src/nonbonded/SlaterIntegrals.hpp

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/***
 * This file was imported from qtpie, found at http://code.google.com/p/qtpie
 * and was modified minimally for use in OpenMD.
 *
 * No author attribution was found in the code, but it presumably is
 * the work of J. Chen and Todd J. Martinez.
 *
 * QTPIE (charge transfer with polarization current equilibration) is
 * a new charge model, similar to other charge models like QEq,
 * fluc-q, EEM or ABEEM. Unlike other existing charge models, however,
 * it is capable of describing both charge transfer and polarization
 * phenomena. It is also unique for its ability to describe
 * intermolecular charge transfer at reasonable computational cost.
 *
 * Good references to cite when using this code are:
 *
 * J. Chen and T. J. Martinez, "QTPIE: Charge transfer with
 * polarization current equalization. A fluctuating charge model with
 * correct asymptotics", Chemical Physics Letters 438 (2007),
 * 315-320. DOI: 10.1016/j.cplett.2007.02.065. Erratum: ibid, 463
 * (2008), 288. DOI: 10.1016/j.cplett.2008.08.060
 *
 * J. Chen, D. Hundertmark and T. J. Martinez, "A unified theoretical
 * framework for fluctuating-charge models in atom-space and in
 * bond-space", Journal of Chemical Physics 129 (2008), 214113. DOI:
 * 10.1063/1.3021400.
 *
 * J. Chen and T. J. Martinez, "Charge conservation in
 * electronegativity equalization and its implications for the
 * electrostatic properties of fluctuating-charge models", Journal of
 * Chemical Physics 131 (2009), 044114. DOI: 10.1063/1.3183167
 *
 * J. Chen and T. J. Martinez, "The dissociation catastrophe in
 * fluctuating-charge models and its implications for the concept of
 * atomic electronegativity", Progress in Theoretical Chemistry and
 * Physics, to appear. arXiv:0812.1543
 *
 * J. Chen, "Theory and applications of fluctuating-charge models",
 * PhD (chemical physics) thesis, University of Illinois at
 * Urbana-Champaign, Department of Chemistry, 2009.
 *
 * J. Chen and T. J. Martinez, "Size-extensive polarizabilities with
 * intermolecular charge transfer in a fluctuating-charge model", in
 * preparation. arXiv:0812.1544
 */

#include "config.h"
#include <cmath>
#include <cstdlib>
#include <iostream>
#include "math/Factorials.hpp"
#include "utils/NumericConstant.hpp"

#ifndef NONBONDED_SLATERINTEGRALS_HPP
#define NONBONDED_SLATERINTEGRALS_HPP

template <typename T> inline T sqr(T t) { return t*t; }
template <typename T> inline T mod(T x, T m)
{ return x<0 ? m - 1 - ((-x) - 1)%m : x%m; }

// #include "parameters.h"

/**
 * @brief Computes Rosen's Guillimer-Zener function A
 *  Computes Rosen's A integral, an auxiliary quantity needed to
 *  compute integrals involving Slater-type orbitals of s symmetry.
 * \f[
 *   A_n(\alpha) = \int_1^\infty x^n e^{-\alpha x}dx
 *   = \frac{n! e^{-\alpha}}{\alpha^{n+1}}\sum_{\nu=0}^n
 *   \frac{\alpha^\nu}{\nu!}
 * \f]
 * @param n - principal quantum number
 * @param alpha - Slater exponent 
 * @return the value of Rosen's A integral
 * @note N. Rosen, Phys. Rev., 38 (1931), 255
 */
inline RealType RosenA(int n, RealType a)
{
  RealType RosenA_ = 0.;
  if (a != 0.)
    {
      RealType Term = 1.;
      RosenA_ = Term;
      for (int nu=1; nu<=n; nu++)
        {
          Term *= a / nu;
          RosenA_ += Term;
        }
      RosenA_ = (RosenA_/Term) * (exp(-a)/a);
    }
  return RosenA_;
}

/**
 * @brief Computes Rosen's Guillimer-Zener function B
 *  Computes Rosen's B integral, an auxiliary quantity needed to
 *  compute integrals involving Slater-type orbitals of s symmetry.
 * \f[
 *  B_n(\alpha) = \int_{-1}^1 x^n e^{-\alpha x} dx
 *              = \frac{n!}{\alpha^{n+1}} 
 *                \sum_{\nu=0}^n \frac{e^\alpha(-\alpha)^\nu
 *                  - e^{-\alpha} \alpha^\nu}{\nu!}
 * \f]
 * @param n - principal quantum number
 * @param alpha - Slater exponent 
 * @return the value of Rosen's B integral
 * @note N. Rosen, Phys. Rev., 38 (1931), 255
 */
inline RealType RosenB(int n, RealType alpha)
{
  RealType TheSum, Term;
  RealType RosenB_, PSinhRosenA, PCoshRosenA, PHyperRosenA;
  bool IsPositive;
  if (alpha != 0.)
    {
      Term = 1.;
      TheSum = 1.;
      IsPositive = true;
                
      // These two expressions are (up to constant factors) equivalent
      // to computing the hyperbolic sine and cosine of a respectively
      // The series consists of adding up these terms in an alternating fashion
      PSinhRosenA =  exp(alpha) - exp(-alpha);
      PCoshRosenA = -exp(alpha) - exp(-alpha);
      TheSum = PSinhRosenA;
      for (unsigned nu=1; nu<=n; nu++)
        {
          if (IsPositive)
            {
              PHyperRosenA = PCoshRosenA;
              IsPositive = false;
            }
          else // term to add should be negative
            {
              PHyperRosenA = PSinhRosenA;
              IsPositive = true;
            }
          Term *= alpha / nu;
          TheSum += Term * PHyperRosenA;
        }
      RosenB_ = TheSum / (alpha*Term);
    }
  else // pathological case of a=0
    {
      printf("WARNING, a = 0 in RosenB\n");
      RosenB_ = (1. - pow(-1., n)) / (n + 1.);
    }
  return RosenB_;
}

/** @brief Computes Rosen's D combinatorial factor
 *  Computes Rosen's D factor, an auxiliary quantity needed to
 *  compute integrals involving Slater-type orbitals of s symmetry.
 *  \f[
 *  RosenD^{mn}_p = \sum_k (-1)^k \frac{m! n!}
 *                  {(p-k)!(m-p+k)!(n-k)!k!}
 *  \f]
 * @return the value of Rosen's D factor
 * @note N. Rosen, Phys. Rev., 38 (1931), 255
 */
inline RealType RosenD(int m, int n, int p)
{
  if (m+n+p > maxFact)
    {
      printf("Error, arguments exceed maximum factorial computed %d > %d\n", m+n+p, maxFact);
      ::exit(0);
    }
        
  RealType RosenD_ = 0;
  for (int k=max(p-m,0); k<=min(n,p); k++)
    {
      if (mod(k,2) == 0)
        RosenD_ += (fact[m] / (fact[p-k] * fact[m-p+k])) * (fact[n] / (fact[n-k] * fact[k]));
      else
        RosenD_ -= (fact[m] / ( fact[p-k] * fact[m-p+k])) * (fact[n] / (fact[n-k] * fact[k]));
    }
  return RosenD_;
}

/** @brief Computes Coulomb integral analytically over s-type STOs
 * Computes the two-center Coulomb integral over Slater-type orbitals of s symmetry.
 * @param a : Slater zeta exponent of first atom in inverse Bohr (au)
 * @param b : Slater zeta exponent of second atom in inverse Bohr (au)
 * @param m : principal quantum number of first atom
 * @param n : principal quantum number of second atom
 * @param R : internuclear distance in atomic units (bohr)
 * @return value of the Coulomb potential energy integral
 * @note N. Rosen, Phys. Rev., 38 (1931), 255
 * @note In Rosen's paper, this integral is known as K2.
 */
inline RealType sSTOCoulInt(RealType a, RealType b, int m, int n, RealType R)
{
  RealType x, K2;
  RealType Factor1, Factor2, Term, OneElectronTerm;
  RealType eps, epsi;
        
  // To speed up calculation, we terminate loop once contributions
  // to integral fall below the bound, epsilon
  RealType epsilon = 0.;

  // x is the argument of the auxiliary RosenA and RosenB functions
  x = 2. * a * R;
        
  // First compute the two-electron component
  RealType sSTOCoulInt_ = 0.;
  if (std::fabs(x) < OpenMD::NumericConstant::epsilon) // Pathological case
    {

      // This solution for the one-center coulomb integrals comes from 
      // Yoshiyuki Hase, Computers & Chemistry 9(4), pp. 285-287 (1985).
      
      RealType Term1 = fact[2*m - 1] / pow(2*a, 2*m);
      RealType Term2 = 0.;
      for (int nu = 1; nu <= 2*n; nu++) {
        Term2 += nu * pow(2*b, 2*n - nu) * fact[2*(m+n)-nu-1] / (fact[2*n-nu]*2*n * pow(2*(a+b), 2*(m+n)-nu));
      }
      sSTOCoulInt_ = pow(2*a, 2*m+1) * (Term1 - Term2) / fact[2*m];

      // Original QTPIE code for the one-center case is below.  Doesn't
      // appear to generate the correct one-center results.
      //
      // if ((a==b) && (m==n))
      //   {
      //     for (int nu=0; nu<=2*n-1; nu++)
      //       {
      //         K2 = 0.;
      //         for (unsigned p=0; p<=2*n+m; p++) K2 += 1. / fact[p];
      //         sSTOCoulInt_ += K2 * fact[2*n+m] / fact[m];
      //       }
      //     sSTOCoulInt_ = 2 * a / (n * fact[2*n]) * sSTOCoulInt_;
      //   }
      // else
      //   {
      //     // Not implemented
      //     printf("ERROR, sSTOCoulInt cannot compute from arguments\n");
      //     printf("a = %lf b = %lf m = %d n = %d R = %lf\n",a, b, m, n, R);
      //     exit(0);
      //   }

    }
  else
    {
      OneElectronTerm = 1./R + pow(x, 2*m)/(fact[2*m]*R)*
        ((x-2*m)*RosenA(2*m-1,x)-exp(-x)) + sSTOCoulInt_;
      eps = epsilon / OneElectronTerm;
      if (a == b)
        {
          // Apply Rosen (48)
          Factor1 = -a*pow(a*R, 2*m)/(n*fact[2*m]);
          for (int nu=0; nu<=2*n-1; nu++)
            {
              Factor2 = (2.*n-nu)/fact[nu]*pow(a*R,nu);
              epsi = eps / fabs(Factor1 * Factor2);
              K2 = 0.;
              for (int p=0; p<=m+(nu-1)/2; p++)
                {
                  Term = RosenD(2*m-1, nu, 2*p)/(2.*p+1.) *RosenA(2*m+nu-1-2*p,x);
                  K2 += Term;
                  if ((Term > 0) && (Term < epsi)) goto label1;
                }
              sSTOCoulInt_ +=  K2 * Factor2;
            }
        label1:
          sSTOCoulInt_ *= Factor1;
        }
      else
        {
          Factor1 = -a*pow(a*R,2*m)/(2.*n*fact[2*m]);
          epsi = eps/fabs(Factor1);
          if (b == 0.)
            printf("WARNING: b = 0 in sSTOCoulInt\n");
          else
            {
              // Apply Rosen (54)
              for (int nu=0; nu<=2*n-1; nu++)
                {
                  K2 = 0;
                  for (int p=0; p<=2*m+nu-1; p++)
                    K2=K2+RosenD(2*m-1,nu,p)*RosenB(p,R*(a-b))
                      *RosenA(2*m+nu-1-p,R*(a+b));
                  Term = K2*(2*n-nu)/fact[nu]*pow(b*R, nu);
                  sSTOCoulInt_ += Term;
                  if (fabs(Term) < epsi) goto label2;
                }
            label2:
              sSTOCoulInt_ *= Factor1;
            }
        }
      // Now add the one-electron term from Rosen (47) = Rosen (53)
      sSTOCoulInt_ += OneElectronTerm;
    }
  return sSTOCoulInt_;
}

/**
 * @brief Computes overlap integral analytically over s-type STOs
 *   Computes the overlap integral over two
 *   Slater-type orbitals of s symmetry.
 * @param a : Slater zeta exponent of first atom in inverse Bohr (au)
 * @param b : Slater zeta exponent of second atom in inverse Bohr (au)
 * @param m : principal quantum number of first atom
 * @param n : principal quantum number of second atom
 * @param R : internuclear distance in atomic units (bohr)
 * @return the value of the sSTOOvInt integral
 * @note N. Rosen, Phys. Rev., 38 (1931), 255
 * @note In the Rosen paper, this integral is known as I.
 */
inline RealType sSTOOvInt(RealType a, RealType b, int m, int n, RealType R)
{
  RealType Factor, Term, eps;
        
  // To speed up calculation, we terminate loop once contributions
  // to integral fall below the bound, epsilon
  RealType epsilon = 0.;
  RealType sSTOOvInt_ = 0.;
        
  if (a == b)
    {
      Factor = pow(a*R, m+n+1)/sqrt(fact[2*m]*fact[2*n]);
      eps = epsilon / fabs(Factor);
      for (int q=0; q<=(m+n)/2; q++)
        {
          Term = RosenD(m,n,2*q)/(2.*q+1.)*RosenA(m+n-2*q,a*R);
          sSTOOvInt_ += Term;
          if (fabs(Term) < eps) exit(0);
        }
      sSTOOvInt_ *= Factor;
    }
  else
    {
      Factor = 0.5*pow(a*R, m+0.5)*pow(b*R,n+0.5)
        /sqrt(fact[2*m]*fact[2*n]);
      eps = epsilon / fabs(Factor);
      for (int q=0; q<=m+n; q++)
        {
          Term = RosenD(m,n,q)*RosenB(q, R/2.*(a-b))
            * RosenA(m+n-q,R/2.*(a+b));
          sSTOOvInt_ += Term;
          if (fabs(Term) < eps) exit(0);
        }
      sSTOOvInt_ *= Factor;
    }
  return sSTOOvInt_;
}

/**
 * @brief Computes kinetic energy integral analytically over s-type STOs
 *   Computes the overlap integral over two Slater-type orbitals of s symmetry.
 * @param a : Slater zeta exponent of first atom in inverse Bohr (au)
 * @param b : Slater zeta exponent of second atom in inverse Bohr (au)
 * @param m : principal quantum number of first atom
 * @param n : principal quantum number of second atom
 * @param R : internuclear distance in atomic units (bohr)
 * @return the value of the kinetic energy integral
 * @note N. Rosen, Phys. Rev., 38 (1931), 255
 * @note untested
 */
inline RealType KinInt(RealType a, RealType b, int m, int n,RealType R)
{
  RealType KinInt_ = -0.5*b*b*sSTOOvInt(a, b, m, n, R);
  if (n > 0)
    {
      KinInt_ += b*b*pow(2*b/(2*b-1),0.5) * sSTOOvInt(a, b, m, n-1, R);
      if (n > 1) KinInt_ += pow(n*(n-1)/((n-0.5)*(n-1.5)), 0.5)
                   * sSTOOvInt(a, b, m, n-2, R);
    }
  return KinInt_;
}

/**
 * @brief Computes derivative of Coulomb integral with respect to the interatomic distance
 *   Computes the two-center Coulomb integral over Slater-type orbitals of s symmetry.
 * @param a: Slater zeta exponent of first atom in inverse Bohr (au)
 * @param b: Slater zeta exponent of second atom in inverse Bohr (au)
 * @param m: principal quantum number of first atom
 * @param n: principal quantum number of second atom
 * @param R: internuclear distance in atomic units (bohr)
 * @return the derivative of the Coulomb potential energy integral
 * @note Derived in QTPIE research notes, May 15 2007
 */
inline RealType sSTOCoulIntGrad(RealType a, RealType b, int m, int n, RealType R)
{
  RealType x, y, z, K2, TheSum;
  // x is the argument of the auxiliary RosenA and RosenB functions
  x = 2. * a * R;
        
  // First compute the two-electron component
  RealType sSTOCoulIntGrad_ = 0.;
  if (x==0) // Pathological case
    {
      printf("WARNING: argument given to sSTOCoulIntGrad is 0\n");
      printf("a = %lf R= %lf\n", a, R);
    }
  else
    {
      if (a == b)
        {
          TheSum = 0.;
          for (int nu=0; nu<=2*(n-1); nu++)
            {
              K2 = 0.;
              for (int p=0; p<=(m+nu)/2; p++)
                K2 += RosenD(2*m-1, nu+1, 2*p)/(2*p + 1.) * RosenA(2*m+nu-1-2*p, x);
              TheSum += (2*n-nu-1)/fact[nu]*pow(a*R, nu) * K2;
            }
          sSTOCoulIntGrad_ = -pow(a, 2*m+2)*pow(R, 2*m) /(n*fact[2*m])*TheSum;
          TheSum = 0.;
          for (int nu=0; nu<=2*n-1; nu++)
            {
              K2 = 0.;
              for (int p=0; p<=(m+nu-1)/2; p++)
                K2 += RosenD(2*m-1, nu, 2*p)/(2*p + 1.) * RosenA(2*m+nu-2*p, x);
              TheSum += (2*n-nu)/fact[nu]*pow(a*R,nu) * K2;
            }
          sSTOCoulIntGrad_ += 2*pow(a, 2*m+2)*pow(R, 2*m) /(n*fact[2*m])*TheSum;
        }
      else
        {
          // Slater exponents are different
          // First calculate some useful arguments
          y = R*(a+b);
          z = R*(a-b);
          TheSum = 0.;
          for (int nu=0; nu<=2*n-1; nu++)
            {
              K2 = 0.;
              for (int p=0; p<=2*m+nu; p++)
                K2 += RosenD(2*m-1, nu+1, p)
                  * RosenB(p,z)*RosenA(2*m+nu-p, y);
              TheSum += (2*n-nu-1)/fact[nu]*pow(b*R,nu) * K2;
            }
          sSTOCoulIntGrad_ = -b*pow(a,2*m+1)*pow(R,2*m)/
            (2*n*fact[2*m])*TheSum;
          TheSum = 0.;
          for (int nu=0; nu<=2*n; nu++)
            {
              K2 = 0.;
              for (int p=0; p<=2*m-1+nu; p++)
                K2 += RosenD(2*m-1, nu, p)
                  * ((a-b)*RosenB(p+1,z)*RosenA(2*m+nu-p-1, y)
                     +(a+b)*RosenB(p  ,z)*RosenA(2*m+nu-p  , y));
              TheSum += (2*n-nu)/fact[nu]*pow(b*R,nu) * K2;
            }
          sSTOCoulIntGrad_ += pow(a,2*m+1)*pow(R,2*m)/(2*n*fact[2*m])*TheSum;
        }
      // Now add one-electron terms and common term
      sSTOCoulIntGrad_ = sSTOCoulIntGrad_ - (2.*m+1.)/sqr(R)
        + 2.*m/R * sSTOCoulInt(a,b,m,n,R)
        + pow(x,2*m)/(fact[2*m]*sqr(R)) * ((2.*m+1.)*exp(-x)
                                           + 2.*m*(1.+2.*m-x)*RosenA(2*m-1,x));
    }
  return sSTOCoulIntGrad_;
}

/** 
 * @brief Computes gradient of overlap integral with respect to the interatomic diatance
 *   Computes the derivative of the overlap integral over two Slater-type orbitals of s symmetry.
 * @param a: Slater zeta exponent of first atom in inverse Bohr (au)
 * @param b: Slater zeta exponent of second atom in inverse Bohr (au)
 * @param m: principal quantum number of first atom
 * @param n: principal quantum number of second atom
 * @param R: internuclear distance in atomic units (bohr)
 * @return the derivative of the sSTOOvInt integral
 * @note Derived in QTPIE research notes, May 15 2007
 */
inline RealType sSTOOvIntGrad(RealType a, RealType b, int m, int n, RealType R)
{
  RealType w, x, y, z, TheSum;
        
  // Calculate first term
  RealType sSTOOvIntGrad_ = (m+n+1.)/R * sSTOOvInt(a, b, m, n, R);
        
  // Calculate remaining terms; answers depend on exponents 
  TheSum = 0.;
  x = a * R;
  if (a == b)
    {
      for (int q=0; q<=(m+n)/2; q++)
        TheSum += RosenD(m,n,2*q) / (2*q + 1.) * RosenA(m+n-2*q+1, x);
      sSTOOvIntGrad_ -= a*pow(x,m+n+1)/ sqrt(fact[2*m]*fact[2*n])*TheSum;
    }
  else
    {
      w = b*R;
      y = 0.5*R*(a+b);
      z = 0.5*R*(a-b);
      for (int q=0; q<m+n; q++)
        TheSum = TheSum + RosenD(m,n,q) *
          ((a-b)*RosenB(q+1,z)*RosenA(m+n-q  ,y)
           +(a+b)*RosenB(q  ,z)*RosenA(m+n-q+1,y));
      sSTOOvIntGrad_ -= 0.25*sqrt((pow(x, 2*m+1)*pow(w, 2*n+1))/(fact[2*m]*fact[2*n]))*TheSum;
    }
  return sSTOOvIntGrad_;
}

/**
 * @brief Calculates a Slater-type orbital exponent based on the hardness parameters
 * @param Hardness: chemical hardness in atomic units
 * @param        n: principal quantum number
 * @note Modified for use with OpenMD by Gezelter and Michalka.
 */
inline RealType getSTOZeta(int n, RealType hardness)
{
  //  Approximate the exact value of the constant of proportionality
  //  by its value at a very small distance epsilon
  //  since the exact R = 0 case has not be programmed
  RealType epsilon = 1.0e-8;
  
  // Assign orbital exponent  
  return pow(sSTOCoulInt(1., 1., n, n, epsilon) / hardness, -1./(3. + 2.*n));
}

#endif
Revision:	1767
Committed:	Fri Jul 6 22:01:58 2012 UTC (13 years ago) by gezelter
File size:	19222 byte(s)
Log Message:	Various fixes required to compile OpenMD with the MS Visual C++ compiler
#	User	Rev	Content
1	gezelter	1718	/***************************************************************************
2			* This program is free software; you can redistribute it and/or modify *
3			* it under the terms of the GNU General Public License as published by *
4			* the Free Software Foundation; either version 3 of the License, or *
5			* (at your option) any later version. *
6			* *
7			* This program is distributed in the hope that it will be useful, *
8			* but WITHOUT ANY WARRANTY; without even the implied warranty of *
9			* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the *
10			* GNU General Public License for more details. *
11			* *
12			* You should have received a copy of the GNU General Public License *
13			* along with this program; if not, see <http://www.gnu.org/licenses/>. *
14			***************************************************************************/
15
16			/***
17			* This file was imported from qtpie, found at http://code.google.com/p/qtpie
18			* and was modified minimally for use in OpenMD.
19			*
20			* No author attribution was found in the code, but it presumably is
21			* the work of J. Chen and Todd J. Martinez.
22			*
23			* QTPIE (charge transfer with polarization current equilibration) is
24			* a new charge model, similar to other charge models like QEq,
25			* fluc-q, EEM or ABEEM. Unlike other existing charge models, however,
26			* it is capable of describing both charge transfer and polarization
27			* phenomena. It is also unique for its ability to describe
28			* intermolecular charge transfer at reasonable computational cost.
29			*
30			* Good references to cite when using this code are:
31			*
32			* J. Chen and T. J. Martinez, "QTPIE: Charge transfer with
33			* polarization current equalization. A fluctuating charge model with
34			* correct asymptotics", Chemical Physics Letters 438 (2007),
35			* 315-320. DOI: 10.1016/j.cplett.2007.02.065. Erratum: ibid, 463
36			* (2008), 288. DOI: 10.1016/j.cplett.2008.08.060
37			*
38			* J. Chen, D. Hundertmark and T. J. Martinez, "A unified theoretical
39			* framework for fluctuating-charge models in atom-space and in
40			* bond-space", Journal of Chemical Physics 129 (2008), 214113. DOI:
41			* 10.1063/1.3021400.
42			*
43			* J. Chen and T. J. Martinez, "Charge conservation in
44			* electronegativity equalization and its implications for the
45			* electrostatic properties of fluctuating-charge models", Journal of
46			* Chemical Physics 131 (2009), 044114. DOI: 10.1063/1.3183167
47			*
48			* J. Chen and T. J. Martinez, "The dissociation catastrophe in
49			* fluctuating-charge models and its implications for the concept of
50			* atomic electronegativity", Progress in Theoretical Chemistry and
51			* Physics, to appear. arXiv:0812.1543
52			*
53			* J. Chen, "Theory and applications of fluctuating-charge models",
54			* PhD (chemical physics) thesis, University of Illinois at
55			* Urbana-Champaign, Department of Chemistry, 2009.
56			*
57			* J. Chen and T. J. Martinez, "Size-extensive polarizabilities with
58			* intermolecular charge transfer in a fluctuating-charge model", in
59			* preparation. arXiv:0812.1544
60			*/
61
62			#include "config.h"
63			#include <cmath>
64	gezelter	1749	#include <cstdlib>
65	jmichalk	1734	#include <iostream>
66	gezelter	1718	#include "math/Factorials.hpp"
67	gezelter	1766	#include "utils/NumericConstant.hpp"
68	gezelter	1718
69			#ifndef NONBONDED_SLATERINTEGRALS_HPP
70			#define NONBONDED_SLATERINTEGRALS_HPP
71
72			template <typename T> inline T sqr(T t) { return t*t; }
73			template <typename T> inline T mod(T x, T m)
74			{ return x<0 ? m - 1 - ((-x) - 1)%m : x%m; }
75
76			// #include "parameters.h"
77
78			/**
79			* @brief Computes Rosen's Guillimer-Zener function A
80			* Computes Rosen's A integral, an auxiliary quantity needed to
81			* compute integrals involving Slater-type orbitals of s symmetry.
82			* \f[
83			* A_n(\alpha) = \int_1^\infty x^n e^{-\alpha x}dx
84			* = \frac{n! e^{-\alpha}}{\alpha^{n+1}}\sum_{\nu=0}^n
85			* \frac{\alpha^\nu}{\nu!}
86			* \f]
87			* @param n - principal quantum number
88			* @param alpha - Slater exponent
89			* @return the value of Rosen's A integral
90			* @note N. Rosen, Phys. Rev., 38 (1931), 255
91			*/
92			inline RealType RosenA(int n, RealType a)
93			{
94			RealType RosenA_ = 0.;
95			if (a != 0.)
96			{
97			RealType Term = 1.;
98			RosenA_ = Term;
99	jmichalk	1734	for (int nu=1; nu<=n; nu++)
100	gezelter	1718	{
101			Term *= a / nu;
102			RosenA_ += Term;
103			}
104			RosenA_ = (RosenA_/Term) * (exp(-a)/a);
105			}
106			return RosenA_;
107			}
108
109			/**
110			* @brief Computes Rosen's Guillimer-Zener function B
111			* Computes Rosen's B integral, an auxiliary quantity needed to
112			* compute integrals involving Slater-type orbitals of s symmetry.
113			* \f[
114			* B_n(\alpha) = \int_{-1}^1 x^n e^{-\alpha x} dx
115			* = \frac{n!}{\alpha^{n+1}}
116			* \sum_{\nu=0}^n \frac{e^\alpha(-\alpha)^\nu
117			* - e^{-\alpha} \alpha^\nu}{\nu!}
118			* \f]
119			* @param n - principal quantum number
120			* @param alpha - Slater exponent
121			* @return the value of Rosen's B integral
122			* @note N. Rosen, Phys. Rev., 38 (1931), 255
123			*/
124			inline RealType RosenB(int n, RealType alpha)
125			{
126			RealType TheSum, Term;
127			RealType RosenB_, PSinhRosenA, PCoshRosenA, PHyperRosenA;
128			bool IsPositive;
129			if (alpha != 0.)
130			{
131			Term = 1.;
132			TheSum = 1.;
133			IsPositive = true;
134
135			// These two expressions are (up to constant factors) equivalent
136			// to computing the hyperbolic sine and cosine of a respectively
137			// The series consists of adding up these terms in an alternating fashion
138			PSinhRosenA = exp(alpha) - exp(-alpha);
139			PCoshRosenA = -exp(alpha) - exp(-alpha);
140			TheSum = PSinhRosenA;
141			for (unsigned nu=1; nu<=n; nu++)
142			{
143			if (IsPositive)
144			{
145			PHyperRosenA = PCoshRosenA;
146			IsPositive = false;
147			}
148			else // term to add should be negative
149			{
150			PHyperRosenA = PSinhRosenA;
151			IsPositive = true;
152			}
153			Term *= alpha / nu;
154			TheSum += Term * PHyperRosenA;
155			}
156			RosenB_ = TheSum / (alpha*Term);
157			}
158			else // pathological case of a=0
159			{
160			printf("WARNING, a = 0 in RosenB\n");
161			RosenB_ = (1. - pow(-1., n)) / (n + 1.);
162			}
163			return RosenB_;
164			}
165
166			/** @brief Computes Rosen's D combinatorial factor
167			* Computes Rosen's D factor, an auxiliary quantity needed to
168			* compute integrals involving Slater-type orbitals of s symmetry.
169			* \f[
170			* RosenD^{mn}_p = \sum_k (-1)^k \frac{m! n!}
171			* {(p-k)!(m-p+k)!(n-k)!k!}
172			* \f]
173			* @return the value of Rosen's D factor
174			* @note N. Rosen, Phys. Rev., 38 (1931), 255
175			*/
176			inline RealType RosenD(int m, int n, int p)
177			{
178			if (m+n+p > maxFact)
179			{
180			printf("Error, arguments exceed maximum factorial computed %d > %d\n", m+n+p, maxFact);
181			::exit(0);
182			}
183
184			RealType RosenD_ = 0;
185			for (int k=max(p-m,0); k<=min(n,p); k++)
186			{
187			if (mod(k,2) == 0)
188			RosenD_ += (fact[m] / (fact[p-k] * fact[m-p+k])) * (fact[n] / (fact[n-k] * fact[k]));
189			else
190			RosenD_ -= (fact[m] / ( fact[p-k] * fact[m-p+k])) * (fact[n] / (fact[n-k] * fact[k]));
191			}
192			return RosenD_;
193			}
194
195			/** @brief Computes Coulomb integral analytically over s-type STOs
196			* Computes the two-center Coulomb integral over Slater-type orbitals of s symmetry.
197			* @param a : Slater zeta exponent of first atom in inverse Bohr (au)
198			* @param b : Slater zeta exponent of second atom in inverse Bohr (au)
199			* @param m : principal quantum number of first atom
200			* @param n : principal quantum number of second atom
201			* @param R : internuclear distance in atomic units (bohr)
202			* @return value of the Coulomb potential energy integral
203			* @note N. Rosen, Phys. Rev., 38 (1931), 255
204			* @note In Rosen's paper, this integral is known as K2.
205			*/
206			inline RealType sSTOCoulInt(RealType a, RealType b, int m, int n, RealType R)
207			{
208			RealType x, K2;
209			RealType Factor1, Factor2, Term, OneElectronTerm;
210			RealType eps, epsi;
211
212			// To speed up calculation, we terminate loop once contributions
213			// to integral fall below the bound, epsilon
214			RealType epsilon = 0.;
215
216			// x is the argument of the auxiliary RosenA and RosenB functions
217			x = 2. * a * R;
218
219			// First compute the two-electron component
220			RealType sSTOCoulInt_ = 0.;
221	gezelter	1766	if (std::fabs(x) < OpenMD::NumericConstant::epsilon) // Pathological case
222	gezelter	1718	{
223	gezelter	1766
224			// This solution for the one-center coulomb integrals comes from
225			// Yoshiyuki Hase, Computers & Chemistry 9(4), pp. 285-287 (1985).
226
227			RealType Term1 = fact[2m - 1] / pow(2a, 2*m);
228			RealType Term2 = 0.;
229			for (int nu = 1; nu <= 2*n; nu++) {
230			Term2 += nu * pow(2b, 2n - nu) * fact[2(m+n)-nu-1] / (fact[2n-nu]2n * pow(2(a+b), 2(m+n)-nu));
231			}
232			sSTOCoulInt_ = pow(2a, 2m+1) * (Term1 - Term2) / fact[2*m];
233
234			// Original QTPIE code for the one-center case is below. Doesn't
235			// appear to generate the correct one-center results.
236			//
237			// if ((a==b) && (m==n))
238			// {
239			// for (int nu=0; nu<=2*n-1; nu++)
240			// {
241			// K2 = 0.;
242			// for (unsigned p=0; p<=2*n+m; p++) K2 += 1. / fact[p];
243			// sSTOCoulInt_ += K2 * fact[2*n+m] / fact[m];
244			// }
245			// sSTOCoulInt_ = 2 * a / (n * fact[2n]) sSTOCoulInt_;
246			// }
247			// else
248			// {
249			// // Not implemented
250			// printf("ERROR, sSTOCoulInt cannot compute from arguments\n");
251			// printf("a = %lf b = %lf m = %d n = %d R = %lf\n",a, b, m, n, R);
252			// exit(0);
253			// }
254
255	gezelter	1718	}
256			else
257			{
258			OneElectronTerm = 1./R + pow(x, 2m)/(fact[2m]R)
259			((x-2m)RosenA(2*m-1,x)-exp(-x)) + sSTOCoulInt_;
260			eps = epsilon / OneElectronTerm;
261			if (a == b)
262			{
263			// Apply Rosen (48)
264			Factor1 = -apow(aR, 2m)/(nfact[2*m]);
265			for (int nu=0; nu<=2*n-1; nu++)
266			{
267			Factor2 = (2.n-nu)/fact[nu]pow(a*R,nu);
268			epsi = eps / fabs(Factor1 * Factor2);
269			K2 = 0.;
270			for (int p=0; p<=m+(nu-1)/2; p++)
271			{
272			Term = RosenD(2m-1, nu, 2p)/(2.p+1.) RosenA(2m+nu-1-2p,x);
273			K2 += Term;
274			if ((Term > 0) && (Term < epsi)) goto label1;
275			}
276			sSTOCoulInt_ += K2 * Factor2;
277			}
278			label1:
279			sSTOCoulInt_ *= Factor1;
280			}
281			else
282			{
283			Factor1 = -apow(aR,2m)/(2.nfact[2m]);
284			epsi = eps/fabs(Factor1);
285			if (b == 0.)
286			printf("WARNING: b = 0 in sSTOCoulInt\n");
287			else
288			{
289			// Apply Rosen (54)
290			for (int nu=0; nu<=2*n-1; nu++)
291			{
292			K2 = 0;
293			for (int p=0; p<=2*m+nu-1; p++)
294			K2=K2+RosenD(2m-1,nu,p)RosenB(p,R*(a-b))
295			RosenA(2m+nu-1-p,R*(a+b));
296			Term = K2(2n-nu)/fact[nu]pow(bR, nu);
297			sSTOCoulInt_ += Term;
298			if (fabs(Term) < epsi) goto label2;
299			}
300			label2:
301			sSTOCoulInt_ *= Factor1;
302			}
303			}
304			// Now add the one-electron term from Rosen (47) = Rosen (53)
305			sSTOCoulInt_ += OneElectronTerm;
306			}
307			return sSTOCoulInt_;
308			}
309
310			/**
311			* @brief Computes overlap integral analytically over s-type STOs
312			* Computes the overlap integral over two
313			* Slater-type orbitals of s symmetry.
314			* @param a : Slater zeta exponent of first atom in inverse Bohr (au)
315			* @param b : Slater zeta exponent of second atom in inverse Bohr (au)
316			* @param m : principal quantum number of first atom
317			* @param n : principal quantum number of second atom
318			* @param R : internuclear distance in atomic units (bohr)
319			* @return the value of the sSTOOvInt integral
320			* @note N. Rosen, Phys. Rev., 38 (1931), 255
321			* @note In the Rosen paper, this integral is known as I.
322			*/
323			inline RealType sSTOOvInt(RealType a, RealType b, int m, int n, RealType R)
324			{
325			RealType Factor, Term, eps;
326
327			// To speed up calculation, we terminate loop once contributions
328			// to integral fall below the bound, epsilon
329			RealType epsilon = 0.;
330			RealType sSTOOvInt_ = 0.;
331
332			if (a == b)
333			{
334			Factor = pow(aR, m+n+1)/sqrt(fact[2m]fact[2n]);
335			eps = epsilon / fabs(Factor);
336			for (int q=0; q<=(m+n)/2; q++)
337			{
338			Term = RosenD(m,n,2q)/(2.q+1.)RosenA(m+n-2q,a*R);
339			sSTOOvInt_ += Term;
340			if (fabs(Term) < eps) exit(0);
341			}
342			sSTOOvInt_ *= Factor;
343			}
344			else
345			{
346			Factor = 0.5pow(aR, m+0.5)pow(bR,n+0.5)
347			/sqrt(fact[2m]fact[2*n]);
348			eps = epsilon / fabs(Factor);
349			for (int q=0; q<=m+n; q++)
350			{
351			Term = RosenD(m,n,q)RosenB(q, R/2.(a-b))
352			* RosenA(m+n-q,R/2.*(a+b));
353			sSTOOvInt_ += Term;
354			if (fabs(Term) < eps) exit(0);
355			}
356			sSTOOvInt_ *= Factor;
357			}
358			return sSTOOvInt_;
359			}
360
361			/**
362			* @brief Computes kinetic energy integral analytically over s-type STOs
363			* Computes the overlap integral over two Slater-type orbitals of s symmetry.
364			* @param a : Slater zeta exponent of first atom in inverse Bohr (au)
365			* @param b : Slater zeta exponent of second atom in inverse Bohr (au)
366			* @param m : principal quantum number of first atom
367			* @param n : principal quantum number of second atom
368			* @param R : internuclear distance in atomic units (bohr)
369			* @return the value of the kinetic energy integral
370			* @note N. Rosen, Phys. Rev., 38 (1931), 255
371			* @note untested
372			*/
373			inline RealType KinInt(RealType a, RealType b, int m, int n,RealType R)
374			{
375			RealType KinInt_ = -0.5bb*sSTOOvInt(a, b, m, n, R);
376			if (n > 0)
377			{
378			KinInt_ += bbpow(2b/(2b-1),0.5) * sSTOOvInt(a, b, m, n-1, R);
379			if (n > 1) KinInt_ += pow(n(n-1)/((n-0.5)(n-1.5)), 0.5)
380			* sSTOOvInt(a, b, m, n-2, R);
381			}
382			return KinInt_;
383			}
384
385			/**
386			* @brief Computes derivative of Coulomb integral with respect to the interatomic distance
387			* Computes the two-center Coulomb integral over Slater-type orbitals of s symmetry.
388			* @param a: Slater zeta exponent of first atom in inverse Bohr (au)
389			* @param b: Slater zeta exponent of second atom in inverse Bohr (au)
390			* @param m: principal quantum number of first atom
391			* @param n: principal quantum number of second atom
392			* @param R: internuclear distance in atomic units (bohr)
393			* @return the derivative of the Coulomb potential energy integral
394			* @note Derived in QTPIE research notes, May 15 2007
395			*/
396			inline RealType sSTOCoulIntGrad(RealType a, RealType b, int m, int n, RealType R)
397			{
398			RealType x, y, z, K2, TheSum;
399			// x is the argument of the auxiliary RosenA and RosenB functions
400			x = 2. * a * R;
401
402			// First compute the two-electron component
403			RealType sSTOCoulIntGrad_ = 0.;
404			if (x==0) // Pathological case
405			{
406			printf("WARNING: argument given to sSTOCoulIntGrad is 0\n");
407			printf("a = %lf R= %lf\n", a, R);
408			}
409			else
410			{
411			if (a == b)
412			{
413			TheSum = 0.;
414			for (int nu=0; nu<=2*(n-1); nu++)
415			{
416			K2 = 0.;
417			for (int p=0; p<=(m+nu)/2; p++)
418			K2 += RosenD(2m-1, nu+1, 2p)/(2p + 1.) RosenA(2m+nu-1-2p, x);
419			TheSum += (2n-nu-1)/fact[nu]pow(aR, nu) K2;
420			}
421			sSTOCoulIntGrad_ = -pow(a, 2m+2)pow(R, 2m) /(nfact[2m])TheSum;
422			TheSum = 0.;
423			for (int nu=0; nu<=2*n-1; nu++)
424			{
425			K2 = 0.;
426			for (int p=0; p<=(m+nu-1)/2; p++)
427			K2 += RosenD(2m-1, nu, 2p)/(2p + 1.) RosenA(2m+nu-2p, x);
428			TheSum += (2n-nu)/fact[nu]pow(aR,nu) K2;
429			}
430			sSTOCoulIntGrad_ += 2pow(a, 2m+2)pow(R, 2m) /(nfact[2m])*TheSum;
431			}
432			else
433			{
434			// Slater exponents are different
435			// First calculate some useful arguments
436			y = R*(a+b);
437			z = R*(a-b);
438			TheSum = 0.;
439			for (int nu=0; nu<=2*n-1; nu++)
440			{
441			K2 = 0.;
442			for (int p=0; p<=2*m+nu; p++)
443			K2 += RosenD(2*m-1, nu+1, p)
444			* RosenB(p,z)RosenA(2m+nu-p, y);
445			TheSum += (2n-nu-1)/fact[nu]pow(bR,nu) K2;
446			}
447			sSTOCoulIntGrad_ = -bpow(a,2m+1)pow(R,2m)/
448			(2nfact[2m])TheSum;
449			TheSum = 0.;
450			for (int nu=0; nu<=2*n; nu++)
451			{
452			K2 = 0.;
453			for (int p=0; p<=2*m-1+nu; p++)
454			K2 += RosenD(2*m-1, nu, p)
455			* ((a-b)RosenB(p+1,z)RosenA(2*m+nu-p-1, y)
456			+(a+b)RosenB(p ,z)RosenA(2*m+nu-p , y));
457			TheSum += (2n-nu)/fact[nu]pow(bR,nu) K2;
458			}
459			sSTOCoulIntGrad_ += pow(a,2m+1)pow(R,2m)/(2nfact[2m])*TheSum;
460			}
461			// Now add one-electron terms and common term
462			sSTOCoulIntGrad_ = sSTOCoulIntGrad_ - (2.*m+1.)/sqr(R)
463			+ 2.m/R sSTOCoulInt(a,b,m,n,R)
464			+ pow(x,2m)/(fact[2m]sqr(R)) ((2.m+1.)exp(-x)
465			+ 2.m(1.+2.m-x)RosenA(2*m-1,x));
466			}
467			return sSTOCoulIntGrad_;
468			}
469
470			/**
471			* @brief Computes gradient of overlap integral with respect to the interatomic diatance
472			* Computes the derivative of the overlap integral over two Slater-type orbitals of s symmetry.
473			* @param a: Slater zeta exponent of first atom in inverse Bohr (au)
474			* @param b: Slater zeta exponent of second atom in inverse Bohr (au)
475			* @param m: principal quantum number of first atom
476			* @param n: principal quantum number of second atom
477			* @param R: internuclear distance in atomic units (bohr)
478			* @return the derivative of the sSTOOvInt integral
479			* @note Derived in QTPIE research notes, May 15 2007
480			*/
481			inline RealType sSTOOvIntGrad(RealType a, RealType b, int m, int n, RealType R)
482			{
483			RealType w, x, y, z, TheSum;
484
485			// Calculate first term
486			RealType sSTOOvIntGrad_ = (m+n+1.)/R * sSTOOvInt(a, b, m, n, R);
487
488			// Calculate remaining terms; answers depend on exponents
489			TheSum = 0.;
490			x = a * R;
491			if (a == b)
492			{
493			for (int q=0; q<=(m+n)/2; q++)
494			TheSum += RosenD(m,n,2q) / (2q + 1.) * RosenA(m+n-2*q+1, x);
495			sSTOOvIntGrad_ -= apow(x,m+n+1)/ sqrt(fact[2m]fact[2n])*TheSum;
496			}
497			else
498			{
499			w = b*R;
500			y = 0.5R(a+b);
501			z = 0.5R(a-b);
502			for (int q=0; q<m+n; q++)
503			TheSum = TheSum + RosenD(m,n,q) *
504			((a-b)RosenB(q+1,z)RosenA(m+n-q ,y)
505			+(a+b)RosenB(q ,z)RosenA(m+n-q+1,y));
506			sSTOOvIntGrad_ -= 0.25sqrt((pow(x, 2m+1)pow(w, 2n+1))/(fact[2m]fact[2n]))TheSum;
507			}
508			return sSTOOvIntGrad_;
509			}
510
511			/**
512			* @brief Calculates a Slater-type orbital exponent based on the hardness parameters
513			* @param Hardness: chemical hardness in atomic units
514			* @param n: principal quantum number
515			* @note Modified for use with OpenMD by Gezelter and Michalka.
516			*/
517			inline RealType getSTOZeta(int n, RealType hardness)
518			{
519			// Approximate the exact value of the constant of proportionality
520			// by its value at a very small distance epsilon
521			// since the exact R = 0 case has not be programmed
522			RealType epsilon = 1.0e-8;
523
524			// Assign orbital exponent
525			return pow(sSTOCoulInt(1., 1., n, n, epsilon) / hardness, -1./(3. + 2.*n));
526			}
527
528			#endif