src/nonbonded/SlaterIntegrals.hpp

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 *   This program is free software; you can redistribute it and/or modify  *
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 *   the Free Software Foundation; either version 3 of the License, or     *
 *   (at your option) any later version.                                   *
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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 a - 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:	1808
Committed:	Mon Oct 22 20:42:10 2012 UTC (12 years, 7 months ago) by gezelter
File size:	19218 byte(s)
Log Message:	A bug fix in the electric field for the new electrostatic code. Also comment fixes for Doxygen
#	Content
1	/***************************************************************************
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	#include <cstdlib>
65	#include <iostream>
66	#include "math/Factorials.hpp"
67	#include "utils/NumericConstant.hpp"
68
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 a - 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	for (int nu=1; nu<=n; nu++)
100	{
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	if (std::fabs(x) < OpenMD::NumericConstant::epsilon) // Pathological case
222	{
223
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	}
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