boys.h

/*
 *  Copyright (C) 2004-2020 Edward F. Valeev
 *
 *  This file is part of Libint.
 *
 *  Libint is free software: you can redistribute it and/or modify
 *  it under the terms of the GNU Lesser General Public License as published by
 *  the Free Software Foundation, either version 3 of the License, or
 *  (at your option) any later version.
 *
 *  Libint 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 Lesser General Public License for more details.
 *
 *  You should have received a copy of the GNU Lesser General Public License
 *  along with Libint.  If not, see <http://www.gnu.org/licenses/>.
 *
 */

#ifndef _libint2_src_lib_libint_boys_h_
#define _libint2_src_lib_libint_boys_h_

#if defined(__cplusplus)

#include <iostream>
#include <cstdlib>
#include <cmath>
#include <stdexcept>
#include <libint2/util/vector.h>
#include <cassert>
#include <vector>
#include <algorithm>
#include <limits>
#include <mutex>
#include <type_traits>
#include <memory>

// from now on at least C++11 is required by default
#include <libint2/util/cxxstd.h>
#if LIBINT2_CPLUSPLUS_STD < 2011
# error "Libint2 C++ API requires C++11 support"
#endif

#include <libint2/boys_fwd.h>
#include <libint2/numeric.h>
#include <libint2/initialize.h>

#if HAVE_LAPACK // use F77-type interface for now, switch to LAPACKE later
extern "C" void dgesv_(const int* n,
                       const int* nrhs, double* A, const int* lda,
                       int* ipiv, double* b, const int* ldb,
                       int* info);
#endif

namespace libint2 {

  /// holds tables of expensive quantities
  template<typename Real>
  class ExpensiveNumbers {
    public:
      ExpensiveNumbers(int ifac = -1, int idf = -1, int ibc = -1) {
        if (ifac >= 0) {
          fac.resize(ifac + 1);
          fac[0] = 1.0;
          for (int i = 1; i <= ifac; i++) {
            fac[i] = i * fac[i - 1];
          }
        }

        if (idf >= 0) {
          df.resize(idf + 1);
          /* df[i] gives (i-1)!!, so that (-1)!! is defined... */
          df[0] = 1.0;
          if (idf >= 1)
            df[1] = 1.0;
          if (idf >= 2)
            df[2] = 1.0;
          for (int i = 3; i <= idf; i++) {
            df[i] = (i - 1) * df[i - 2];
          }
        }

        if (ibc >= 0) {
          bc_.resize((ibc+1)*(ibc+1));
          std::fill(bc_.begin(), bc_.end(), Real(0));
          bc.resize(ibc+1);
          bc[0] = &bc_[0];
          for(int i=1; i<=ibc; ++i)
            bc[i] = bc[i-1] + (ibc+1);

          for(int i=0; i<=ibc; i++)
            bc[i][0] = 1.0;
          for(int i=0; i<=ibc; i++)
            for(int j=1; j<=i; ++j)
              bc[i][j] = bc[i][j-1] * Real(i-j+1) / Real(j);
        }

        for (int i = 0; i < 128; i++) {
          twoi1[i] = 1.0 / (Real(2.0) * i + Real(1.0));
          ihalf[i] = Real(i) - Real(0.5);
        }

      }

      ~ExpensiveNumbers() {
      }

      std::vector<Real> fac;
      std::vector<Real> df;
      std::vector<Real*> bc;

      // these quantitites are needed with indices <= mmax
      // 64 is sufficient to handle up to 4 center integrals with up to L=15 basis functions
      // but need higher values for Yukawa integrals ...
      Real twoi1[128]; /* 1/(2 i + 1); needed for downward recursion */
      Real ihalf[128]; /* i - 0.5, needed for upward recursion */

    private:
      std::vector<Real> bc_;
  };

#define _local_min_macro(a,b) ((a) > (b) ? (a) : (b))

  /** Computes the Boys function, \f$ F_m (T) = \int_0^1 u^{2m} \exp(-T u^2) \, {\rm d}u \f$,
    * using single algorithm (asymptotic expansion). Slow for the sake of precision control.
    * Useful in two cases:
    * <ul>
    *   <li> for reference purposes, if \c Real supports high/arbitrary precision, and </li>
    *   <li> for moderate values of \f$ T \f$, if \c Real is a low-precision floating-point type.
    *        N.B. FmEval_Reference2 , which can compute for all practical values of \f$ T \f$ and \f$ m \f$, is recommended
    *        with standard \c Real types (\c double and \c float). </li>
    * </ul>
    *
    * \note Precision is controlled heuristically, i.e. cannot be guaranteed mathematically;
    *       will stop if absolute precision is reached, or precision of \c Real is exhausted.
    *       It is important that \c std::numeric_limits<Real> is defined appropriately.
    *
    * @tparam Real the type to use for all floating-point computations.
    *         Must be able to compute logarithm and exponential, i.e.
    *         log(x) and exp(x), where x is Real, must be valid expressions.
    */
  template<typename Real>
  struct FmEval_Reference {

      static std::shared_ptr<const FmEval_Reference> instance(int /* mmax */, Real /* precision */) {

        // thread-safe per C++11 standard [6.7.4]
        static auto instance_ = std::make_shared<const FmEval_Reference>();

        return instance_;
      }

      /// computes a single value of \f$ F_m(T) \f$ using MacLaurin series to full precision of @c Real
      static Real eval(Real T, size_t m) {
        assert(m < 100);
        static const Real T_crit = std::numeric_limits<Real>::is_bounded == true ? -log( std::numeric_limits<Real>::min() * 100.5 / 2. ) : Real(0) ;
        if (std::numeric_limits<Real>::is_bounded && T > T_crit)
          throw std::overflow_error("FmEval_Reference<Real>::eval: Real lacks precision for the given value of argument T");
        static const Real half = Real(1)/2;
        Real denom = (m + half);
        using std::exp;
        Real term = exp(-T) / (2 * denom);
        Real old_term = 0;
        Real sum = term;
        const Real epsilon = get_epsilon(T);
        const Real epsilon_divided_10 = epsilon / 10;
        do {
          denom += 1;
          old_term = term;
          term = old_term * T / denom;
          sum += term;
          //rel_error = term / sum , hence iterate until rel_error = epsilon
          // however, must ensure that contributions are decreasing to ensure that omitted contributions are smaller than epsilon
        } while (term > sum * epsilon_divided_10 || old_term < term);

        return sum;
      }

      /// fills up an array of Fm(T) for m in [0,mmax]
      /// @param[out] Fm array to be filled in with the Boys function values, must be at least mmax+1 elements long
      /// @param[in] T the Boys function argument
      /// @param[in] mmax the maximum value of m for which Boys function will be computed;
      static void eval(Real* Fm, Real T, size_t mmax) {

        // evaluate for mmax using MacLaurin series
        // it converges fastest for the largest m -> use it to compute Fmmax(T)
        //  see JPC 94, 5564 (1990).
        for(size_t m=0; m<=mmax; ++m)
          Fm[m] = eval(T, m);
        return;
      }

  };

  /** Computes the Boys function, \$ F_m (T) = \int_0^1 u^{2m} \exp(-T u^2) \, {\rm d}u \$,
    * using multi-algorithm approach (upward recursion for T>=117, and asymptotic summation for T<117).
    * This is slow and should be used for reference purposes, e.g. computing the interpolation tables.
    * Precision is not always guaranteed as it is limited by the precision of \c Real type.
    * When \c Real is \c double, can maintain absolute precision of epsilon for up to m=40.
    *
    * @tparam Real the type to use for all floating-point computations.
    *         Must be able to compute logarithm, exponential, square root, and error function, i.e.
    *         log(x), exp(x), sqrt(x), and erf(x), where x is Real, must be valid expressions.
    */
  template<typename Real>
  struct FmEval_Reference2 {

      static std::shared_ptr<const FmEval_Reference2> instance(int /* mmax */, Real /* precision */) {

        // thread-safe per C++11 standard [6.7.4]
        static auto instance_ = std::make_shared<const FmEval_Reference2>();

        return instance_;
      }

      /// fills up an array of Fm(T) for m in [0,mmax]
      /// @param[out] Fm array to be filled in with the Boys function values, must be at least mmax+1 elements long
      /// @param[in] t the Boys function argument
      /// @param[in] mmax the maximum value of m for which Boys function will be computed;
      static void eval(Real* Fm, Real t, size_t mmax) {

        if (t < Real(117)) {
          FmEval_Reference<Real>::eval(Fm,t,mmax);
        }
        else {
          const Real two_over_sqrt_pi{1.128379167095512573896158903121545171688101258657997713688171443421284936882986828973487320404214727};
          const Real K = 1/two_over_sqrt_pi;

          auto t2 = 2*t;
          auto et = exp(-t);
          auto sqrt_t = sqrt(t);
          Fm[0] = K*erf(sqrt_t)/sqrt_t;
          if (mmax > 0)
          for(size_t m=0; m<=mmax-1; m++) {
            Fm[m+1] = ((2*m + 1)*Fm[m] - et)/(t2);
          }
        }
      }

  };

  /** Computes the Boys function, \$ F_m (T) = \int_0^1 u^{2m} \exp(-T u^2) \, {\rm d}u \$,
    * using 7-th order Chebyshev interpolation.
    */
  template <typename Real = double>
  class FmEval_Chebyshev7 {

#include <libint2/boys_cheb7.h>

      static_assert(std::is_same<Real,double>::value, "FmEval_Chebyshev7 only supports double as the real type");

      static constexpr const int ORDER = interpolation_order;   //!, interpolation order
      static constexpr const int ORDERp1 = ORDER+1;   //!< ORDER + 1

      static constexpr const Real T_crit = cheb_table_tmax;          //!< critical value of T above which safe to use upward recusion
      static constexpr Real delta = cheb_table_delta;           //!< interval size
      static constexpr Real one_over_delta = 1/delta;  //! 1/delta

      int mmax;                   //!< the maximum m that is tabulated
      ExpensiveNumbers<double> numbers_;
      Real *c; /* the Chebyshev coefficients table, T_crit*one_over_delta by mmax*ORDERp1 */

    public:
      /// \param m_max maximum value of the Boys function index; set to -1 to skip initialization
      /// \param precision the desired relative precision
      /// \throw std::invalid_argument if \c m_max is greater than \c cheb_table_mmax (see boys_cheb7.h)
      /// \throw std::invalid_argument if \c precision is smaller than std::numeric_limits<double>::epsilon()
      FmEval_Chebyshev7(int m_max, double precision = std::numeric_limits<double>::epsilon()) :
          mmax(m_max), numbers_(14) {
#if defined(__x86_64__) || defined(_M_X64) || defined(__i386) || defined(_M_IX86)
# if !defined(__AVX__) && defined(NDEBUG)
        if (libint2::verbose()) {
          static bool printed_performance_warning = false;
          if (!printed_performance_warning) {
            libint2::verbose_stream()
                << "libint2::FmEval_Chebyshev7 on x86(-64) platforms needs AVX support for best performance"
                << std::endl;
            printed_performance_warning = true;
          }
        }
# endif
#endif
        if (precision < std::numeric_limits<double>::epsilon())
          throw std::invalid_argument(std::string("FmEval_Chebyshev7 does not support precision smaller than ") + std::to_string(std::numeric_limits<double>::epsilon()));
        if (mmax > cheb_table_mmax)
          throw std::invalid_argument(
              "FmEval_Chebyshev7::init() : requested mmax exceeds the "
              "hard-coded mmax");
        if (m_max >= 0)
          init_table();
      }
      ~FmEval_Chebyshev7() {
        if (mmax >= 0) {
          free(c);
        }
      }

      /// Singleton interface allows to manage the lone instance; adjusts max m values as needed in thread-safe fashion
      static std::shared_ptr<const FmEval_Chebyshev7> instance(int m_max, double = 0.0) {

        assert(m_max >= 0);
        // thread-safe per C++11 standard [6.7.4]
        static auto instance_ = std::make_shared<const FmEval_Chebyshev7>(m_max);

        while (instance_->max_m() < m_max) {
          static std::mutex mtx;
          std::lock_guard<std::mutex> lck(mtx);
          if (instance_->max_m() < m_max) {
            auto new_instance = std::make_shared<const FmEval_Chebyshev7>(m_max);
            instance_ = new_instance;
          }
        }

        return instance_;
      }

      /// @return the maximum value of m for which the Boys function can be computed with this object
      int max_m() const { return mmax; }

      /// fills in Fm with computed Boys function values for m in [0,mmax]
      /// @param[out] Fm array to be filled in with the Boys function values, must be at least mmax+1 elements long
      /// @param[in] x the Boys function argument
      /// @param[in] mmax the maximum value of m for which Boys function will be computed; mmax must be <= the value returned by max_m
      inline void eval(Real* Fm, Real x, int m_max) const {

        // large T => use upward recursion
        // cost = 1 div + 1 sqrt + (1 + 2*(m-1)) muls
        if (x > T_crit) {
          const double one_over_x = 1/x;
          Fm[0] = 0.88622692545275801365 * sqrt(one_over_x); // see Eq. (9.8.9) in Helgaker-Jorgensen-Olsen
          if (m_max == 0)
            return;
          // this upward recursion formula omits - e^(-x)/(2x), which for x>T_crit is small enough to guarantee full double precision
          for (int i = 1; i <= m_max; i++)
            Fm[i] = Fm[i - 1] * numbers_.ihalf[i] * one_over_x; // see Eq. (9.8.13)
          return;
        }

        // ---------------------------------------------
        // small and intermediate arguments => interpolate Fm and (optional) downward recursion
        // ---------------------------------------------
        // which interval does this x fall into?
        const Real x_over_delta = x * one_over_delta;
        const int iv = int(x_over_delta); // the interval index
        const Real xd = x_over_delta - (Real)iv - 0.5; // this ranges from -0.5 to 0.5
        const int m_min = 0;

#if defined(__AVX__)
        const auto x2 = xd*xd;
        const auto x3 = x2*xd;
        const auto x4 = x2*x2;
        const auto x5 = x2*x3;
        const auto x6 = x3*x3;
        const auto x7 = x3*x4;
        libint2::simd::VectorAVXDouble x0vec(1., xd, x2, x3);
        libint2::simd::VectorAVXDouble x1vec(x4, x5, x6, x7);
#endif // AVX

        const Real *d = c + (ORDERp1) * (iv * (mmax+1) + m_min); // ptr to the interpolation data for m=mmin
        int m = m_min;
#if defined(__AVX__)
        if (m_max-m >=3) {
          const int unroll_size = 4;
          const int m_fence = (m_max + 2 - unroll_size);
          for(; m<m_fence; m+=unroll_size, d+=ORDERp1*unroll_size) {
            libint2::simd::VectorAVXDouble d00v, d01v, d10v, d11v,
                                           d20v, d21v, d30v, d31v;
            d00v.load_aligned(d);            d01v.load_aligned((d+4));
            d10v.load_aligned(d+ORDERp1);    d11v.load_aligned((d+4)+ORDERp1);
            d20v.load_aligned(d+2*ORDERp1);  d21v.load_aligned((d+4)+2*ORDERp1);
            d30v.load_aligned(d+3*ORDERp1);  d31v.load_aligned((d+4)+3*ORDERp1);
            libint2::simd::VectorAVXDouble fm0 = d00v * x0vec + d01v * x1vec;
            libint2::simd::VectorAVXDouble fm1 = d10v * x0vec + d11v * x1vec;
            libint2::simd::VectorAVXDouble fm2 = d20v * x0vec + d21v * x1vec;
            libint2::simd::VectorAVXDouble fm3 = d30v * x0vec + d31v * x1vec;
            libint2::simd::VectorAVXDouble sum0123 = horizontal_add(fm0, fm1, fm2, fm3);
            sum0123.convert(&Fm[m]);
          }
        } // unroll_size=4
        if (m_max-m >=1) {
          const int unroll_size = 2;
          const int m_fence = (m_max + 2 - unroll_size);
          for(; m<m_fence; m+=unroll_size, d+=ORDERp1*unroll_size) {
            libint2::simd::VectorAVXDouble d00v, d01v, d10v, d11v;
            d00v.load_aligned(d);
            d01v.load_aligned((d+4));
            d10v.load_aligned(d+ORDERp1);
            d11v.load_aligned((d+4)+ORDERp1);
            libint2::simd::VectorAVXDouble fm0 = d00v * x0vec + d01v * x1vec;
            libint2::simd::VectorAVXDouble fm1 = d10v * x0vec + d11v * x1vec;
            libint2::simd::VectorSSEDouble sum01 = horizontal_add(fm0, fm1);
            sum01.convert(&Fm[m]);
          }
        } // unroll_size=2
        { // no unrolling
          for(; m<=m_max; ++m, d+=ORDERp1) {
            libint2::simd::VectorAVXDouble d0v, d1v;
            d0v.load_aligned(d);
            d1v.load_aligned(d+4);
            Fm[m] = horizontal_add(d0v * x0vec + d1v * x1vec);
          }
        }
#else // AVX not available
        for(m=m_min; m<=m_max; ++m, d+=ORDERp1) {
          Fm[m] = d[0]
                + xd * (d[1]
                + xd * (d[2]
                + xd * (d[3]
                + xd * (d[4]
                + xd * (d[5]
                + xd * (d[6]
                + xd * (d[7])))))));

          //        // check against the reference value
          //        if (false) {
          //          double refvalue = FmEval_Reference2<double>::eval(x, m); // compute F(T)
          //          if (abs(refvalue - Fm[m]) > 1e-10) {
          //            std::cout << "T = " << x << " m = " << m << " cheb = "
          //                      << Fm[m] << " ref = " << refvalue << std::endl;
          //          }
          //        }
        }
#endif


      } // eval()

    private:

      void init_table() {

        // get memory
        void* result;
        int status = posix_memalign(&result, ORDERp1*sizeof(Real), (mmax + 1) * cheb_table_nintervals * ORDERp1 * sizeof(Real));
        if (status != 0) {
          if (status == EINVAL)
            throw std::logic_error(
              "FmEval_Chebyshev7::init() : posix_memalign failed, alignment must be a power of 2 at least as large as sizeof(void *)");
          if (status == ENOMEM)
            throw std::bad_alloc();
          abort();  // should be unreachable
        }
        c = static_cast<Real*>(result);

        // copy contents of static table into c
        // need all intervals
        for (std::size_t iv = 0; iv < cheb_table_nintervals; ++iv) {
          // but only values of m up to mmax
          std::copy(cheb_table[iv], cheb_table[iv]+(mmax+1)*ORDERp1, c+(iv*(mmax+1))*ORDERp1);
        }
      }

  }; // FmEval_Chebyshev7

#if LIBINT2_CONSTEXPR_STATICS
  template <typename Real>
  constexpr
  double FmEval_Chebyshev7<Real>::cheb_table[FmEval_Chebyshev7<Real>::cheb_table_nintervals][(FmEval_Chebyshev7<Real>::cheb_table_mmax+1)*(FmEval_Chebyshev7<Real>::interpolation_order+1)];
#else
  // clang needs an explicit specifalization declaration to avoid warning
#  ifdef __clang__
  template <typename Real>
  double FmEval_Chebyshev7<Real>::cheb_table[FmEval_Chebyshev7<Real>::cheb_table_nintervals][(FmEval_Chebyshev7<Real>::cheb_table_mmax+1)*(FmEval_Chebyshev7<Real>::interpolation_order+1)];
#  endif
#endif

#ifndef STATIC_OON
#define STATIC_OON
  namespace {
    const double oon[] = {0.0, 1.0, 1.0/2.0, 1.0/3.0, 1.0/4.0, 1.0/5.0, 1.0/6.0, 1.0/7.0, 1.0/8.0, 1.0/9.0, 1.0/10.0, 1.0/11.0};
  }
#endif

  /** Computes the Boys function, \$ F_m (T) = \int_0^1 u^{2m} \exp(-T u^2) \, {\rm d}u \$,
    * using Taylor interpolation of up to 8-th order.
    * @tparam Real the type to use for all floating-point computations. Following expressions must be valid: exp(Real), pow(Real), fabs(Real), max(Real), and floor(Real).
    * @tparam INTERPOLATION_ORDER the interpolation order. The higher the order the less memory this object will need, but the computational cost will increase (usually very slightly)
    */
  template<typename Real = double, int INTERPOLATION_ORDER = 7>
  class FmEval_Taylor {
    public:
      static_assert(std::is_same<Real,double>::value, "FmEval_Taylor only supports double as the real type");

      static const int max_interp_order = 8;
      static const bool INTERPOLATION_AND_RECURSION = false; // compute F_lmax(T) and then iterate down to F_0(T)? Else use interpolation only
      const double soft_zero_;

      /// Constructs the object to be able to compute Boys funcion for m in [0,mmax], with relative \c precision
      FmEval_Taylor(unsigned int mmax, Real precision = std::numeric_limits<Real>::epsilon()) :
          soft_zero_(1e-6), cutoff_(precision), numbers_(
              INTERPOLATION_ORDER + 1, 2 * (mmax + INTERPOLATION_ORDER - 1)) {

#if defined(__x86_64__) || defined(_M_X64) || defined(__i386) || defined(_M_IX86)
# if !defined(__AVX__) && defined(NDEBUG)
        if (libint2::verbose()) {
          static bool printed_performance_warning = false;
          if (!printed_performance_warning) {
            libint2::verbose_stream()
                << "libint2::FmEval_Taylor on x86(-64) platforms needs AVX support for best performance"
                << std::endl;
            printed_performance_warning = true;
          }
        }
# endif
#endif

        assert(mmax <= 63);

        const double sqrt_pi = std::sqrt(M_PI);

        /*---------------------------------------
         We are doing Taylor interpolation with
         n=TAYLOR_ORDER terms here:
         error <= delT^n/(n+1)!
         ---------------------------------------*/
        delT_ = 2.0
            * std::pow(cutoff_ * numbers_.fac[INTERPOLATION_ORDER + 1],
                       1.0 / INTERPOLATION_ORDER);
        oodelT_ = 1.0 / delT_;
        max_m_ = mmax + INTERPOLATION_ORDER - 1;

        T_crit_ = new Real[max_m_ + 1]; /*--- m=0 is included! ---*/
        max_T_ = 0;
        /*--- Figure out T_crit for each m and put into the T_crit ---*/
        for (int m = max_m_; m >= 0; --m) {
          /*------------------------------------------
           Damped Newton-Raphson method to solve
           T^{m-0.5}*exp(-T) = epsilon*Gamma(m+0.5)
           The solution is the max T for which to do
           the interpolation
           ------------------------------------------*/
          double T = -log(cutoff_);
          const double egamma = cutoff_ * sqrt_pi * numbers_.df[2 * m]
              / std::pow(2.0, m);
          double T_new = T;
          double func;
          do {
            const double damping_factor = 0.2;
            T = T_new;
            /* f(T) = the difference between LHS and RHS of the equation above */
            func = std::pow(T, m - 0.5) * std::exp(-T) - egamma;
            const double dfuncdT = ((m - 0.5) * std::pow(T, m - 1.5)
                - std::pow(T, m - 0.5)) * std::exp(-T);
            /* f(T) has 2 roots and has a maximum in between. If f'(T) > 0 we are to the left of the hump. Make a big step to the right. */
            if (dfuncdT > 0.0) {
              T_new *= 2.0;
            } else {
              /* damp the step */
              double deltaT = -func / dfuncdT;
              const double sign_deltaT = (deltaT > 0.0) ? 1.0 : -1.0;
              const double max_deltaT = damping_factor * T;
              if (std::fabs(deltaT) > max_deltaT)
                deltaT = sign_deltaT * max_deltaT;
              T_new = T + deltaT;
            }
            if (T_new <= 0.0) {
              T_new = T / 2.0;
            }
          } while (std::fabs(func / egamma) >= soft_zero_);
          T_crit_[m] = T_new;
          const int T_idx = (int) std::floor(T_new / delT_);
          max_T_ = std::max(max_T_, T_idx);
        }

        // allocate the grid (see the comments below)
        {
          const int nrow = max_T_ + 1;
          const int ncol = max_m_ + 1;
          grid_ = new Real*[nrow];
          grid_[0] = new Real[nrow * ncol];
          //std::cout << "Allocated interpolation table of " << nrow * ncol << " reals" << std::endl;
          for (int r = 1; r < nrow; ++r)
            grid_[r] = grid_[r - 1] + ncol;
        }

        /*-------------------------------------------------------
         Tabulate the gamma function from t=delT to T_crit[m]:
         1) include T=0 though the table is empty for T=0 since
         Fm(0) is simple to compute
         -------------------------------------------------------*/
        /*--- do the mmax first ---*/
        for (int T_idx = max_T_; T_idx >= 0; --T_idx) {
          const double T = T_idx * delT_;
          libint2::FmEval_Reference2<double>::eval(grid_[T_idx], T, max_m_);
        }
      }

      ~FmEval_Taylor() {
        delete[] T_crit_;
        delete[] grid_[0];
        delete[] grid_;
      }

      /// Singleton interface allows to manage the lone instance;
      /// adjusts max m and precision values as needed in thread-safe fashion
      static std::shared_ptr<const FmEval_Taylor> instance(unsigned int mmax, Real precision = std::numeric_limits<Real>::epsilon()) {
        assert(mmax >= 0);
        assert(precision >= 0);
        // thread-safe per C++11 standard [6.7.4]
        static auto instance_ = std::make_shared<const FmEval_Taylor>(mmax, precision);

        while (instance_->max_m() < mmax || instance_->precision() > precision) {
          static std::mutex mtx;
          std::lock_guard<std::mutex> lck(mtx);
          if (instance_->max_m() < mmax || instance_->precision() > precision) {
            auto new_instance = std::make_shared<const FmEval_Taylor>(mmax, precision);
            instance_ = new_instance;
          }
        }

        return instance_;
      }

      /// @return the maximum value of m for which this object can compute the Boys function
      int max_m() const { return max_m_ - INTERPOLATION_ORDER + 1; }
      /// @return the precision with which this object can compute the Boys function
      Real precision() const { return cutoff_; }

      /// computes Boys function values with m index in range [0,mmax]
      /// @param[out] Fm array to be filled in with the Boys function values, must be at least mmax+1 elements long
      /// @param[in] x the Boys function argument
      /// @param[in] mmax the maximum value of m for which Boys function will be computed;
      ///                  it must be <= the value returned by max_m() (this is not checked)
      void eval(Real* Fm, Real T, int mmax) const {
        const double sqrt_pio2 = 1.2533141373155002512;
        const double two_T = 2.0 * T;

        // stop recursion at mmin
        const int mmin = INTERPOLATION_AND_RECURSION ? mmax : 0;
        /*-------------------------------------
         Compute Fm(T) from mmax down to mmin
         -------------------------------------*/
        const bool use_upward_recursion = true;
        if (use_upward_recursion) {
//          if (T > 30.0) {
          if (T > T_crit_[0]) {
            const double one_over_x = 1.0/T;
            Fm[0] = 0.88622692545275801365 * sqrt(one_over_x); // see Eq. (9.8.9) in Helgaker-Jorgensen-Olsen
            if (mmax == 0)
              return;
            // this upward recursion formula omits - e^(-x)/(2x), which for x>T_crit is <1e-15
            for (int i = 1; i <= mmax; i++)
              Fm[i] = Fm[i - 1] * numbers_.ihalf[i] * one_over_x; // see Eq. (9.8.13)
            return;
          }
        }

        // since Tcrit grows with mmax, this condition only needs to be determined once
        if (T > T_crit_[mmax]) {
          double pow_two_T_to_minusjp05 = std::pow(two_T, -mmax - 0.5);
          for (int m = mmax; m >= mmin; --m) {
            /*--- Asymptotic formula ---*/
            Fm[m] = numbers_.df[2 * m] * sqrt_pio2 * pow_two_T_to_minusjp05;
            pow_two_T_to_minusjp05 *= two_T;
          }
        }
        else
        {
          const int T_ind = (int) (0.5 + T * oodelT_);
          const Real h = T_ind * delT_ - T;
          const Real* F_row = grid_[T_ind] + mmin;

#if defined (__AVX__)
          libint2::simd::VectorAVXDouble h0123, h4567;
          if (INTERPOLATION_ORDER == 3 || INTERPOLATION_ORDER == 7) {
            const double h2 = h*h*oon[2];
            const double h3 = h2*h*oon[3];
            h0123 = libint2::simd::VectorAVXDouble (1.0, h, h2, h3);
            if (INTERPOLATION_ORDER == 7) {
              const double h4 = h3*h*oon[4];
              const double h5 = h4*h*oon[5];
              const double h6 = h5*h*oon[6];
              const double h7 = h6*h*oon[7];
              h4567 = libint2::simd::VectorAVXDouble (h4, h5, h6, h7);
            }
          }
          //          libint2::simd::VectorAVXDouble h0123(1.0);
          //          libint2::simd::VectorAVXDouble h4567(1.0);
#endif

          int m = mmin;
          if (mmax-m >=1) {
            const int unroll_size = 2;
            const int m_fence = (mmax + 2 - unroll_size);
            for(; m<m_fence; m+=unroll_size, F_row+=unroll_size) {

#if defined(__AVX__)
              if (INTERPOLATION_ORDER == 3 || INTERPOLATION_ORDER == 7) {
                 libint2::simd::VectorAVXDouble fr0_0123; fr0_0123.load(F_row);
                 libint2::simd::VectorAVXDouble fr1_0123; fr1_0123.load(F_row+1);
                 libint2::simd::VectorSSEDouble fm01 = horizontal_add(fr0_0123*h0123, fr1_0123*h0123);
                 if (INTERPOLATION_ORDER == 7) {
                   libint2::simd::VectorAVXDouble fr0_4567; fr0_4567.load(F_row+4);
                   libint2::simd::VectorAVXDouble fr1_4567; fr1_4567.load(F_row+5);
                   fm01 += horizontal_add(fr0_4567*h4567, fr1_4567*h4567);
                 }
                 fm01.convert(&Fm[m]);
              }
              else {
#endif
              Real total0 = 0.0, total1 = 0.0;
              for(int i=INTERPOLATION_ORDER; i>=1; --i) {
                total0 = oon[i]*h*(F_row[i] + total0);
                total1 = oon[i]*h*(F_row[i+1] + total1);
              }
              Fm[m] = F_row[0] + total0;
              Fm[m+1] = F_row[1] + total1;
#if defined(__AVX__)
              }
#endif
            }
          } // unroll_size = 2
          if (m<=mmax) { // unroll_size = 1
#if defined(__AVX__)
            if (INTERPOLATION_ORDER == 3 || INTERPOLATION_ORDER == 7) {
              libint2::simd::VectorAVXDouble fr0123; fr0123.load(F_row);
              if (INTERPOLATION_ORDER == 7) {
                libint2::simd::VectorAVXDouble fr4567; fr4567.load(F_row+4);
//                libint2::simd::VectorSSEDouble fm = horizontal_add(fr0123*h0123, fr4567*h4567);
//                Fm[m] = horizontal_add(fm);
                Fm[m] = horizontal_add(fr0123*h0123 + fr4567*h4567);
              }
              else { // INTERPOLATION_ORDER == 3
                Fm[m] = horizontal_add(fr0123*h0123);
              }
            }
            else {
#endif
            Real total = 0.0;
            for(int i=INTERPOLATION_ORDER; i>=1; --i) {
              total = oon[i]*h*(F_row[i] + total);
            }
            Fm[m] = F_row[0] + total;
#if defined(__AVX__)
            }
#endif
          } // unroll_size = 1

          // check against the reference value
//          if (false) {
//            double refvalue = FmEval_Reference2<double>::eval(T, mmax); // compute F(T) with m=mmax
//            if (abs(refvalue - Fm[mmax]) > 1e-14) {
//              std::cout << "T = " << T << " m = " << mmax << " cheb = "
//                  << Fm[mmax] << " ref = " << refvalue << std::endl;
//            }
//          }

        } // if T < T_crit

        /*------------------------------------
         And then do downward recursion in j
         ------------------------------------*/
        if (INTERPOLATION_AND_RECURSION && mmin > 0) {
          const Real exp_mT = std::exp(-T);
          for (int m = mmin - 1; m >= 0; --m) {
            Fm[m] = (exp_mT + two_T * Fm[m+1]) * numbers_.twoi1[m];
          }
        }
      }

    private:
      Real **grid_; /* Table of "exact" Fm(T) values. Row index corresponds to
       values of T (max_T+1 rows), column index to values
       of m (max_m+1 columns) */
      Real delT_; /* The step size for T, depends on cutoff */
      Real oodelT_; /* 1.0 / delT_, see above */
      Real cutoff_; /* Tolerance cutoff used in all computations of Fm(T) */
      int max_m_; /* Maximum value of m in the table, depends on cutoff
       and the number of terms in Taylor interpolation */
      int max_T_; /* Maximum index of T in the table, depends on cutoff
       and m */
      Real *T_crit_; /* Maximum T for each row, depends on cutoff;
       for a given m and T_idx <= max_T_idx[m] use Taylor interpolation,
       for a given m and T_idx > max_T_idx[m] use the asymptotic formula */

      ExpensiveNumbers<double> numbers_;

      /**
       * Power series estimate of the error introduced by replacing
       * \f$ F_m(T) = \int_0^1 \exp(-T t^2) t^{2 m} \, \mathrm{d} t \f$ with analytically
       * integrable \f$ \int_0^\infty \exp(-T t^2) t^{2 m} \, \mathrm{d} t = \frac{(2m-1)!!}{2^{m+1}} \sqrt{\frac{\pi}{T^{2m+1}}} \f$
       * @param m
       * @param T
       * @return the error estimate
       */
      static double truncation_error(unsigned int m, double T) {
        const double m2= m * m;
        const double m3= m2 * m;
        const double m4= m2 * m2;
        const double T2= T * T;
        const double T3= T2 * T;
        const double T4= T2 * T2;
        const double T5= T2 * T3;

        const double result = exp(-T) * (105 + 16*m4 + 16*m3*(T - 8) - 30*T + 12*T2
            - 8*T3 + 16*T4 + 8*m2*(43 - 9*T + 2*T2) +
            4*m*(-88 + 23*T - 8*T2 + 4*T3))/(32*T5);
        return result;
      }
      /**
       * Leading-order estimate of the error introduced by replacing
       * \f$ F_m(T) = \int_0^1 \exp(-T t^2) t^{2 m} \, \mathrm{d} t \f$ with analytically
       * integrable \f$ \int_0^\infty \exp(-T t^2) t^{2 m} \, \mathrm{d} t = \frac{(2m-1)!!}{2^{m+1}} \sqrt{\frac{\pi}{T^{2m+1}}} \f$
       * @param m
       * @param T
       * @return the error estimate
       */
      static double truncation_error(double T) {
        const double result = exp(-T) /(2*T);
        return result;
      }
  };


  namespace detail {
  constexpr double pow10(long exp) {
    return (exp == 0) ? 1. : ((exp > 0) ? 10. * pow10(exp-1) : 0.1 * pow10(exp+1));
  }
  }

  //////////////////////////////////////////////////////////
  /// core integral for Yukawa and exponential interactions
  //////////////////////////////////////////////////////////

  /**
   * Evaluates core integral for Gaussian integrals over the Yukawa potential \f$ \exp(- \zeta r) / r \f$ and
   * the exponential interaction \f$ \exp(- \zeta r) \f$
   * @tparam Real real type
   * @warning only @p Real = double is supported
   * @warning guarantees absolute precision of only about 1e-14
   */
  template<typename Real = double>
  struct TennoGmEval {

  private:

    #include <libint2/tenno_cheb.h>

      static_assert(std::is_same<Real,double>::value, "TennoGmEval only supports double as the real type");

      static const int mmin_ = -1;
      static constexpr const Real Tmax = (1 << cheb_table_tmaxlog2); //!< critical value of T above which use upward recursion
      static constexpr const Real Umax = detail::pow10(cheb_table_umaxlog10); //!< max value of U for which to interpolate
      static constexpr const Real Umin = detail::pow10(cheb_table_uminlog10); //!< min value of U for which to interpolate
      static constexpr const std::size_t ORDERp1 = interpolation_order + 1;
      static constexpr const Real maxabsprecision = 1.4e-14;  //!< guaranteed abs precision of the interpolation table for m>0

  public:
      /// \param m_max maximum value of the Gm function index
      /// \param precision the desired *absolute* precision (relative precision for most intervals will be below epsilon, but for large T/U values and high m relative precision is low
      /// \throw std::invalid_argument if \c m_max is greater than \c cheb_table_mmax (see tenno_cheb.h)
      /// \throw std::invalid_argument if \c precision is smaller than \c maxabsprecision
      TennoGmEval(unsigned int mmax, Real precision = -1) :
        mmax_(mmax), precision_(precision),
        numbers_() {
#if defined(__x86_64__) || defined(_M_X64) || defined(__i386) || defined(_M_IX86)
# if !defined(__AVX__) && defined(NDEBUG)
        if (libint2::verbose()) {
          static bool printed_performance_warning = false;
          if (!printed_performance_warning) {
            libint2::verbose_stream()
                << "libint2::TennoGmEval on x86(-64) platforms needs AVX support for best performance"
                << std::endl;
            printed_performance_warning = true;
          }
        }
# endif
#endif

//        if (precision_ < maxabsprecision)
//          throw std::invalid_argument(
//              std::string(
//                  "TennoGmEval does not support precision smaller than ") +
//              std::to_string(maxabsprecision));

        if (mmax > cheb_table_mmax)
          throw std::invalid_argument(
              "TennoGmEval::init() : requested mmax exceeds the "
              "hard-coded mmax");
        init_table();
      }

      ~TennoGmEval() {
        if (c_ != nullptr)
          free(c_);
      }

      /// Singleton interface allows to manage the lone instance; adjusts max m values as needed in thread-safe fashion
      static std::shared_ptr<const TennoGmEval> instance(int m_max, double = 0) {

        assert(m_max >= 0);
        // thread-safe per C++11 standard [6.7.4]
        static auto instance_ = std::make_shared<const TennoGmEval>(m_max);

        while (instance_->max_m() < m_max) {
          static std::mutex mtx;
          std::lock_guard<std::mutex> lck(mtx);
          if (instance_->max_m() < m_max) {
            auto new_instance = std::make_shared<const TennoGmEval>(m_max);
            instance_ = new_instance;
          }
        }

        return instance_;
      }

      unsigned int max_m() const { return mmax_; }
      /// @return the precision with which this object can compute the result
      Real precision() const { return precision_; }

      /// @param[in] Gm pointer to array of @c mmax+1 @c Real elements, on
      ///               return this contains the core integral for Yukawa
      ///               interaction, \f$ \exp(-zeta r_{12})/r_{12} \f$ ,
      ///               namely \f$ G_{m}(T,U) = \int_0^1 t^{2m} \exp(U(1-t^{-2}) - Tt^2) dt, m \in [0,m_\mathrm{max}] \f$, where
      ///               \f$ T = \rho |\vec{P} - \vec{Q}|^2 \f$ and
      ///               \f$ U = \zeta^2 / (4 \rho) \f$
      /// @param[in] one_over_rho \f$ 1/\rho \f$
      /// @param[in] T \f$ T \f$
      /// @param[in] mmax \f$ m_\mathrm{max} \f$
      /// @param[in] zeta \f$ \zeta \f$
      void eval_yukawa(Real* Gm, Real one_over_rho, Real T, size_t mmax, Real zeta) const {
        assert(mmax <= mmax_);
        assert(T >= 0);
        const auto U = 0.25 * zeta * zeta * one_over_rho;
        assert(U >= 0);
        if (T > Tmax || U < Umin) {
          eval_Gm_urr(Gm, T, U, mmax, 0); // no need for G_-1
        } else {
          interpolate_Gm<false>(Gm, T, U, 0, mmax);
        }
      }
      /// @param[in] Gm pointer to array of @c mmax+1 @c Real elements, on
      ///               return this contains the core integral for Slater-type
      ///               geminal, \f$ \exp(-zeta r_{12}) \f$ ,
      ///               namely \f$ \sqrt{U} (G_{m-1}(T,U) - G_m(T,U)), m \in [0,m_\mathrm{max}] \f$ where\
      //                \f$ G_{m}(T,U) = \int_0^1 t^{2m} \exp(U(1-t^{-2}) - Tt^2) dt \f$, where
      ///               \f$ T = \rho |\vec{P} - \vec{Q}|^2 \f$ and
      ///               \f$ U = \zeta^2 / (4 \rho) \f$
      /// @param[in] one_over_rho \f$ 1/\rho \f$
      /// @param[in] T \f$ T \f$
      /// @param[in] mmax \f$ m_\mathrm{max} \f$
      /// @param[in] zeta \f$ \zeta \f$
      void eval_slater(Real* Gm, Real one_over_rho, Real T, size_t mmax, Real zeta) const {
        assert(mmax <= mmax_);
        assert(T >= 0);
        const auto U = 0.25 * zeta * zeta * one_over_rho;
        assert(U > 0);  // integral factorizes into 2 overlaps for U = 0
        const auto zeta_over_two_rho = 0.5 * zeta * one_over_rho;
        if (T > Tmax) {
          eval_Gm_urr(Gm, T, U, mmax, -1);
        } else {
          interpolate_Gm<true>(Gm, T, U, zeta_over_two_rho, mmax);
        }
      }

  private:

      /// Computes G_0 and (optionally) G_{-1}
      /// @tparam compute_Gm1, if true, compute G_0 and G_{-1}, otherwise G_0 only
      /// @param[in] T the value of \f$ T \f$
      /// @param[in] U the value of \f$ U \f$
      /// @return if @c compute_Gm1==true return {G_0,G_{-1}}, otherwise {G_0,0}
      template <bool compute_Gm1>
      static inline std::tuple<Real,Real> eval_G0_and_maybe_Gm1(Real T, Real U) {
        assert(T >= 0);
        assert(U >= 0);
        const Real sqrtPi(1.7724538509055160272981674833411451827975494561224);

        Real G_m1 = 0;
        Real G_0 = 0;
        if (U == 0) {  // \sqrt{U} G_{-1} is finite, need to handle that case separately
          assert(compute_Gm1 == false);
          if (T < std::numeric_limits<Real>::epsilon()) {
            G_0 = 1;
          }
          else {
            const Real sqrt_T = sqrt(T);
            const Real sqrtPi_over_2 = sqrtPi * 0.5;
            G_0 = sqrtPi_over_2 * erf(sqrt_T) / sqrt_T;
          }
        }
        else if (T > std::numeric_limits<Real>::epsilon()) {  // U > 0
          const Real sqrt_U = sqrt(U);
          const Real sqrt_T = sqrt(T);
          const Real oo_sqrt_T = 1 / sqrt_T;
          const Real oo_sqrt_U = compute_Gm1 ? (1 / sqrt_U) : 0;
          const Real kappa = sqrt_U - sqrt_T;
          const Real lambda = sqrt_U + sqrt_T;
          const Real sqrtPi_over_4 = sqrtPi * 0.25;
          const Real pfac = sqrtPi_over_4;
          const Real erfc_k = exp(kappa * kappa - T) * erfc(kappa);
          const Real erfc_l = exp(lambda * lambda - T) * erfc(lambda);

          G_m1 = compute_Gm1 ? pfac * (erfc_k + erfc_l) * oo_sqrt_U : 0.0;
          G_0 = pfac * (erfc_k - erfc_l) * oo_sqrt_T;
        }
        else {  // T = 0, U > 0
          const Real exp_U = exp(U);
          const Real sqrt_U = sqrt(U);
          if (compute_Gm1) {
            const Real sqrtPi_over_2 = sqrtPi * 0.5;
            const Real oo_sqrt_U = compute_Gm1 ? (1 / sqrt_U) : 0;

            G_m1 = exp_U * sqrtPi_over_2 * erfc(sqrt_U) * oo_sqrt_U;
          }
          G_0 = 1 - exp_U * sqrtPi * erfc(sqrt_U) * sqrt_U;
        }

        return std::make_tuple(G_0, G_m1);
      }

      /// Computes core integrals by upward recursion from \f$ G_{-1} (T,U) \f$ and \f$ G_0 (T,U) \f$
      /// this is unstable for small T, so use when interpolation does not apply.
      /// @param[out] Gm \f$ G_m(T,U), m=mmin..mmax \f$ , i.e. @c Gm[m] \f$ = G_{\mathrm{mmin} + m}(T,U) \f$
      /// @param[in] T the value of \f$ T \f$
      /// @param[in] U the value of \f$ U \f$
      /// @param[in] mmax the maximum value of \f$ m \f$
      /// @param[in] mmin the minimum value of \f$ m \f$ ; the only valid values are 0 and -1
      static void eval_Gm_urr(Real* Gm, Real T, Real U, size_t mmax, long mmin) {
        assert(mmin == 0 || mmin == -1);
        assert(T > 0);
        assert(U > 0);

        const Real sqrt_U = sqrt(U);
        const Real sqrt_T = sqrt(T);
        const Real oo_sqrt_T = 1 / sqrt_T;
        const Real oo_sqrt_U = 1 / sqrt_U;
        const Real kappa = sqrt_U - sqrt_T;
        const Real lambda = sqrt_U + sqrt_T;
        const Real sqrtPi_over_4(0.44311346272637900682454187083528629569938736403060);
        const Real pfac = sqrtPi_over_4;
        const Real erfc_k = exp(kappa*kappa - T) * erfc(kappa);
        const Real erfc_l = exp(lambda*lambda - T) * erfc(lambda);

        Real G_m1_value;
        Real& G_m1 = (mmin == -1) ? Gm[0] : G_m1_value;
        G_m1 = pfac * (erfc_k + erfc_l) * oo_sqrt_U;
        Real& G_0 = (mmin == -1) ? Gm[1] : Gm[0];
        G_0 = pfac * (erfc_k - erfc_l) * oo_sqrt_T;

        if (mmax > 0) {

          // first application of URR
          const Real oo_two_T = 0.5 / T;
          const Real two_U = 2.0 * U;
          const Real exp_mT = exp(-T);

          Real* Gmm1 = &G_m1;
          Real* Gm0 = &G_0;
          Real* Gmp1 = Gm0 + 1;
          for(unsigned int m=1, two_m_minus_1=1; m<=mmax; ++m, two_m_minus_1+=2, ++Gmp1) {
            *Gmp1 = oo_two_T * ( two_m_minus_1 * *Gm0 + two_U * *Gmm1 - exp_mT);
            Gmm1 = Gm0;
            Gm0 = Gmp1;
          }
        }

        return;
      }

      /// compute core integrals by Chebyshev interpolation
      /// @tparam[in] exp if true, will compute core integrals of the exponential operator, otherwise will compute Yukawa
      /// @param[out] Gm_vec if @c exp==false Gm_vec[m]=\f$ G_m(T,U), m=0..mmax \f$, otherwise Gm_vec[m]=\f$ \frac{\zeta}{2 \rho} \left( G_{m-1}(T,U) - G_m(T,U) \right), m=0..mmax \f$ ; must be at least @c mmax+1 elements long
      /// @param[in] T the value of \f$ T \f$
      /// @param[in] U the value of \f$ U \f$
      /// @param[in] zeta_over_two_rho the value of \f$ \frac{\zeta}{2 \rho} \f$, only used if @c exp==true
      /// @param[in] mmax the maximum value of \f$ m \f$
      template <bool exp>
      void interpolate_Gm(Real* Gm_vec, Real T, Real U, Real zeta_over_two_rho, long mmax) const {
        assert(T >= 0);
        assert(U >= Umin && U <= Umax);

        // maps x in [0,2] to [-1/2,1/2] in a linear fashion
        auto linear_map_02 = [](Real x) {
          assert(x >= 0 && x <= 2);
          return (x - 1) * 0.5;
        };
        // maps x in [a, 2*a] to [-1/2,1/2] in a logarithmic fashion
        auto log2_map = [](Real x, Real one_over_a) {
          return std::log2(x * one_over_a) - 0.5;
        };
        // maps x in [a, 10*a] to [-1/2,1/2] in a logarithmic fashion
        auto log10_map = [](Real x, Real one_over_a) {
          return std::log10(x * one_over_a) - 0.5;
        };

        // which interval does this T fall into?
        const int Tint = T < 2 ? 0 : int(std::floor(std::log2(T)));  assert(Tint >= 0 && Tint < 10);
        // precomputed 1 / 2^K
        const constexpr Real one_over_2K[] = {1, 0.5, 0.25, 0.125, 0.0625, 0.03125, 0.015625, 0.0078125, 0.00390625, .001953125};
        // which interval does this U fall into?
        const int Uint = int(std::floor(std::log10(U / Umin))); assert(Uint >= 0 && Uint < 10);
        // precomputed 1 / 10^(cheb_table_uminlog10 + K)
        const constexpr Real one_over_10K[] = {1. / detail::pow10(cheb_table_uminlog10),
                                               1. / detail::pow10(cheb_table_uminlog10+1),
                                               1. / detail::pow10(cheb_table_uminlog10+2),
                                               1. / detail::pow10(cheb_table_uminlog10+3),
                                               1. / detail::pow10(cheb_table_uminlog10+4),
                                               1. / detail::pow10(cheb_table_uminlog10+5),
                                               1. / detail::pow10(cheb_table_uminlog10+6),
                                               1. / detail::pow10(cheb_table_uminlog10+7),
                                               1. / detail::pow10(cheb_table_uminlog10+8),
                                               1. / detail::pow10(cheb_table_uminlog10+9)};
        const Real t = Tint == 0 ? linear_map_02(T) : log2_map(T, one_over_2K[Tint]); // this ranges from -0.5 to 0.5
        const Real u = log10_map(U, one_over_10K[Uint]); // this ranges from -0.5 to 0.5

        const int interval = Tint * 10 + Uint;

#if defined(__AVX__)
        assert(ORDERp1 == 16);
        const auto t2 = t*t;
        const auto t3 = t2*t;
        const auto t4 = t2*t2;
        const auto t5 = t2*t3;
        const auto t6 = t3*t3;
        const auto t7 = t3*t4;
        const auto t8 = t4*t4;
        const auto t9 = t4*t5;
        const auto t10 = t5*t5;
        const auto t11 = t5*t6;
        const auto t12 = t6*t6;
        const auto t13 = t6*t7;
        const auto t14 = t7*t7;
        const auto t15 = t7*t8;
        const auto u2 = u*u;
        const auto u3 = u2*u;
        const auto u4 = u2*u2;
        const auto u5 = u2*u3;
        const auto u6 = u3*u3;
        const auto u7 = u3*u4;
        const auto u8 = u4*u4;
        const auto u9 = u4*u5;
        const auto u10 = u5*u5;
        const auto u11 = u5*u6;
        const auto u12 = u6*u6;
        const auto u13 = u6*u7;
        const auto u14 = u7*u7;
        const auto u15 = u7*u8;
        libint2::simd::VectorAVXDouble u0vec(1., u, u2, u3);
        libint2::simd::VectorAVXDouble u1vec(u4, u5, u6, u7);
        libint2::simd::VectorAVXDouble u2vec(u8, u9, u10, u11);
        libint2::simd::VectorAVXDouble u3vec(u12, u13, u14, u15);
#endif // AVX

        Real Gmm1 = 0.0; // will track the previous value, only used for exp==false

        constexpr const bool compute_Gmm10 = true;
        long mmin_interp;
        if (compute_Gmm10) {
          // precision of interpolation for m=-1,0 can be insufficient, just evaluate explicitly
          Real G0;
          std::tie(exp ? G0 : Gm_vec[0], Gmm1) = eval_G0_and_maybe_Gm1<exp>(T, U);
          if (exp) {
            Gm_vec[0] = (Gmm1 - G0) * zeta_over_two_rho;
            Gmm1 = G0;
          }
          mmin_interp = 1;
        }
        else
          mmin_interp = (exp == true) ? -1 : 0;

        // now compute the rest
        for(long m=mmin_interp; m<=mmax; ++m){
            const Real *c_tuint = c_ + (ORDERp1) * (ORDERp1) * (interval * (mmax_ - mmin_ + 1) + (m - mmin_)); // ptr to the interpolation data for m=mmin
#if defined(__AVX__)
            libint2::simd::VectorAVXDouble c00v, c01v, c02v, c03v, c10v, c11v, c12v, c13v,
                                           c20v, c21v, c22v, c23v, c30v, c31v, c32v, c33v,
                                           c40v, c41v, c42v, c43v, c50v, c51v, c52v, c53v,
                                           c60v, c61v, c62v, c63v, c70v, c71v, c72v, c73v;
            libint2::simd::VectorAVXDouble c80v, c81v, c82v, c83v, c90v, c91v, c92v, c93v,
                                           ca0v, ca1v, ca2v, ca3v, cb0v, cb1v, cb2v, cb3v,
                                           cc0v, cc1v, cc2v, cc3v, cd0v, cd1v, cd2v, cd3v,
                                           ce0v, ce1v, ce2v, ce3v, cf0v, cf1v, cf2v, cf3v;
            c00v.load_aligned(c_tuint);                c01v.load_aligned((c_tuint+4));
            c02v.load_aligned(c_tuint+8);              c03v.load_aligned((c_tuint+12));
            libint2::simd::VectorAVXDouble t0vec(1, 1, 1, 1);
            libint2::simd::VectorAVXDouble t0 = t0vec * (c00v * u0vec + c01v * u1vec + c02v * u2vec + c03v * u3vec);
            c10v.load_aligned(c_tuint      +ORDERp1);  c11v.load_aligned((c_tuint+4)   +ORDERp1);
            c12v.load_aligned((c_tuint+8)  +ORDERp1);  c13v.load_aligned((c_tuint+12)  +ORDERp1);
            libint2::simd::VectorAVXDouble t1vec(t, t, t, t);
            libint2::simd::VectorAVXDouble t1 = t1vec * (c10v * u0vec + c11v * u1vec + c12v * u2vec + c13v * u3vec);
            c20v.load_aligned(c_tuint    +2*ORDERp1);  c21v.load_aligned((c_tuint+4) +2*ORDERp1);
            c22v.load_aligned((c_tuint+8)+2*ORDERp1);  c23v.load_aligned((c_tuint+12)+2*ORDERp1);
            libint2::simd::VectorAVXDouble t2vec(t2, t2, t2, t2);
            libint2::simd::VectorAVXDouble t2 = t2vec * (c20v * u0vec + c21v * u1vec + c22v * u2vec + c23v * u3vec);
            c30v.load_aligned(c_tuint    +3*ORDERp1);  c31v.load_aligned((c_tuint+4) +3*ORDERp1);
            c32v.load_aligned((c_tuint+8)+3*ORDERp1);  c33v.load_aligned((c_tuint+12)+3*ORDERp1);
            libint2::simd::VectorAVXDouble t3vec(t3, t3, t3, t3);
            libint2::simd::VectorAVXDouble t3 = t3vec * (c30v * u0vec + c31v * u1vec + c32v * u2vec + c33v * u3vec);
            libint2::simd::VectorAVXDouble t0123 = horizontal_add(t0, t1, t2, t3);

            c40v.load_aligned(c_tuint    +4*ORDERp1);  c41v.load_aligned((c_tuint+4) +4*ORDERp1);
            c42v.load_aligned((c_tuint+8)+4*ORDERp1);  c43v.load_aligned((c_tuint+12)+4*ORDERp1);
            libint2::simd::VectorAVXDouble t4vec(t4, t4, t4, t4);
            libint2::simd::VectorAVXDouble t4 = t4vec * (c40v * u0vec + c41v * u1vec + c42v * u2vec + c43v * u3vec);
            c50v.load_aligned(c_tuint    +5*ORDERp1);  c51v.load_aligned((c_tuint+4) +5*ORDERp1);
            c52v.load_aligned((c_tuint+8)+5*ORDERp1);  c53v.load_aligned((c_tuint+12)+5*ORDERp1);
           libint2::simd::VectorAVXDouble t5vec(t5, t5, t5, t5);
            libint2::simd::VectorAVXDouble t5 = t5vec * (c50v * u0vec + c51v * u1vec + c52v * u2vec + c53v * u3vec);
            c60v.load_aligned(c_tuint    +6*ORDERp1);  c61v.load_aligned((c_tuint+4) +6*ORDERp1);
            c62v.load_aligned((c_tuint+8)+6*ORDERp1);  c63v.load_aligned((c_tuint+12)+6*ORDERp1);
            libint2::simd::VectorAVXDouble t6vec(t6, t6, t6, t6);
            libint2::simd::VectorAVXDouble t6 = t6vec * (c60v * u0vec + c61v * u1vec + c62v * u2vec + c63v * u3vec);
            c70v.load_aligned(c_tuint    +7*ORDERp1);  c71v.load_aligned((c_tuint+4) +7*ORDERp1);
            c72v.load_aligned((c_tuint+8)+7*ORDERp1);  c73v.load_aligned((c_tuint+12)+7*ORDERp1);
            libint2::simd::VectorAVXDouble t7vec(t7, t7, t7, t7);
            libint2::simd::VectorAVXDouble t7 = t7vec * (c70v * u0vec + c71v * u1vec + c72v * u2vec + c73v * u3vec);
            libint2::simd::VectorAVXDouble t4567 = horizontal_add(t4, t5, t6, t7);

            c80v.load_aligned(c_tuint    +8*ORDERp1);  c81v.load_aligned((c_tuint+4) +8*ORDERp1);
            c82v.load_aligned((c_tuint+8)+8*ORDERp1);  c83v.load_aligned((c_tuint+12)+8*ORDERp1);
            libint2::simd::VectorAVXDouble t8vec(t8, t8, t8, t8);
            libint2::simd::VectorAVXDouble t8 = t8vec * (c80v * u0vec + c81v * u1vec + c82v * u2vec + c83v * u3vec);
            c90v.load_aligned(c_tuint    +9*ORDERp1);  c91v.load_aligned((c_tuint+4) +9*ORDERp1);
            c92v.load_aligned((c_tuint+8)+9*ORDERp1);  c93v.load_aligned((c_tuint+12)+9*ORDERp1);
            libint2::simd::VectorAVXDouble t9vec(t9, t9, t9, t9);
            libint2::simd::VectorAVXDouble t9 = t9vec * (c90v * u0vec + c91v * u1vec + c92v * u2vec + c93v * u3vec);
            ca0v.load_aligned(c_tuint    +10*ORDERp1);  ca1v.load_aligned((c_tuint+4) +10*ORDERp1);
            ca2v.load_aligned((c_tuint+8)+10*ORDERp1);  ca3v.load_aligned((c_tuint+12)+10*ORDERp1);
            libint2::simd::VectorAVXDouble tavec(t10, t10, t10, t10);
            libint2::simd::VectorAVXDouble ta = tavec * (ca0v * u0vec + ca1v * u1vec + ca2v * u2vec + ca3v * u3vec);
            cb0v.load_aligned(c_tuint    +11*ORDERp1);  cb1v.load_aligned((c_tuint+4) +11*ORDERp1);
            cb2v.load_aligned((c_tuint+8)+11*ORDERp1);  cb3v.load_aligned((c_tuint+12)+11*ORDERp1);
            libint2::simd::VectorAVXDouble tbvec(t11, t11, t11, t11);
            libint2::simd::VectorAVXDouble tb = tbvec * (cb0v * u0vec + cb1v * u1vec + cb2v * u2vec + cb3v * u3vec);
            libint2::simd::VectorAVXDouble t89ab = horizontal_add(t8, t9, ta, tb);

            cc0v.load_aligned(c_tuint    +12*ORDERp1);  cc1v.load_aligned((c_tuint+4) +12*ORDERp1);
            cc2v.load_aligned((c_tuint+8)+12*ORDERp1);  cc3v.load_aligned((c_tuint+12)+12*ORDERp1);
            libint2::simd::VectorAVXDouble tcvec(t12, t12, t12, t12);
            libint2::simd::VectorAVXDouble tc = tcvec * (cc0v * u0vec + cc1v * u1vec + cc2v * u2vec + cc3v * u3vec);
            cd0v.load_aligned(c_tuint    +13*ORDERp1);  cd1v.load_aligned((c_tuint+4) +13*ORDERp1);
            cd2v.load_aligned((c_tuint+8)+13*ORDERp1);  cd3v.load_aligned((c_tuint+12)+13*ORDERp1);
            libint2::simd::VectorAVXDouble tdvec(t13, t13, t13, t13);
            libint2::simd::VectorAVXDouble td = tdvec * (cd0v * u0vec + cd1v * u1vec + cd2v * u2vec + cd3v * u3vec);
            ce0v.load_aligned(c_tuint    +14*ORDERp1);  ce1v.load_aligned((c_tuint+4) +14*ORDERp1);
            ce2v.load_aligned((c_tuint+8)+14*ORDERp1);  ce3v.load_aligned((c_tuint+12)+14*ORDERp1);
            libint2::simd::VectorAVXDouble tevec(t14, t14, t14, t14);
            libint2::simd::VectorAVXDouble te = tevec * (ce0v * u0vec + ce1v * u1vec + ce2v * u2vec + ce3v * u3vec);
            cf0v.load_aligned(c_tuint    +15*ORDERp1);  cf1v.load_aligned((c_tuint+4)+15*ORDERp1);
            cf2v.load_aligned((c_tuint+8)+15*ORDERp1);  cf3v.load_aligned((c_tuint+4)+15*ORDERp1);
            libint2::simd::VectorAVXDouble tfvec(t15, t15, t15, t15);
            libint2::simd::VectorAVXDouble tf = tfvec * (cf0v * u0vec + cf1v * u1vec + cf2v * u2vec + cf3v * u3vec);
            libint2::simd::VectorAVXDouble tcdef = horizontal_add(tc, td, te, tf);

            auto tall = horizontal_add(t0123, t4567, t89ab, tcdef);
            const auto Gm = horizontal_add(tall);
#else  // AVX
            // no AVX, use explicit loops (probably slow)
            double uvec[16];
            double tvec[16];
            uvec[0] = 1;
            tvec[0] = 1;
            for(int i=1; i!=16; ++i) {
              uvec[i] = uvec[i-1] * u;
              tvec[i] = tvec[i-1] * t;
            }
            double Gm = 0.0;
            for(int i=0, ij=0; i!=16; ++i) {
              for (int j = 0; j != 16; ++j, ++ij) {
                Gm += c_tuint[ij] * tvec[i] * uvec[j];
              }
            }
#endif // AVX

            if (exp == false) {
              Gm_vec[m] = Gm;
            }
            else {
              if (m != -1) {
                Gm_vec[m] = (Gmm1 - Gm) * zeta_over_two_rho;
              }
              Gmm1 = Gm;
            }
        }

        return;
      }

    private:
      unsigned int mmax_;
      Real precision_;
      ExpensiveNumbers<Real> numbers_;
      Real* c_ = nullptr;

      void init_table() {

        // get memory
        void* result;
        int status = posix_memalign(&result, std::max(sizeof(Real), 32ul), (mmax_ - mmin_ + 1) * cheb_table_nintervals * ORDERp1 * ORDERp1 * sizeof(Real));
        if (status != 0) {
          if (status == EINVAL)
            throw std::logic_error(
              "TennoGmEval::init() : posix_memalign failed, alignment must be a power of 2 at least as large as sizeof(void *)");
          if (status == ENOMEM)
            throw std::bad_alloc();
          abort();  // should be unreachable
        }
        c_ = static_cast<Real*>(result);

        // copy contents of static table into c
        // need all intervals
        for (std::size_t iv = 0; iv < cheb_table_nintervals; ++iv) {
          // but only values of m up to mmax
          std::copy(cheb_table[iv], cheb_table[iv]+((mmax_-mmin_)+1)*ORDERp1*ORDERp1, c_+(iv*((mmax_-mmin_)+1))*ORDERp1*ORDERp1);
        }
      }


  };  // TennoGmEval

#if LIBINT2_CONSTEXPR_STATICS
  template <typename Real>
  constexpr
  double TennoGmEval<Real>::cheb_table[TennoGmEval<Real>::cheb_table_nintervals][(TennoGmEval<Real>::cheb_table_mmax+2)*(TennoGmEval<Real>::interpolation_order+1)*(TennoGmEval<Real>::interpolation_order+1)];
#else
  // clang needs an explicit specifalization declaration to avoid warning
#  ifdef __clang__
  template <typename Real>
  double TennoGmEval<Real>::cheb_table[TennoGmEval<Real>::cheb_table_nintervals][(TennoGmEval<Real>::cheb_table_mmax+2)*(TennoGmEval<Real>::interpolation_order+1)*(TennoGmEval<Real>::interpolation_order+1)];
#  endif
#endif

  template<typename Real, int k>
    struct GaussianGmEval;

  namespace detail {

    /// some evaluators need thread-local scratch, but most don't
    template <typename CoreEval> struct CoreEvalScratch {
      CoreEvalScratch(const CoreEvalScratch&) = default;
      CoreEvalScratch(CoreEvalScratch&&) = default;
      explicit CoreEvalScratch(int) { }
    };
    /// GaussianGmEval<Real,-1> needs extra scratch data
    template <typename Real>
    struct CoreEvalScratch<GaussianGmEval<Real, -1>> {
      std::vector<Real> Fm_;
      std::vector<Real> g_i;
      std::vector<Real> r_i;
      std::vector<Real> oorhog_i;
      CoreEvalScratch(const CoreEvalScratch&) = default;
      CoreEvalScratch(CoreEvalScratch&&) = default;
      explicit CoreEvalScratch(int mmax) {
        init(mmax);
      }
      private:
      void init(int mmax) {
        Fm_.resize(mmax+1);
        g_i.resize(mmax+1);
        r_i.resize(mmax+1);
        oorhog_i.resize(mmax+1);
        g_i[0] = 1.0;
        r_i[0] = 1.0;
      }
    };
  } // namespace libint2::detail

  //////////////////////////////////////////////////////////
  /// core integrals r12^k \sum_i \exp(- a_i r_12^2)
  //////////////////////////////////////////////////////////

  /**
   * Evaluates core integral \$ G_m(\rho, T) = \left( - \frac{\partial}{\partial T} \right)^n G_0(\rho,T) \f$,
   * \f$ G_0(\rho,T) = \int \exp(-\rho |\vec{r} - \vec{P} + \vec{Q}|^2) g(r) \, {\rm d}\vec{r} \f$
   * over a general contracted
   * Gaussian geminal \f$ g(r_{12}) = r_{12}^k \sum_i c_i \exp(- a_i r_{12}^2), \quad k = -1, 0, 2 \f$ .
   * The integrals are needed in R12/F12 methods with STG-nG correlation factors.
   * Specifically, for a correlation factor \f$ f(r_{12}) = \sum_i c_i \exp(- a_i r_{12}^2) \f$
   * integrals with the following kernels are needed:
   * <ul>
   *   <li> \f$ f(r_{12}) \f$  (k=0) </li>
   *   <li> \f$ f(r_{12}) / r_{12} \f$  (k=-1) </li>
   *   <li> \f$ f(r_{12})^2 \f$ (k=0, @sa GaussianGmEval::eval ) </li>
   *   <li> \f$ [f(r_{12}), [\hat{T}_1, f(r_{12})]] \f$ (k=2, @sa GaussianGmEval::eval ) </li>
   * </ul>
   *
   * N.B. ``Asymmetric'' kernels, \f$ f(r_{12}) g(r_{12}) \f$ and
   *   \f$ [f(r_{12}), [\hat{T}_1, g(r_{12})]] \f$, where f and g are two different geminals,
   *   can also be handled straightforwardly.
   *
   * \note for more details see DOI: 10.1039/b605188j
   */
  template<typename Real, int k>
  struct GaussianGmEval : private detail::CoreEvalScratch<GaussianGmEval<Real,k>> // N.B. empty-base optimization
  {
#ifndef LIBINT_USER_DEFINED_REAL
      using FmEvalType = libint2::FmEval_Chebyshev7<double>;
#else
      using FmEvalType = libint2::FmEval_Reference<scalar_type>;
#endif

      /**
       * @param[in] mmax the evaluator will be used to compute Gm(T) for 0 <= m <= mmax
       */
      GaussianGmEval(int mmax, Real precision) :
          detail::CoreEvalScratch<GaussianGmEval<Real, k>>(mmax), mmax_(mmax),
          precision_(precision), fm_eval_(),
          numbers_(-1,-1,mmax) {
        assert(k == -1 || k == 0 || k == 2);
        // for k=-1 need to evaluate the Boys function
        if (k == -1) {
          fm_eval_ = FmEvalType::instance(mmax_, precision_);
        }
      }

      ~GaussianGmEval() {
      }

      /// Singleton interface allows to manage the lone instance;
      /// adjusts max m and precision values as needed in thread-safe fashion
      static std::shared_ptr<GaussianGmEval> instance(unsigned int mmax, Real precision = std::numeric_limits<Real>::epsilon()) {
        assert(mmax >= 0);
        assert(precision >= 0);
        // thread-safe per C++11 standard [6.7.4]
        static auto instance_ = std::make_shared<GaussianGmEval>(mmax, precision);

        while (instance_->max_m() < mmax || instance_->precision() > precision) {
          static std::mutex mtx;
          std::lock_guard<std::mutex> lck(mtx);
          if (instance_->max_m() < mmax || instance_->precision() > precision) {
            auto new_instance = std::make_shared<GaussianGmEval>(mmax, precision);
            instance_ = new_instance;
          }
        }

        return instance_;
      }

      /// @return the maximum value of m for which the \f$ G_m(\rho, T) \f$ can be computed with this object
      int max_m() const { return mmax_; }
      /// @return the precision with which this object can compute the Boys function
      Real precision() const { return precision_; }

      /** computes \f$ G_m(\rho, T) \f$ using downward recursion.
       *
       * @warning NOT reentrant if \c k == -1 and C++11 is not available
       *
       * @param[out] Gm array to be filled in with the \f$ Gm(\rho, T) \f$ values, must be at least mmax+1 elements long
       * @param[in] rho
       * @param[in] T
       * @param[in] mmax mmax the maximum value of m for which Boys function will be computed;
       *                 it must be <= the value returned by max_m() (this is not checked)
       * @param[in] geminal the Gaussian geminal for which the core integral \f$ Gm(\rho, T) \f$ is computed; a contracted Gaussian geminal is represented as a vector of pairs, each of which specifies the exponent (first) and contraction coefficient (second) of the primitive Gaussian geminal
       * @param[in] scr if \c k ==-1 and need this to be reentrant, must provide ptr to
       *                the per-thread \c libint2::detail::CoreEvalScratch<GaussianGmEval<Real,-1>> object;
       *                no need to specify \c scr otherwise
       */
      template <typename AnyReal>
      void eval(Real* Gm, Real rho, Real T, size_t mmax,
                const std::vector<std::pair<AnyReal, AnyReal> >& geminal,
                void* scr = 0) {

        std::fill(Gm, Gm+mmax+1, Real(0));

        const auto sqrt_rho = sqrt(rho);
        const auto oo_sqrt_rho = 1/sqrt_rho;
        if (k == -1) {
          void* _scr = (scr == 0) ? this : scr;
          auto& scratch = *(reinterpret_cast<detail::CoreEvalScratch<GaussianGmEval<Real, -1>>*>(_scr));
          for(int i=1; i<=mmax; i++) {
            scratch.r_i[i] = scratch.r_i[i-1] * rho;
          }
        }

        typedef typename std::vector<std::pair<AnyReal, AnyReal> >::const_iterator citer;
        const citer gend = geminal.end();
        for(citer i=geminal.begin(); i!= gend; ++i) {

          const auto gamma = i->first;
          const auto gcoef = i->second;
          const auto rhog = rho + gamma;
          const auto oorhog = 1/rhog;

          const auto gorg = gamma * oorhog;
          const auto rorg = rho * oorhog;
          const auto sqrt_rho_org = sqrt_rho * oorhog;
          const auto sqrt_rhog = sqrt(rhog);
          const auto sqrt_rorg = sqrt_rho_org * sqrt_rhog;

          /// (ss|g12|ss)
          constexpr Real const_SQRTPI_2(0.88622692545275801364908374167057259139877472806119); /* sqrt(pi)/2 */
          const auto SS_K0G12_SS = gcoef * oo_sqrt_rho * const_SQRTPI_2 * rorg * sqrt_rorg * exp(-gorg*T);

          if (k == -1) {
            void* _scr = (scr == 0) ? this : scr;
            auto& scratch = *(reinterpret_cast<detail::CoreEvalScratch<GaussianGmEval<Real, -1>>*>(_scr));

            const auto rorgT = rorg * T;
            fm_eval_->eval(&scratch.Fm_[0], rorgT, mmax);

#if 1
            constexpr Real const_2_SQRTPI(1.12837916709551257389615890312154517);   /* 2/sqrt(pi)     */
            const auto pfac = const_2_SQRTPI * sqrt_rhog * SS_K0G12_SS;
            scratch.oorhog_i[0] = pfac;
            for(int i=1; i<=mmax; i++) {
              scratch.g_i[i] = scratch.g_i[i-1] * gamma;
              scratch.oorhog_i[i] = scratch.oorhog_i[i-1] * oorhog;
            }
            for(int m=0; m<=mmax; m++) {
              Real ssss = 0.0;
              Real* bcm = numbers_.bc[m];
              for(int n=0; n<=m; n++) {
                ssss += bcm[n] * scratch.r_i[n] * scratch.g_i[m-n] * scratch.Fm_[n];
              }
              Gm[m] += ssss * scratch.oorhog_i[m];
            }
#endif
          }

          if (k == 0) {
            auto ss_oper_ss_m = SS_K0G12_SS;
            Gm[0] += ss_oper_ss_m;
            for(int m=1; m<=mmax; ++m) {
              ss_oper_ss_m *= gorg;
              Gm[m] += ss_oper_ss_m;
            }
          }

          if (k == 2) {

            /// (ss|g12*r12^2|ss)
            const auto rorgT = rorg * T;
            const auto SS_K2G12_SS_0 = (1.5 + rorgT) * (SS_K0G12_SS * oorhog);
            const auto SS_K2G12_SS_m1 = rorg * (SS_K0G12_SS * oorhog);

            auto SS_K2G12_SS_gorg_m = SS_K2G12_SS_0 ;
            auto SS_K2G12_SS_gorg_m1 = SS_K2G12_SS_m1;
            Gm[0] += SS_K2G12_SS_gorg_m;
            for(int m=1; m<=mmax; ++m) {
              SS_K2G12_SS_gorg_m *= gorg;
              Gm[m] += SS_K2G12_SS_gorg_m - m * SS_K2G12_SS_gorg_m1;
              SS_K2G12_SS_gorg_m1 *= gorg;
            }
          }

        }

      }

    private:
      int mmax_;
      Real precision_; //< absolute precision
      std::shared_ptr<const FmEvalType> fm_eval_;

      ExpensiveNumbers<Real> numbers_;
  };

  template <typename GmEvalFunction>
  struct GenericGmEval : private GmEvalFunction {

    typedef typename GmEvalFunction::value_type Real;

      GenericGmEval(int mmax, Real precision) : GmEvalFunction(mmax, precision),
          mmax_(mmax), precision_(precision) {}

      static std::shared_ptr<const GenericGmEval> instance(int mmax, Real precision = std::numeric_limits<Real>::epsilon()) {
        return std::make_shared<const GenericGmEval>(mmax, precision);
      }

      template <typename Real, typename... ExtraArgs>
      void eval(Real* Gm, Real rho, Real T, int mmax, ExtraArgs... args) const {
        assert(mmax <= mmax_);
        (GmEvalFunction(*this))(Gm, rho, T, mmax, std::forward<ExtraArgs>(args)...);
      }

      /// @return the maximum value of m for which the \f$ G_m(\rho, T) \f$ can be computed with this object
      int max_m() const { return mmax_; }
      /// @return the precision with which this object can compute the Boys function
      Real precision() const { return precision_; }

    private:
      int mmax_;
      Real precision_;
  };

  // these Gm engines need extra scratch data
  namespace os_core_ints {
  template <typename Real, int K> struct r12_xx_K_gm_eval;
  template <typename Real> struct erfc_coulomb_gm_eval;
  }

  namespace detail {
  /// r12_xx_K_gm_eval<1> needs extra scratch data
  template <typename Real>
  struct CoreEvalScratch<os_core_ints::r12_xx_K_gm_eval<Real, 1>> {
    std::vector<Real> Fm_;
    CoreEvalScratch(const CoreEvalScratch&) = default;
    CoreEvalScratch(CoreEvalScratch&&) = default;
    // need to store Fm(T) for m = 0 .. mmax+1
    explicit CoreEvalScratch(int mmax) { Fm_.resize(mmax+2); }
  };
  /// erfc_coulomb_gm_eval needs extra scratch data
  template <typename Real>
  struct CoreEvalScratch<os_core_ints::erfc_coulomb_gm_eval<Real>> {
    std::vector<Real> Fm_;
    CoreEvalScratch(const CoreEvalScratch&) = default;
    CoreEvalScratch(CoreEvalScratch&&) = default;
    // need to store Fm(T) for m = 0 .. mmax
    explicit CoreEvalScratch(int mmax) { Fm_.resize(mmax+1); }
  };
  }

  /// Obara-Saika core ints code
  namespace os_core_ints {

    /// core integral evaluator delta function kernels
  template <typename Real>
  struct delta_gm_eval {
    typedef Real value_type;

    delta_gm_eval(unsigned int, Real) {}
    void operator()(Real* Gm, Real rho, Real T, int mmax) const {
      constexpr static auto one_over_two_pi = 1.0 / (2.0 * M_PI);
      const auto G0 = exp(-T) * rho * one_over_two_pi;
      std::fill(Gm, Gm + mmax + 1, G0);
    }
  };

  /// core integral evaluator for \f$ r_{12}^K \f$ kernel
  /// @tparam K currently supported \c K=1 (use Boys engine directly for \c K=-1)
  /// @note need extra scratch for Boys function values when \c K==1,
  ///       the Gm vector is not long enough for scratch

  template <typename Real, int K>
  struct r12_xx_K_gm_eval;

  template <typename Real>
  struct r12_xx_K_gm_eval<Real, 1>
      : private detail::CoreEvalScratch<r12_xx_K_gm_eval<Real, 1>> {
    typedef detail::CoreEvalScratch<r12_xx_K_gm_eval<Real, 1>> base_type;
    typedef Real value_type;

#ifndef LIBINT_USER_DEFINED_REAL
    using FmEvalType = libint2::FmEval_Chebyshev7<double>;
#else
    using FmEvalType = libint2::FmEval_Reference<scalar_type>;
#endif

    r12_xx_K_gm_eval(unsigned int mmax, Real precision)
        : base_type(mmax) {
      fm_eval_ = FmEvalType::instance(mmax + 1, precision);
    }
    void operator()(Real* Gm, Real rho, Real T, int mmax) {
      fm_eval_->eval(&base_type::Fm_[0], T, mmax + 1);
      auto T_plus_m_plus_one = T + 1.0;
      Gm[0] = T_plus_m_plus_one * base_type::Fm_[0] - T * base_type::Fm_[1];
      auto minus_m = -1.0;
      T_plus_m_plus_one += 1.0;
      for (auto m = 1; m <= mmax;
           ++m, minus_m -= 1.0, T_plus_m_plus_one += 1.0) {
        Gm[m] =
            minus_m * base_type::Fm_[m - 1] + T_plus_m_plus_one * base_type::Fm_[m] - T * base_type::Fm_[m + 1];
      }
    }

   private:
    std::shared_ptr<const FmEvalType> fm_eval_;  // need for odd K
  };

  /// core integral evaluator for \f$ \mathrm{erf}(\omega r) / r \f$ kernel
  template <typename Real>
  struct erf_coulomb_gm_eval {
    typedef Real value_type;

#ifndef LIBINT_USER_DEFINED_REAL
    using FmEvalType = libint2::FmEval_Chebyshev7<double>;
#else
    using FmEvalType = libint2::FmEval_Reference<scalar_type>;
#endif

    erf_coulomb_gm_eval(unsigned int mmax, Real precision) {
      fm_eval_ = FmEvalType::instance(mmax, precision);
    }
    void operator()(Real* Gm, Real rho, Real T, int mmax, Real omega) const {
      if (omega > 0) {
        auto omega2 = omega * omega;
        auto omega2_over_omega2_plus_rho = omega2 / (omega2 + rho);
        fm_eval_->eval(Gm, T * omega2_over_omega2_plus_rho,
                       mmax);

        using std::sqrt;
        auto ooversqrto2prho_exp_2mplus1 =
            sqrt(omega2_over_omega2_plus_rho);
        for (auto m = 0; m <= mmax;
             ++m, ooversqrto2prho_exp_2mplus1 *= omega2_over_omega2_plus_rho) {
          Gm[m] *= ooversqrto2prho_exp_2mplus1;
        }
      }
      else {
        std::fill(Gm, Gm+mmax+1, Real{0});
      }
    }

     private:
      std::shared_ptr<const FmEvalType> fm_eval_;  // need for odd K
  };

  /// core integral evaluator for \f$ \mathrm{erfc}(\omega r) / r \f$ kernel
  /// @note need extra scratch for Boys function values,
  ///       since need to call Boys engine twice
  template <typename Real>
  struct erfc_coulomb_gm_eval : private
  detail::CoreEvalScratch<erfc_coulomb_gm_eval<Real>> {
    typedef detail::CoreEvalScratch<erfc_coulomb_gm_eval<Real>> base_type;
    typedef Real value_type;

    #ifndef LIBINT_USER_DEFINED_REAL
    using FmEvalType = libint2::FmEval_Chebyshev7<double>;
#else
    using FmEvalType = libint2::FmEval_Reference<scalar_type>;
#endif

    erfc_coulomb_gm_eval(unsigned int mmax, Real precision)
        : base_type(mmax) {
      fm_eval_ = FmEvalType::instance(mmax, precision);
    }
    void operator()(Real* Gm, Real rho, Real T, int mmax, Real omega) {
      fm_eval_->eval(&base_type::Fm_[0], T, mmax);
      std::copy(base_type::Fm_.cbegin(), base_type::Fm_.cbegin() + mmax + 1, Gm);
      if (omega > 0) {
        auto omega2 = omega * omega;
        auto omega2_over_omega2_plus_rho = omega2 / (omega2 + rho);
        fm_eval_->eval(&base_type::Fm_[0], T * omega2_over_omega2_plus_rho,
                       mmax);

        using std::sqrt;
        auto ooversqrto2prho_exp_2mplus1 =
            sqrt(omega2_over_omega2_plus_rho);
        for (auto m = 0; m <= mmax;
             ++m, ooversqrto2prho_exp_2mplus1 *= omega2_over_omega2_plus_rho) {
          Gm[m] -= ooversqrto2prho_exp_2mplus1 * base_type::Fm_[m];
        }
      }
    }

     private:
      std::shared_ptr<const FmEvalType> fm_eval_;  // need for odd K
  };

  }  // namespace os_core_ints

  /*
   *  Slater geminal fitting is available only if have LAPACK
   */
#if HAVE_LAPACK
  /*
  f[x_] := - Exp[-\[Zeta] x] / \[Zeta];

  ff[cc_, aa_, x_] := Sum[cc[[i]]*Exp[-aa[[i]] x^2], {i, 1, n}];
  */
  template <typename Real>
  Real
  fstg(Real zeta,
       Real x) {
    return -std::exp(-zeta*x)/zeta;
  }

  template <typename Real>
  Real
  fngtg(const std::vector<Real>& cc,
        const std::vector<Real>& aa,
        Real x) {
    Real value = 0.0;
    const Real x2 = x * x;
    const unsigned int n = cc.size();
    for(unsigned int i=0; i<n; ++i)
      value += cc[i] * std::exp(- aa[i] * x2);
    return value;
  }

  // --- weighting functions ---
  // L2 error is weighted by ww(x)
  // hence error is weighted by sqrt(ww(x))
  template <typename Real>
  Real
  wwtewklopper(Real x) {
    const Real x2 = x * x;
    return x2 * std::exp(-2 * x2);
  }
  template <typename Real>
  Real
  wwcusp(Real x) {
    const Real x2 = x * x;
    const Real x6 = x2 * x2 * x2;
    return std::exp(-0.005 * x6);
  }
  // default is Tew-Klopper
  template <typename Real>
  Real
  ww(Real x) {
    //return wwtewklopper(x);
    return wwcusp(x);
  }

  template <typename Real>
  Real
  norm(const std::vector<Real>& vec) {
    Real value = 0.0;
    const unsigned int n = vec.size();
    for(unsigned int i=0; i<n; ++i)
      value += vec[i] * vec[i];
    return value;
  }

  template <typename Real>
  void LinearSolveDamped(const std::vector<Real>& A,
                         const std::vector<Real>& b,
                         Real lambda,
                         std::vector<Real>& x) {
    const size_t n = b.size();
    std::vector<Real> Acopy(A);
    for(size_t m=0; m<n; ++m) Acopy[m*n + m]  *= (1 + lambda);
    std::vector<Real> e(b);

    //int info = LAPACKE_dgesv( LAPACK_ROW_MAJOR, n, 1, &Acopy[0], n, &ipiv[0], &e[0], n );
    {
      std::vector<int> ipiv(n);
      int n = b.size();
      int one = 1;
      int info;
      dgesv_(&n, &one, &Acopy[0], &n, &ipiv[0], &e[0], &n, &info);
      assert (info == 0);
    }

    x = e;
  }

  /**
   * computes a least-squares fit of \f$ -exp(-\zeta r_{12})/\zeta = \sum_{i=1}^n c_i exp(-a_i r_{12}^2) \f$
   * on \f$ r_{12} \in [0, x_{\rm max}] \f$ discretized to npts.
   * @param[in] n
   * @param[in] zeta
   * @param[out] geminal
   * @param[in] xmin
   * @param[in] xmax
   * @param[in] npts
   */
  template <typename Real>
  void stg_ng_fit(unsigned int n,
                 Real zeta,
                 std::vector< std::pair<Real, Real> >& geminal,
                 Real xmin = 0.0,
                 Real xmax = 10.0,
                 unsigned int npts = 1001) {

    // initial guess
    std::vector<Real> cc(n, 1.0); // coefficients
    std::vector<Real> aa(n); // exponents
    for(unsigned int i=0; i<n; ++i)
      aa[i] = std::pow(3.0, (i + 2 - (n + 1)/2.0));

    // first rescale cc for ff[x] to match the norm of f[x]
    Real ffnormfac = 0.0;
    using std::sqrt;
    for(unsigned int i=0; i<n; ++i)
      for(unsigned int j=0; j<n; ++j)
        ffnormfac += cc[i] * cc[j]/sqrt(aa[i] + aa[j]);
    const Real Nf = sqrt(2.0 * zeta) * zeta;
    const Real Nff = sqrt(2.0) / (sqrt(ffnormfac) *
        sqrt(sqrt(M_PI)));
    for(unsigned int i=0; i<n; ++i) cc[i] *= -Nff/Nf;

    Real lambda0 = 1000; // damping factor is initially set to 1000, eventually should end up at 0
    const Real nu = 3.0; // increase/decrease the damping factor scale it by this
    const Real epsilon = 1e-15; // convergence
    const unsigned int maxniter = 200;

    // grid points on which we will fit
    std::vector<Real> xi(npts);
    for(unsigned int i=0; i<npts; ++i) xi[i] = xmin + (xmax - xmin)*i/(npts - 1);

    std::vector<Real> err(npts);

    const size_t nparams = 2*n; // params = expansion coefficients + gaussian exponents
    std::vector<Real> J( npts * nparams );
    std::vector<Real> delta(nparams);

//    std::cout << "iteration 0" << std::endl;
//    for(unsigned int i=0; i<n; ++i)
//      std::cout << cc[i] << " " << aa[i] << std::endl;

    Real errnormI;
    Real errnormIm1 = 1e3;
    bool converged = false;
    unsigned int iter = 0;
    while (!converged && iter < maxniter) {
//      std::cout << "Iteration " << ++iter << ": lambda = " << lambda0/nu << std::endl;

        for(unsigned int i=0; i<npts; ++i) {
          const Real x = xi[i];
          err[i] = (fstg(zeta, x) - fngtg(cc, aa, x)) * sqrt(ww(x));
        }
        errnormI = norm(err)/sqrt((Real)npts);

//        std::cout << "|err|=" << errnormI << std::endl;
        converged = std::abs((errnormI - errnormIm1)/errnormIm1) <= epsilon;
        if (converged) break;
        errnormIm1 = errnormI;

        for(unsigned int i=0; i<npts; ++i) {
          const Real x2 = xi[i] * xi[i];
          const Real sqrt_ww_x = sqrt(ww(xi[i]));
          const unsigned int ioffset = i * nparams;
          for(unsigned int j=0; j<n; ++j)
            J[ioffset+j] = (std::exp(-aa[j] * x2)) * sqrt_ww_x;
          const unsigned int ioffsetn = ioffset+n;
          for(unsigned int j=0; j<n; ++j)
            J[ioffsetn+j] = -  sqrt_ww_x * x2 * cc[j] * std::exp(-aa[j] * x2);
        }

        std::vector<Real> A( nparams * nparams);
        for(size_t r=0, rc=0; r<nparams; ++r) {
          for(size_t c=0; c<nparams; ++c, ++rc) {
            double Arc = 0.0;
            for(size_t i=0, ir=r, ic=c; i<npts; ++i, ir+=nparams, ic+=nparams)
              Arc += J[ir] * J[ic];
            A[rc] = Arc;
          }
        }

        std::vector<Real> b( nparams );
        for(size_t r=0; r<nparams; ++r) {
          Real br = 0.0;
          for(size_t i=0, ir=r; i<npts; ++i, ir+=nparams)
            br += J[ir] * err[i];
          b[r] = br;
        }

        // try decreasing damping first
        // if not successful try increasing damping until it results in a decrease in the error
        lambda0 /= nu;
        for(int l=-1; l<1000; ++l) {

          LinearSolveDamped(A, b, lambda0, delta );

          std::vector<double> cc_0(cc); for(unsigned int i=0; i<n; ++i) cc_0[i] += delta[i];
          std::vector<double> aa_0(aa); for(unsigned int i=0; i<n; ++i) aa_0[i] += delta[i+n];

          // if any of the exponents are negative the step is too large and need to increase damping
          bool step_too_large = false;
          for(unsigned int i=0; i<n; ++i)
            if (aa_0[i] < 0.0) {
              step_too_large = true;
              break;
            }
          if (!step_too_large) {
            std::vector<double> err_0(npts);
            for(unsigned int i=0; i<npts; ++i) {
              const double x = xi[i];
              err_0[i] = (fstg(zeta, x) - fngtg(cc_0, aa_0, x)) * sqrt(ww(x));
            }
            const double errnorm_0 = norm(err_0)/sqrt((double)npts);
            if (errnorm_0 < errnormI) {
              cc = cc_0;
              aa = aa_0;
              break;
            }
            else // step lead to increase of the error -- try dampening a bit more
              lambda0 *= nu;
          }
          else // too large of a step
            lambda0 *= nu;
        } // done adjusting the damping factor

      } // end of iterative minimization

      // if reached max # of iterations throw if the error is too terrible
      assert(not (iter == maxniter && errnormI > 1e-10));

      for(unsigned int i=0; i<n; ++i)
        geminal[i] = std::make_pair(aa[i], cc[i]);
    }
#endif

} // end of namespace libint2

#endif // C++ only
#endif // header guard