1e-10) TT = 22;
        else if (eps >= 1e-12) TT = 26;
        else TT = 30;
 
        if ((mu > 0) and (not fix_interval)) {
//      if (restrict_interval) {
            slo = -0.5*log(4.0*TT/(mu*mu));
        }
        else {
            slo = log(eps/hi) - 1.0;
        }
        shi = 0.5*log(TT/(lo*lo));
                if (shi <= slo) throw "bsh_fit: logic error in slo,shi";
 
        // Resolution required for quadrature over s
        double h = 1.0/(0.2-.50*log10(eps)); // was 0.5 was 0.47
 
        // Truncate the number of binary digits in h's mantissa
        // so that rounding does not occur when performing
        // manipulations to determine the quadrature points and
        // to limit the number of distinct values in case of
        // multiple precisions being used at the same time.
        h = floor(64.0*h)/64.0;
 
 
        // Round shi/lo up/down to an integral multiple of quadrature points
        shi = ceil(shi/h)*h;
        slo = floor(slo/h)*h;
 
        long npt = long((shi-slo)/h+0.5);
 
        //if (prnt)
                //std::cout << "mu " << mu << " slo " << slo << " shi " << shi << " npt " << npt << " h " << h << " " << eps << std::endl;
 
        Tensor<double> coeff(npt), expnt(npt);
 
        for (int i=0; i<npt; ++i) {
            double s = slo + h*(npt-i); // i+1
            coeff[i] = h*2.0/sqrt(constants::pi)*exp(-mu*mu*exp(-2.0*s)/4.0)*exp(s);
            coeff[i] = coeff[i]/(4.0*constants::pi);
            expnt[i] = exp(2.0*s);
        }
 
#if ONE_TERM
        npt=1;
        double s=1.0;
        coeff[0]=1.0;
        expnt[0] = exp(2.0*s);
        coeff=coeff(Slice(0,0));
        expnt=expnt(Slice(0,0));
        print("only one term in gfit",s,coeff[0],expnt[0]);
 
 
#endif
 
        // Prune large exponents from the fit ... never necessary due to construction
 
        // Prune small exponents from Coulomb fit.  Evaluate a gaussian at
        // the range midpoint, and replace it there with the next most
        // diffuse gaussian.  Then examine the resulting error at the two
        // end points ... if this error is less than the desired
        // precision, can discard the diffuse gaussian.
 
        if ((mu == 0.0) and (not fix_interval)) {
//      if (restrict_interval) {
            GFit<double,3>::prune_small_coefficients(eps,lo,hi,coeff,expnt);
        }
 
        // Modify the coeffs of the largest exponents to satisfy the moment conditions
        //
        // SETTING NMOM>1 TURNS OUT TO BE A BAD IDEA (AS CURRENTLY IMPLEMENTED)
        // [It is accurate and efficient for a one-shot application but it seems to
        //  introduce fine-scale noise that amplifies during iterative solution of
        //  the SCF and DFT equations ... the symptom is negative coeffs in the fit]
        //
        // SET NMOM=0 or 1 (1 recommended) unless you are doing a one-shot application
        //
        // Determine the effective range of the four largest exponents and compute
        // moments of the exact and remainder of the fit.  Then adjust the coeffs
        // to reproduce the exact moments in that volume.
        //
        // If nmom!=4 we assume that we will eliminate n=-1 which is stored first
        // in the moment list
        //
        // <r^i|gj> cj = <r^i|exact-remainder>
        const long nmom = 0;
        if (nmom > 0) {
            Tensor<double> q(4), qg(4);
            double range = sqrt(-log(1e-6)/expnt[nmom-1]);
            if (prnt) print("exponent(nmom-1)",expnt[nmom-1],"has range", range);
 
            bsh_spherical_moments(mu, range, q);
            Tensor<double> M(nmom,nmom);
            for (int i=nmom; i<npt; ++i) {
                Tensor<double> qt(4);
                gaussian_spherical_moments(expnt[i], range, qt);
                qg += qt*coeff[i];
            }
            if (nmom != 4) {
                q = q(Slice(1,nmom));
                qg = qg(Slice(1,nmom));
            }
            if (prnt) {
                print("moments", q);
                print("moments", qg);
            }
            q = q - qg;
            for (int j=0; j<nmom; ++j) {
                Tensor<double> qt(4);
                gaussian_spherical_moments(expnt[j], range, qt);
                if (nmom != 4) qt = qt(Slice(1,nmom));
                for (int i=0; i<nmom; ++i) {
                    M(i,j) = qt[i];
                }
            }
            Tensor<double> ncoeff;
            gesv(M, q, ncoeff);
            if (prnt) {
                print("M\n",M);
                print("old coeffs", coeff(Slice(0,nmom-1)));
                print("new coeffs", ncoeff);
            }
 
            coeff(Slice(0,nmom-1)) = ncoeff;
        }
 
        pcoeff = coeff;
        pexpnt = expnt;
    }
 
    /// fit a Slater function using a sum of Gaussians
 
    /// formula inspired by the BSH fit, with the roles of r and mu exchanged
    /// see also Eq. (A3) in
    /// S. Ten-no, Initiation of explicitly correlated Slater-type geminal theory,
    /// Chem. Phys. Lett., vol. 398, no. 1, pp. 56-61, 2004.
    static void slater_fit(double gamma, double lo, double hi, double eps,
            Tensor<double>& pcoeff, Tensor<double>& pexpnt, bool prnt) {
 
        MADNESS_CHECK(gamma >0.0);
        // empirical number TT for the upper integration limit
        double TT;
        if (eps >= 1e-2) TT = 5;
        else if (eps >= 1e-4) TT = 10;
        else if (eps >= 1e-6) TT = 14;
        else if (eps >= 1e-8) TT = 18;
        else if (eps >= 1e-10) TT = 22;
        else if (eps >= 1e-12) TT = 26;
        else TT = 30;
 
        // integration limits for quadrature over s: slo and shi
        // slo and shi must not depend on gamma!!!
        double slo=0.5 * log(eps) - 1.0;
        double shi=log(TT/(lo*lo))*0.5;
 
        // Resolution required for quadrature over s
        double h = 1.0/(0.2-.5*log10(eps)); // was 0.5 was 0.47
 
        // Truncate the number of binary digits in h's mantissa
        // so that rounding does not occur when performing
        // manipulations to determine the quadrature points and
        // to limit the number of distinct values in case of
        // multiple precisions being used at the same time.
        h = floor(64.0*h)/64.0;
 
        // Round shi/lo up/down to an integral multiple of quadrature points
        shi = ceil(shi/h)*h;
        slo = floor(slo/h)*h;
 
        long npt = long((shi-slo)/h+0.5);
 
        Tensor<double> coeff(npt), expnt(npt);
 
        for (int i=0; i<npt; ++i) {
            const double s = slo + h*(npt-i);   // i+1
            coeff[i] = h*exp(-gamma*gamma*exp(2.0*s) + s);
            coeff[i]*=2.0*gamma/sqrt(constants::pi);
            expnt[i] = 0.25*exp(-2.0*s);
        }
 
        if (prnt) {
            std::cout << "weights and roots for a Slater function with gamma=" << gamma << std::endl;
            for (int i=0; i<npt; ++i)
                std::cout << i << " " << coeff[i] << " " << expnt[i] << std::endl;
 
            long npt = 300;
            //double hi = 1.0;
            //if (mu) hi = min(1.0,30.0/mu);
            std::cout << "       x         value   abserr   relerr" << std::endl;
            std::cout << "  ------------  ------- -------- -------- " << std::endl;
            double step = exp(log(hi/lo)/(npt+1));
            for (int i=0; i<=npt; ++i) {
                double r = lo*(pow(step,i+0.5));
                double exact = exp(-gamma*r);
                double test = 0.0;
                for (int j=0; j<coeff.dim(0); ++j)
                    test += coeff[j]*exp(-r*r*expnt[j]);
                double err = 0.0;
                if (exact) err = (exact-test)/exact;
                printf("  %.6e %8.1e %8.1e %8.1e\n",r, exact, exact-test, err);
            }
        }
        pcoeff = coeff;
        pexpnt = expnt;
    }
 
 
    /// fit a correlation factor (1- exp(-mu r))
 
    /// use the Slater fit with an additional term: 1*exp(-0 r^2)
    static void f12_fit(double gamma, double lo, double hi, double eps,
                           Tensor<double>& pcoeff, Tensor<double>& pexpnt, bool prnt) {
        Tensor<double> coeff,expnt;
        slater_fit(gamma, lo, hi, eps, coeff, expnt, prnt);
 
        pcoeff=Tensor<double>(coeff.size()+1);
        pcoeff(Slice(1,-1,1))=-coeff(_);
        pexpnt=Tensor<double>(expnt.size()+1);
        pexpnt(Slice(1,-1,1))=expnt(_);
 
        pcoeff(0l)=1.0;
        pexpnt(0l)=1.e-10;
    }
 
 
    /// fit a correlation factor f12^2 = (1- exp(-mu r))^2 = 1 - 2 exp(-mu r) + exp(-2 mu r)
 
    /// no factor 1/(2 mu) or square of it included!
    /// use the Slater fit with an additional term: 1*exp(-0 r^2)
    static void f12sq_fit(double gamma, double lo, double hi, double eps,
                        Tensor<double>& pcoeff, Tensor<double>& pexpnt, bool prnt) {
        Tensor<double> coeff1,expnt1, coeff2, expnt2;
        slater_fit(gamma, lo, hi, eps, coeff1, expnt1, prnt);
        slater_fit(2.0*gamma, lo, hi, eps, coeff2, expnt2, prnt);
 
        // check exponents are the same
        MADNESS_CHECK((expnt1-expnt2).normf()/expnt1.size()<1.e-12);
 
        // add exponential terms
        auto coeff=coeff2-2.0*coeff1;
        pcoeff=Tensor<double>(coeff1.size()+1);
        pcoeff(Slice(1,-1,1))=coeff(_);
        pexpnt=Tensor<double>(expnt1.size()+1);
        pexpnt(Slice(1,-1,1))=expnt1(_);
 
        // add constant term
        pcoeff(0l)=1.0;
        pexpnt(0l)=1.e-10;
    }
 
 
    void static bsh_fit_ndim(int ndim, double mu, double lo, double hi, double eps,
            Tensor<double>& pcoeff, Tensor<double>& pexpnt, bool prnt) {
 
        if (mu > 0) {
            // Restrict hi according to the exponential decay
            double r = -log(4*constants::pi*0.01*eps);
            r = -log(r * 4*constants::pi*0.01*eps);
            if (hi > r) hi = r;
        }
 
 
        // Determine range of quadrature by estimating when
        // kernel drops to exp(-100)
 
        double slo, shi;
        if (mu > 0) {
            slo = -0.5*log(4.0*100.0/(mu*mu));
            slo = -0.5*log(4.0*(slo*ndim - 2.0*slo + 100.0)/(mu*mu));
        }
        else {
            slo = log(eps/hi) - 1.0;
        }
        shi = 0.5*log(100.0/(lo*lo));
 
        // Resolution required for quadrature over s
        double h = 1.0/(0.2-.50*log10(eps)); // was 0.5 was 0.47
 
        // Truncate the number of binary digits in h's mantissa
        // so that rounding does not occur when performing
        // manipulations to determine the quadrature points and
        // to limit the number of distinct values in case of
        // multiple precisions being used at the same time.
        h = floor(64.0*h)/64.0;
 
 
        // Round shi/lo up/down to an integral multiple of quadrature points
        shi = ceil(shi/h)*h;
        slo = floor(slo/h)*h;
 
        long npt = long((shi-slo)/h+0.5);
 
        if (prnt)
            std::cout << "bsh: mu " << mu << " lo " << lo << " hi " << hi
            << " eps " << eps << " slo " << slo << " shi " << shi
            << " npt " << npt << " h " << h << std::endl;
 
 
        // Compute expansion pruning small coeffs and large exponents
        Tensor<double> coeff(npt), expnt(npt);
        int nnpt=0;
        for (int i=0; i<npt; ++i) {
            double s = slo + h*(npt-i); // i+1
            double c = exp(-0.25*mu*mu*exp(-2.0*s)+(ndim-2)*s)*0.5/pow(constants::pi,0.5*ndim);
            double p = exp(2.0*s);
            c = c*h;
            if (c*exp(-p*lo*lo) > eps) {
                coeff(nnpt) = c;
                expnt(nnpt) = p;
                ++nnpt;
            }
        }
        npt = nnpt;
#if ONE_TERM
        npt=1;
        double s=1.0;
        coeff[0]=1.0;
        expnt[0] = exp(2.0*s);
        coeff=coeff(Slice(0,0));
        expnt=expnt(Slice(0,0));
        print("only one term in gfit",s,coeff[0],expnt[0]);
 
#endif
 
 
        // Prune small exponents from Coulomb fit.  Evaluate a gaussian at
        // the range midpoint, and replace it there with the next most
        // diffuse gaussian.  Then examine the resulting error at the two
        // end points ... if this error is less than the desired
        // precision, can discard the diffuse gaussian.
 
        if (mu == 0.0) {
            double mid = lo + (hi-lo)*0.5;
            long i;
            for (i=npt-1; i>0; --i) {
                double cnew = coeff[i]*exp(-(expnt[i]-expnt[i-1])*mid*mid);
                double errlo = coeff[i]*exp(-expnt[i]*lo*lo) -
                        cnew*exp(-expnt[i-1]*lo*lo);
                double errhi = coeff[i]*exp(-expnt[i]*hi*hi) -
                        cnew*exp(-expnt[i-1]*hi*hi);
                if (std::max(std::abs(errlo),std::abs(errhi)) > 0.03*eps) break;
                npt--;
                coeff[i-1] = coeff[i-1] + cnew;
            }
        }
 
        // Shrink array to correct size
        coeff = coeff(Slice(0,npt-1));
        expnt = expnt(Slice(0,npt-1));
 
 
        if (prnt) {
            for (int i=0; i<npt; ++i)
                std::cout << i << " " << coeff[i] << " " << expnt[i] << std::endl;
 
            long npt = 300;
            std::cout << "       x         value" << std::endl;
            std::cout << "  ------------  ---------------------" << std::endl;
            double step = exp(log(hi/lo)/(npt+1));
            for (int i=0; i<=npt; ++i) {
                double r = lo*(pow(step,i+0.5));
                double test = 0.0;
                for (int j=0; j<coeff.dim(0); ++j)
                    test += coeff[j]*exp(-r*r*expnt[j]);
                printf("  %.6e %20.10e\n",r, test);
            }
        }
 
        pcoeff = coeff;
        pexpnt = expnt;
    }
 
    // Returns in q[0..4] int(r^2(n+1)*exp(-alpha*r^2),r=0..R) n=-1,0,1,2
    static void gaussian_spherical_moments(double alpha, double R, Tensor<double>& q) {
        q[0] = -(-0.1e1 + exp(-alpha * R*R)) / alpha / 0.2e1;
        q[1] = (-0.2e1 * R * pow(alpha, 0.3e1 / 0.2e1) + sqrt(constants::pi)
                * erf(R * sqrt(alpha)) * alpha * exp(alpha * R*R))
                * pow(alpha, -0.5e1 / 0.2e1) * exp(-alpha * R*R) / 0.4e1;
        q[2] = -(-0.1e1 + exp(-alpha * R*R) + exp(-alpha * R*R) * alpha * R*R)
                * pow(alpha, -0.2e1) / 0.2e1;
        q[3] = -(-0.3e1 * sqrt(constants::pi) * erf(R * sqrt(alpha)) * pow(alpha, 0.2e1)
                * exp(alpha * R*R) + 0.6e1 * R * pow(alpha, 0.5e1 / 0.2e1)
                + 0.4e1 * pow(R, 0.3e1) * pow(alpha, 0.7e1 / 0.2e1))
                * pow(alpha, -0.9e1 / 0.2e1) * exp(-alpha * R*R) / 0.8e1;
    }
 
    // Returns in q[0..4] int(r^2(n+1)*exp(-mu*r)/(4*constants::pi*r),r=0..R) n=-1,0,1,2
    static void bsh_spherical_moments(double mu, double R, Tensor<double>& q) {
        if (mu == 0.0) {
            q[0] = R / constants::pi  / 0.4e1;
            q[1] = pow(R, 0.2e1) / constants::pi / 0.8e1;
            q[2] = pow(R, 0.3e1) / constants::pi / 0.12e2;
            q[3] = pow(R, 0.4e1) / constants::pi / 0.16e2;
        }
        else  {
            q[0] = (exp(mu * R) - 0.1e1) / mu * exp(-mu * R) / constants::pi / 0.4e1;
            q[1] = -(-exp(mu * R) + 0.1e1 + mu * R) * pow(mu, -0.2e1) / constants::pi
                    * exp(-mu * R) / 0.4e1;
            q[2] = -(-0.2e1 * exp(mu * R) + 0.2e1 + 0.2e1 * mu * R + R*R *
                    pow(mu, 0.2e1))*pow(mu, -0.3e1) / constants::pi * exp(-mu * R) / 0.4e1;
            q[3] = -(-0.6e1 * exp(mu * R) + 0.6e1 + 0.6e1 * mu * R + 0.3e1 * R*R
                    * pow(mu, 0.2e1) + pow(R, 0.3e1) * pow(mu, 0.3e1))
                    * pow(mu, -0.4e1) / constants::pi * exp(-mu * R) / 0.4e1;
        }
    }
 
};
 
 
}
 
#endif // MADNESS_MRA_GFIT_H__INCLUDED