src/nonbonded/SlaterIntegrals.hpp

/***************************************************************************
 *   This program is free software; you can redistribute it and/or modify  *
 *   it under the terms of the GNU General Public License as published by  *
 *   the Free Software Foundation; either version 3 of the License, or     *
 *   (at your option) any later version.                                   *
 *                                                                         *
 *   This program is distributed in the hope that it will be useful,       *
 *   but WITHOUT ANY WARRANTY; without even the implied warranty of        *
 *   MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the         *
 *   GNU General Public License for more details.                          *
 *                                                                         *
 *   You should have received a copy of the GNU General Public License     *
 *   along with this program; if not, see <http://www.gnu.org/licenses/>.  *
 ***************************************************************************/

/***
 * 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 <iostream>
#include "math/Factorials.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;
  int nu;
  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)
{
  int nu, p;
  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 (x == 0.) // Pathological case
    {
      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
          cerr << "ERROR, sSTOCoulInt cannot compute from arguments\n";
          cerr << "a = " << a << "\tb = " << b << "\tm = " << m <<"\tn = " << n << "\t R = " << R << "\n";
          //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:	1734
Committed:	Tue Jun 5 17:49:36 2012 UTC (12 years, 11 months ago) by jmichalk
File size:	18663 byte(s)
Log Message:	Fixed the Slater overlap integral code, added dVdFq to the self interaction.
#	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 <iostream>
65	#include "math/Factorials.hpp"
66
67	#ifndef NONBONDED_SLATERINTEGRALS_HPP
68	#define NONBONDED_SLATERINTEGRALS_HPP
69
70	template <typename T> inline T sqr(T t) { return t*t; }
71	template <typename T> inline T mod(T x, T m)
72	{ return x<0 ? m - 1 - ((-x) - 1)%m : x%m; }
73
74	// #include "parameters.h"
75
76	/**
77	* @brief Computes Rosen's Guillimer-Zener function A
78	* Computes Rosen's A integral, an auxiliary quantity needed to
79	* compute integrals involving Slater-type orbitals of s symmetry.
80	* \f[
81	* A_n(\alpha) = \int_1^\infty x^n e^{-\alpha x}dx
82	* = \frac{n! e^{-\alpha}}{\alpha^{n+1}}\sum_{\nu=0}^n
83	* \frac{\alpha^\nu}{\nu!}
84	* \f]
85	* @param n - principal quantum number
86	* @param alpha - Slater exponent
87	* @return the value of Rosen's A integral
88	* @note N. Rosen, Phys. Rev., 38 (1931), 255
89	*/
90	inline RealType RosenA(int n, RealType a)
91	{
92	RealType RosenA_ = 0.;
93	if (a != 0.)
94	{
95	RealType Term = 1.;
96	RosenA_ = Term;
97	for (int nu=1; nu<=n; nu++)
98	{
99	Term *= a / nu;
100	RosenA_ += Term;
101	}
102	RosenA_ = (RosenA_/Term) * (exp(-a)/a);
103	}
104	return RosenA_;
105	}
106
107	/**
108	* @brief Computes Rosen's Guillimer-Zener function B
109	* Computes Rosen's B integral, an auxiliary quantity needed to
110	* compute integrals involving Slater-type orbitals of s symmetry.
111	* \f[
112	* B_n(\alpha) = \int_{-1}^1 x^n e^{-\alpha x} dx
113	* = \frac{n!}{\alpha^{n+1}}
114	* \sum_{\nu=0}^n \frac{e^\alpha(-\alpha)^\nu
115	* - e^{-\alpha} \alpha^\nu}{\nu!}
116	* \f]
117	* @param n - principal quantum number
118	* @param alpha - Slater exponent
119	* @return the value of Rosen's B integral
120	* @note N. Rosen, Phys. Rev., 38 (1931), 255
121	*/
122	inline RealType RosenB(int n, RealType alpha)
123	{
124	RealType TheSum, Term;
125	RealType RosenB_, PSinhRosenA, PCoshRosenA, PHyperRosenA;
126	int nu;
127	bool IsPositive;
128	if (alpha != 0.)
129	{
130	Term = 1.;
131	TheSum = 1.;
132	IsPositive = true;
133
134	// These two expressions are (up to constant factors) equivalent
135	// to computing the hyperbolic sine and cosine of a respectively
136	// The series consists of adding up these terms in an alternating fashion
137	PSinhRosenA = exp(alpha) - exp(-alpha);
138	PCoshRosenA = -exp(alpha) - exp(-alpha);
139	TheSum = PSinhRosenA;
140	for (unsigned nu=1; nu<=n; nu++)
141	{
142	if (IsPositive)
143	{
144	PHyperRosenA = PCoshRosenA;
145	IsPositive = false;
146	}
147	else // term to add should be negative
148	{
149	PHyperRosenA = PSinhRosenA;
150	IsPositive = true;
151	}
152	Term *= alpha / nu;
153	TheSum += Term * PHyperRosenA;
154	}
155	RosenB_ = TheSum / (alpha*Term);
156	}
157	else // pathological case of a=0
158	{
159	printf("WARNING, a = 0 in RosenB\n");
160	RosenB_ = (1. - pow(-1., n)) / (n + 1.);
161	}
162	return RosenB_;
163	}
164
165	/** @brief Computes Rosen's D combinatorial factor
166	* Computes Rosen's D factor, an auxiliary quantity needed to
167	* compute integrals involving Slater-type orbitals of s symmetry.
168	* \f[
169	* RosenD^{mn}_p = \sum_k (-1)^k \frac{m! n!}
170	* {(p-k)!(m-p+k)!(n-k)!k!}
171	* \f]
172	* @return the value of Rosen's D factor
173	* @note N. Rosen, Phys. Rev., 38 (1931), 255
174	*/
175	inline RealType RosenD(int m, int n, int p)
176	{
177	if (m+n+p > maxFact)
178	{
179	printf("Error, arguments exceed maximum factorial computed %d > %d\n", m+n+p, maxFact);
180	::exit(0);
181	}
182
183	RealType RosenD_ = 0;
184	for (int k=max(p-m,0); k<=min(n,p); k++)
185	{
186	if (mod(k,2) == 0)
187	RosenD_ += (fact[m] / (fact[p-k] * fact[m-p+k])) * (fact[n] / (fact[n-k] * fact[k]));
188	else
189	RosenD_ -= (fact[m] / ( fact[p-k] * fact[m-p+k])) * (fact[n] / (fact[n-k] * fact[k]));
190	}
191	return RosenD_;
192	}
193
194	/** @brief Computes Coulomb integral analytically over s-type STOs
195	* Computes the two-center Coulomb integral over Slater-type orbitals of s symmetry.
196	* @param a : Slater zeta exponent of first atom in inverse Bohr (au)
197	* @param b : Slater zeta exponent of second atom in inverse Bohr (au)
198	* @param m : principal quantum number of first atom
199	* @param n : principal quantum number of second atom
200	* @param R : internuclear distance in atomic units (bohr)
201	* @return value of the Coulomb potential energy integral
202	* @note N. Rosen, Phys. Rev., 38 (1931), 255
203	* @note In Rosen's paper, this integral is known as K2.
204	*/
205	inline RealType sSTOCoulInt(RealType a, RealType b, int m, int n, RealType R)
206	{
207	int nu, p;
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 (x == 0.) // Pathological case
222	{
223	if ((a==b) && (m==n))
224	{
225	for (int nu=0; nu<=2*n-1; nu++)
226	{
227	K2 = 0.;
228	for (unsigned p=0; p<=2*n+m; p++) K2 += 1. / fact[p];
229	sSTOCoulInt_ += K2 * fact[2*n+m] / fact[m];
230	}
231	sSTOCoulInt_ = 2 * a / (n * fact[2n]) sSTOCoulInt_;
232	}
233	else
234	{
235	// Not implemented
236	cerr << "ERROR, sSTOCoulInt cannot compute from arguments\n";
237	cerr << "a = " << a << "\tb = " << b << "\tm = " << m <<"\tn = " << n << "\t R = " << R << "\n";
238	//printf("ERROR, sSTOCoulInt cannot compute from arguments\n");
239	//printf("a = %lf b = %lf m = %d n = %d R = %lf\n",a, b, m, n, R);
240	//exit(0);
241	}
242	}
243	else
244	{
245	OneElectronTerm = 1./R + pow(x, 2m)/(fact[2m]R)
246	((x-2m)RosenA(2*m-1,x)-exp(-x)) + sSTOCoulInt_;
247	eps = epsilon / OneElectronTerm;
248	if (a == b)
249	{
250	// Apply Rosen (48)
251	Factor1 = -apow(aR, 2m)/(nfact[2*m]);
252	for (int nu=0; nu<=2*n-1; nu++)
253	{
254	Factor2 = (2.n-nu)/fact[nu]pow(a*R,nu);
255	epsi = eps / fabs(Factor1 * Factor2);
256	K2 = 0.;
257	for (int p=0; p<=m+(nu-1)/2; p++)
258	{
259	Term = RosenD(2m-1, nu, 2p)/(2.p+1.) RosenA(2m+nu-1-2p,x);
260	K2 += Term;
261	if ((Term > 0) && (Term < epsi)) goto label1;
262	}
263	sSTOCoulInt_ += K2 * Factor2;
264	}
265	label1:
266	sSTOCoulInt_ *= Factor1;
267	}
268	else
269	{
270	Factor1 = -apow(aR,2m)/(2.nfact[2m]);
271	epsi = eps/fabs(Factor1);
272	if (b == 0.)
273	printf("WARNING: b = 0 in sSTOCoulInt\n");
274	else
275	{
276	// Apply Rosen (54)
277	for (int nu=0; nu<=2*n-1; nu++)
278	{
279	K2 = 0;
280	for (int p=0; p<=2*m+nu-1; p++)
281	K2=K2+RosenD(2m-1,nu,p)RosenB(p,R*(a-b))
282	RosenA(2m+nu-1-p,R*(a+b));
283	Term = K2(2n-nu)/fact[nu]pow(bR, nu);
284	sSTOCoulInt_ += Term;
285	if (fabs(Term) < epsi) goto label2;
286	}
287	label2:
288	sSTOCoulInt_ *= Factor1;
289	}
290	}
291	// Now add the one-electron term from Rosen (47) = Rosen (53)
292	sSTOCoulInt_ += OneElectronTerm;
293	}
294	return sSTOCoulInt_;
295	}
296
297	/**
298	* @brief Computes overlap integral analytically over s-type STOs
299	* Computes the overlap integral over two
300	* Slater-type orbitals of s symmetry.
301	* @param a : Slater zeta exponent of first atom in inverse Bohr (au)
302	* @param b : Slater zeta exponent of second atom in inverse Bohr (au)
303	* @param m : principal quantum number of first atom
304	* @param n : principal quantum number of second atom
305	* @param R : internuclear distance in atomic units (bohr)
306	* @return the value of the sSTOOvInt integral
307	* @note N. Rosen, Phys. Rev., 38 (1931), 255
308	* @note In the Rosen paper, this integral is known as I.
309	*/
310	inline RealType sSTOOvInt(RealType a, RealType b, int m, int n, RealType R)
311	{
312	RealType Factor, Term, eps;
313
314	// To speed up calculation, we terminate loop once contributions
315	// to integral fall below the bound, epsilon
316	RealType epsilon = 0.;
317	RealType sSTOOvInt_ = 0.;
318
319	if (a == b)
320	{
321	Factor = pow(aR, m+n+1)/sqrt(fact[2m]fact[2n]);
322	eps = epsilon / fabs(Factor);
323	for (int q=0; q<=(m+n)/2; q++)
324	{
325	Term = RosenD(m,n,2q)/(2.q+1.)RosenA(m+n-2q,a*R);
326	sSTOOvInt_ += Term;
327	if (fabs(Term) < eps) exit(0);
328	}
329	sSTOOvInt_ *= Factor;
330	}
331	else
332	{
333	Factor = 0.5pow(aR, m+0.5)pow(bR,n+0.5)
334	/sqrt(fact[2m]fact[2*n]);
335	eps = epsilon / fabs(Factor);
336	for (int q=0; q<=m+n; q++)
337	{
338	Term = RosenD(m,n,q)RosenB(q, R/2.(a-b))
339	* RosenA(m+n-q,R/2.*(a+b));
340	sSTOOvInt_ += Term;
341	if (fabs(Term) < eps) exit(0);
342	}
343	sSTOOvInt_ *= Factor;
344	}
345	return sSTOOvInt_;
346	}
347
348	/**
349	* @brief Computes kinetic energy integral analytically over s-type STOs
350	* Computes the overlap integral over two Slater-type orbitals of s symmetry.
351	* @param a : Slater zeta exponent of first atom in inverse Bohr (au)
352	* @param b : Slater zeta exponent of second atom in inverse Bohr (au)
353	* @param m : principal quantum number of first atom
354	* @param n : principal quantum number of second atom
355	* @param R : internuclear distance in atomic units (bohr)
356	* @return the value of the kinetic energy integral
357	* @note N. Rosen, Phys. Rev., 38 (1931), 255
358	* @note untested
359	*/
360	inline RealType KinInt(RealType a, RealType b, int m, int n,RealType R)
361	{
362	RealType KinInt_ = -0.5bb*sSTOOvInt(a, b, m, n, R);
363	if (n > 0)
364	{
365	KinInt_ += bbpow(2b/(2b-1),0.5) * sSTOOvInt(a, b, m, n-1, R);
366	if (n > 1) KinInt_ += pow(n(n-1)/((n-0.5)(n-1.5)), 0.5)
367	* sSTOOvInt(a, b, m, n-2, R);
368	}
369	return KinInt_;
370	}
371
372	/**
373	* @brief Computes derivative of Coulomb integral with respect to the interatomic distance
374	* Computes the two-center Coulomb integral over Slater-type orbitals of s symmetry.
375	* @param a: Slater zeta exponent of first atom in inverse Bohr (au)
376	* @param b: Slater zeta exponent of second atom in inverse Bohr (au)
377	* @param m: principal quantum number of first atom
378	* @param n: principal quantum number of second atom
379	* @param R: internuclear distance in atomic units (bohr)
380	* @return the derivative of the Coulomb potential energy integral
381	* @note Derived in QTPIE research notes, May 15 2007
382	*/
383	inline RealType sSTOCoulIntGrad(RealType a, RealType b, int m, int n, RealType R)
384	{
385	RealType x, y, z, K2, TheSum;
386	// x is the argument of the auxiliary RosenA and RosenB functions
387	x = 2. * a * R;
388
389	// First compute the two-electron component
390	RealType sSTOCoulIntGrad_ = 0.;
391	if (x==0) // Pathological case
392	{
393	printf("WARNING: argument given to sSTOCoulIntGrad is 0\n");
394	printf("a = %lf R= %lf\n", a, R);
395	}
396	else
397	{
398	if (a == b)
399	{
400	TheSum = 0.;
401	for (int nu=0; nu<=2*(n-1); nu++)
402	{
403	K2 = 0.;
404	for (int p=0; p<=(m+nu)/2; p++)
405	K2 += RosenD(2m-1, nu+1, 2p)/(2p + 1.) RosenA(2m+nu-1-2p, x);
406	TheSum += (2n-nu-1)/fact[nu]pow(aR, nu) K2;
407	}
408	sSTOCoulIntGrad_ = -pow(a, 2m+2)pow(R, 2m) /(nfact[2m])TheSum;
409	TheSum = 0.;
410	for (int nu=0; nu<=2*n-1; nu++)
411	{
412	K2 = 0.;
413	for (int p=0; p<=(m+nu-1)/2; p++)
414	K2 += RosenD(2m-1, nu, 2p)/(2p + 1.) RosenA(2m+nu-2p, x);
415	TheSum += (2n-nu)/fact[nu]pow(aR,nu) K2;
416	}
417	sSTOCoulIntGrad_ += 2pow(a, 2m+2)pow(R, 2m) /(nfact[2m])*TheSum;
418	}
419	else
420	{
421	// Slater exponents are different
422	// First calculate some useful arguments
423	y = R*(a+b);
424	z = R*(a-b);
425	TheSum = 0.;
426	for (int nu=0; nu<=2*n-1; nu++)
427	{
428	K2 = 0.;
429	for (int p=0; p<=2*m+nu; p++)
430	K2 += RosenD(2*m-1, nu+1, p)
431	* RosenB(p,z)RosenA(2m+nu-p, y);
432	TheSum += (2n-nu-1)/fact[nu]pow(bR,nu) K2;
433	}
434	sSTOCoulIntGrad_ = -bpow(a,2m+1)pow(R,2m)/
435	(2nfact[2m])TheSum;
436	TheSum = 0.;
437	for (int nu=0; nu<=2*n; nu++)
438	{
439	K2 = 0.;
440	for (int p=0; p<=2*m-1+nu; p++)
441	K2 += RosenD(2*m-1, nu, p)
442	* ((a-b)RosenB(p+1,z)RosenA(2*m+nu-p-1, y)
443	+(a+b)RosenB(p ,z)RosenA(2*m+nu-p , y));
444	TheSum += (2n-nu)/fact[nu]pow(bR,nu) K2;
445	}
446	sSTOCoulIntGrad_ += pow(a,2m+1)pow(R,2m)/(2nfact[2m])*TheSum;
447	}
448	// Now add one-electron terms and common term
449	sSTOCoulIntGrad_ = sSTOCoulIntGrad_ - (2.*m+1.)/sqr(R)
450	+ 2.m/R sSTOCoulInt(a,b,m,n,R)
451	+ pow(x,2m)/(fact[2m]sqr(R)) ((2.m+1.)exp(-x)
452	+ 2.m(1.+2.m-x)RosenA(2*m-1,x));
453	}
454	return sSTOCoulIntGrad_;
455	}
456
457	/**
458	* @brief Computes gradient of overlap integral with respect to the interatomic diatance
459	* Computes the derivative of the overlap integral over two Slater-type orbitals of s symmetry.
460	* @param a: Slater zeta exponent of first atom in inverse Bohr (au)
461	* @param b: Slater zeta exponent of second atom in inverse Bohr (au)
462	* @param m: principal quantum number of first atom
463	* @param n: principal quantum number of second atom
464	* @param R: internuclear distance in atomic units (bohr)
465	* @return the derivative of the sSTOOvInt integral
466	* @note Derived in QTPIE research notes, May 15 2007
467	*/
468	inline RealType sSTOOvIntGrad(RealType a, RealType b, int m, int n, RealType R)
469	{
470	RealType w, x, y, z, TheSum;
471
472	// Calculate first term
473	RealType sSTOOvIntGrad_ = (m+n+1.)/R * sSTOOvInt(a, b, m, n, R);
474
475	// Calculate remaining terms; answers depend on exponents
476	TheSum = 0.;
477	x = a * R;
478	if (a == b)
479	{
480	for (int q=0; q<=(m+n)/2; q++)
481	TheSum += RosenD(m,n,2q) / (2q + 1.) * RosenA(m+n-2*q+1, x);
482	sSTOOvIntGrad_ -= apow(x,m+n+1)/ sqrt(fact[2m]fact[2n])*TheSum;
483	}
484	else
485	{
486	w = b*R;
487	y = 0.5R(a+b);
488	z = 0.5R(a-b);
489	for (int q=0; q<m+n; q++)
490	TheSum = TheSum + RosenD(m,n,q) *
491	((a-b)RosenB(q+1,z)RosenA(m+n-q ,y)
492	+(a+b)RosenB(q ,z)RosenA(m+n-q+1,y));
493	sSTOOvIntGrad_ -= 0.25sqrt((pow(x, 2m+1)pow(w, 2n+1))/(fact[2m]fact[2n]))TheSum;
494	}
495	return sSTOOvIntGrad_;
496	}
497
498	/**
499	* @brief Calculates a Slater-type orbital exponent based on the hardness parameters
500	* @param Hardness: chemical hardness in atomic units
501	* @param n: principal quantum number
502	* @note Modified for use with OpenMD by Gezelter and Michalka.
503	*/
504	inline RealType getSTOZeta(int n, RealType hardness)
505	{
506	// Approximate the exact value of the constant of proportionality
507	// by its value at a very small distance epsilon
508	// since the exact R = 0 case has not be programmed
509	RealType epsilon = 1.0e-8;
510
511	// Assign orbital exponent
512	return pow(sSTOCoulInt(1., 1., n, n, epsilon) / hardness, -1./(3. + 2.*n));
513	}
514
515	#endif