ReferenceGBVI.cpp

/* Portions copyright (c) 2006-2009 Stanford University and Simbios.
 * Contributors: Pande Group
 *
 * Permission is hereby granted, free of charge, to any person obtaining
 * a copy of this software and associated documentation files (the
 * "Software"), to deal in the Software without restriction, including
 * without limitation the rights to use, copy, modify, merge, publish,
 * distribute, sublicense, and/or sell copies of the Software, and to
 * permit persons to whom the Software is furnished to do so, subject
 * to the following conditions:
 *
 * The above copyright notice and this permission notice shall be included
 * in all copies or substantial portions of the Software.
 *
 * THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS
 * OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF
 * MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT.
 * IN NO EVENT SHALL THE AUTHORS, CONTRIBUTORS OR COPYRIGHT HOLDERS BE
 * LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION
 * OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION
 * WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.
 */

#include <string.h>
#include <math.h>
#include <sstream>
#include <stdio.h>

#include "ReferenceForce.h"
#include "ReferenceGBVI.h"

using namespace std;
using namespace OpenMM;

/**---------------------------------------------------------------------------------------

    ReferenceGBVI constructor

    gbviParameters      gbviParameters object
    
    --------------------------------------------------------------------------------------- */

ReferenceGBVI::ReferenceGBVI(GBVIParameters* gbviParameters) : _gbviParameters(gbviParameters) {
    _switchDeriviative.resize(gbviParameters->getNumberOfAtoms());
}

/**---------------------------------------------------------------------------------------

    ReferenceGBVI destructor

    --------------------------------------------------------------------------------------- */

ReferenceGBVI::~ReferenceGBVI() {
}

/**---------------------------------------------------------------------------------------

    Get GBVIParameters reference

    @return GBVIParameters reference

    --------------------------------------------------------------------------------------- */

GBVIParameters* ReferenceGBVI::getGBVIParameters() const {
    return _gbviParameters;
}

/**---------------------------------------------------------------------------------------

    Set GBVIParameters reference

    @param GBVIParameters reference

    --------------------------------------------------------------------------------------- */

void ReferenceGBVI::setGBVIParameters(GBVIParameters* gbviParameters) {
    _gbviParameters = gbviParameters;
}

/**---------------------------------------------------------------------------------------

    Return OBC chain derivative: size = _obcParameters->getNumberOfAtoms()

    @return array

    --------------------------------------------------------------------------------------- */

vector<RealOpenMM>& ReferenceGBVI::getSwitchDeriviative() {
    return _switchDeriviative;
}

/**---------------------------------------------------------------------------------------

    Compute quintic spline value and associated derviative

    @param x                   value to compute spline at
    @param rl                  lower cutoff value
    @param ru                  upper cutoff value
    @param outValue            value of spline at x
    @param outDerivative       value of derivative of spline at x

    --------------------------------------------------------------------------------------- */

void ReferenceGBVI::quinticSpline(RealOpenMM x, RealOpenMM rl, RealOpenMM ru,
                            RealOpenMM* outValue, RealOpenMM* outDerivative) {

    // ---------------------------------------------------------------------------------------

    static const RealOpenMM one           = static_cast<RealOpenMM>(  1.0);
    static const RealOpenMM minusSix      = static_cast<RealOpenMM>( -6.0);
    static const RealOpenMM minusTen      = static_cast<RealOpenMM>(-10.0);
    static const RealOpenMM minusThirty   = static_cast<RealOpenMM>(-30.0);
    static const RealOpenMM fifteen       = static_cast<RealOpenMM>( 15.0);
    static const RealOpenMM sixty         = static_cast<RealOpenMM>( 60.0);

    // ---------------------------------------------------------------------------------------

    RealOpenMM numerator    = x  - rl;
    RealOpenMM denominator  = ru - rl;
    RealOpenMM ratio        = numerator/denominator;
    RealOpenMM ratio2       = ratio*ratio;
    RealOpenMM ratio3       = ratio2*ratio;

    *outValue               = one + ratio3*(minusTen + fifteen*ratio + minusSix*ratio2);
    *outDerivative          = ratio2*(minusThirty + sixty*ratio + minusThirty*ratio2)/denominator;
}

/**---------------------------------------------------------------------------------------

    Compute Born radii based on Eq. 3 of Labute paper [JCC 29 p. 1693-1698 2008])
    and quintic splice switching function

    @param atomicRadius3       atomic radius cubed
    @param bornSum             Born sum (volume integral)
    @param gbviParameters      Gbvi parameters (parameters used in spline
                               QuinticLowerLimitFactor & QuinticUpperBornRadiusLimit)
    @param bornRadius          output Born radius
    @param switchDeriviative   output switching function deriviative

    --------------------------------------------------------------------------------------- */

void ReferenceGBVI::computeBornRadiiUsingQuinticSpline(RealOpenMM atomicRadius3, RealOpenMM bornSum,
                                                 GBVIParameters* gbviParameters, 
                                                 RealOpenMM* bornRadius, RealOpenMM* switchDeriviative) {

    // ---------------------------------------------------------------------------------------

    static const RealOpenMM zero          = static_cast<RealOpenMM>( 0.0);
    static const RealOpenMM one           = static_cast<RealOpenMM>( 1.0);
    static const RealOpenMM minusOne      = static_cast<RealOpenMM>(-1.0);
    static const RealOpenMM minusThree    = static_cast<RealOpenMM>(-3.0);
    static const RealOpenMM oneEighth     = static_cast<RealOpenMM>( 0.125);
    static const RealOpenMM minusOneThird = static_cast<RealOpenMM>((-1.0/3.0));
    static const RealOpenMM three         = static_cast<RealOpenMM>( 3.0);

    // ---------------------------------------------------------------------------------------

    // R                = [ S(V)*(A - V) ]**(-1/3)

    // S(V)             = 1                                 V < L
    // S(V)             = qSpline + U/(A-V)                 L < V < A
    // S(V)             = U/(A-V)                           U < V 

    // dR/dr            = (-1/3)*[ S(V)*(A - V) ]**(-4/3)*[ d{ S(V)*(A-V) }/dr

    // d{ S(V)*(A-V) }/dr   = (dV/dr)*[ (A-V)*dS/dV - S(V) ]

    //  (A - V)*dS/dV - S(V)  = 0 - 1                             V < L

    //  (A - V)*dS/dV - S(V)  = (A-V)*d(qSpline) + (A-V)*U/(A-V)**2 - qSpline - U/(A-V) 

    //                        = (A-V)*d(qSpline) - qSpline        L < V < A**(-3)

    //  (A - V)*dS/dV - S(V)  = (A-V)*U*/(A-V)**2 - U/(A-V) = 0   U < V

    RealOpenMM splineL          = gbviParameters->getQuinticLowerLimitFactor()*atomicRadius3;
    RealOpenMM sum;
    if (bornSum > splineL) {
        if (bornSum < atomicRadius3) {
            RealOpenMM splineValue, splineDerivative;
            quinticSpline(bornSum, splineL, atomicRadius3, &splineValue, &splineDerivative); 
            sum                 = (atomicRadius3 - bornSum)*splineValue + gbviParameters->getQuinticUpperBornRadiusLimit();
            *switchDeriviative  = splineValue - (atomicRadius3 - bornSum)*splineDerivative;
        } else {   
            sum                = gbviParameters->getQuinticUpperBornRadiusLimit();
            *switchDeriviative = zero;
        }
    } else {
        sum                = atomicRadius3 - bornSum; 
        *switchDeriviative = one;
    }
    *bornRadius = POW(sum, minusOneThird);
}

/**---------------------------------------------------------------------------------------

    Get Born radii based on Eq. 3 of Labute paper [JCC 29 p. 1693-1698 2008])

    @param atomCoordinates     atomic coordinates
    @param bornRadii           output array of Born radii
    @param chain               not used here

    --------------------------------------------------------------------------------------- */

void ReferenceGBVI::computeBornRadii(const vector<RealVec>& atomCoordinates, vector<RealOpenMM>& bornRadii) {

    // ---------------------------------------------------------------------------------------

    static const RealOpenMM zero          = static_cast<RealOpenMM>(0.0);
    static const RealOpenMM one           = static_cast<RealOpenMM>(1.0);
    static const RealOpenMM minusThree    = static_cast<RealOpenMM>(-3.0);
    static const RealOpenMM oneEighth     = static_cast<RealOpenMM>(0.125);
    static const RealOpenMM minusOneThird = static_cast<RealOpenMM>((-1.0/3.0));
    static const RealOpenMM three         = static_cast<RealOpenMM>(3.0);

    // ---------------------------------------------------------------------------------------

    GBVIParameters* gbviParameters        = getGBVIParameters();
    int numberOfAtoms                     = gbviParameters->getNumberOfAtoms();
    const vector<RealOpenMM>& atomicRadii   = gbviParameters->getAtomicRadii();
    const vector<RealOpenMM>& scaledRadii   = gbviParameters->getScaledRadii();

    vector<RealOpenMM>& switchDeriviatives  = getSwitchDeriviative();

    // ---------------------------------------------------------------------------------------

    // calculate Born radii

    for (int atomI = 0; atomI < numberOfAtoms; atomI++) {
      
        RealOpenMM radiusI         = atomicRadii[atomI];
        RealOpenMM sum             = zero;
 
        // sum over volumes

        for (int atomJ = 0; atomJ < numberOfAtoms; atomJ++) {

            if (atomJ != atomI) {
  
                RealOpenMM deltaR[ReferenceForce::LastDeltaRIndex];
                if (_gbviParameters->getPeriodic())
                    ReferenceForce::getDeltaRPeriodic(atomCoordinates[atomI], atomCoordinates[atomJ], _gbviParameters->getPeriodicBox(), deltaR);
                else
                    ReferenceForce::getDeltaR(atomCoordinates[atomI], atomCoordinates[atomJ], deltaR);
   
                RealOpenMM r               = deltaR[ReferenceForce::RIndex];
   
                if (_gbviParameters->getUseCutoff() && r > _gbviParameters->getCutoffDistance())
                    continue;
   
                sum  += ReferenceGBVI::getVolume(r, radiusI, scaledRadii[atomJ]);
   
            }
        }

        RealOpenMM atomicRadius3 = POW(radiusI, minusThree);
        if (_gbviParameters->getBornRadiusScalingMethod() != GBVIParameters::QuinticSpline) {
           sum                        = atomicRadius3 - sum; 
           bornRadii[atomI]           = POW(sum, minusOneThird);
           switchDeriviatives[atomI]  = one; 
        } else {
           RealOpenMM bornRadius, switchDeriviative;
           computeBornRadiiUsingQuinticSpline(atomicRadius3, sum, gbviParameters, 
                                              &bornRadius, &switchDeriviative);
           bornRadii[atomI]           = bornRadius;
           switchDeriviatives[atomI]  = switchDeriviative;
        }    
    }
}

/**---------------------------------------------------------------------------------------

    Get volume Eq. 4 of Labute paper [JCC 29 p. 1693-1698 2008])

    @param r                   distance between atoms i & j
    @param R                   atomic radius
    @param S                   scaled atomic radius

    @return volume

    --------------------------------------------------------------------------------------- */

RealOpenMM ReferenceGBVI::getVolume(RealOpenMM r, RealOpenMM R, RealOpenMM S) {

    // ---------------------------------------------------------------------------------------

    static const RealOpenMM zero         = static_cast<RealOpenMM>( 0.0);
    static const RealOpenMM minusThree   = static_cast<RealOpenMM>(-3.0);

    RealOpenMM              diff         = (S - R);
    if (FABS(diff) < r) {

        RealOpenMM lowerBound = (R > (r - S)) ? R : (r - S);
        return (ReferenceGBVI::getL(r, (r + S),    S) -
                ReferenceGBVI::getL(r, lowerBound, S));
 
    } else if (r <= diff) {

        return ReferenceGBVI::getL(r, (r + S), S) -
               ReferenceGBVI::getL(r, (r - S), S) + 
               POW(R, minusThree);

    } else {
        return zero;
    }
}

/**---------------------------------------------------------------------------------------

    Get L (used in analytical solution for volume integrals) 

    @param r                   distance between atoms i & j
    @param R                   atomic radius
    @param S                   scaled atomic radius

    @return L value (Eq. 4 of Labute paper [JCC 29 p. 1693-1698 2008])

    --------------------------------------------------------------------------------------- */

RealOpenMM ReferenceGBVI::getL(RealOpenMM r, RealOpenMM x, RealOpenMM S) {

    // ---------------------------------------------------------------------------------------

    static const RealOpenMM one           = static_cast<RealOpenMM>(1.0);
    static const RealOpenMM threeHalves   = static_cast<RealOpenMM>(1.5);
    static const RealOpenMM third         = static_cast<RealOpenMM>((1.0/3.0));
    static const RealOpenMM fourth        = static_cast<RealOpenMM>(0.25);
    static const RealOpenMM eighth        = static_cast<RealOpenMM>(0.125);

    // ---------------------------------------------------------------------------------------

    RealOpenMM rInv   = one/r;

    RealOpenMM xInv   = one/x;
    RealOpenMM xInv2  = xInv*xInv;
    RealOpenMM xInv3  = xInv2*xInv;

    RealOpenMM diff2  = (r + S)*(r - S);

    return (threeHalves*xInv)*((xInv*fourth*rInv) - (xInv2*third) + (diff2*xInv3*eighth*rInv));
}

/**---------------------------------------------------------------------------------------

    Get partial derivative of L wrt r

    @param r                   distance between atoms i & j
    @param R                   atomic radius
    @param S                   scaled atomic radius

    @return partial derivative based on Eq. 4 of Labute paper [JCC 29 p. 1693-1698 2008])

    --------------------------------------------------------------------------------------- */

RealOpenMM ReferenceGBVI::dL_dr(RealOpenMM r, RealOpenMM x, RealOpenMM S) {

    // ---------------------------------------------------------------------------------------

    static const RealOpenMM one           = static_cast<RealOpenMM>(1.0);
    static const RealOpenMM threeHalves   = static_cast<RealOpenMM>(1.5);
    static const RealOpenMM threeEights   = static_cast<RealOpenMM>(0.375);
    static const RealOpenMM third         = static_cast<RealOpenMM>((1.0/3.0));
    static const RealOpenMM fourth        = static_cast<RealOpenMM>(0.25);
    static const RealOpenMM eighth        = static_cast<RealOpenMM>(0.125);

    // ---------------------------------------------------------------------------------------

    RealOpenMM rInv   = one/r;
    RealOpenMM rInv2  = rInv*rInv;

    RealOpenMM xInv   = one/x;
    RealOpenMM xInv2  = xInv*xInv;
    RealOpenMM xInv3  = xInv2*xInv;

    RealOpenMM diff2  = (r + S)*(r - S);

    return ((-threeHalves*xInv2*rInv2)*(fourth + eighth*diff2*xInv2) + threeEights*xInv3*xInv);
}

/**---------------------------------------------------------------------------------------

    Get partial derivative of L wrt x

    @param r                   distance between atoms i & j
    @param R                   atomic radius
    @param S                   scaled atomic radius

    @return partial derivative based on Eq. 4 of Labute paper [JCC 29 p. 1693-1698 2008])

    --------------------------------------------------------------------------------------- */

RealOpenMM ReferenceGBVI::dL_dx(RealOpenMM r, RealOpenMM x, RealOpenMM S) {

    // ---------------------------------------------------------------------------------------

    static const RealOpenMM one           = static_cast<RealOpenMM>( 1.0);
    static const RealOpenMM half          = static_cast<RealOpenMM>( 0.5);
    static const RealOpenMM threeHalvesM  = static_cast<RealOpenMM>(-1.5);
    static const RealOpenMM third         = static_cast<RealOpenMM>( (1.0/3.0));

    // ---------------------------------------------------------------------------------------

    RealOpenMM rInv   = one/r;

    RealOpenMM xInv   = one/x;
    RealOpenMM xInv2  = xInv*xInv;
    RealOpenMM xInv3  = xInv2*xInv;

    RealOpenMM diff   = (r + S)*(r - S);

    return (threeHalvesM*xInv3)*((half*rInv) - xInv + (half*diff*xInv2*rInv));
}

/**---------------------------------------------------------------------------------------

    Sgb function

    @param t                   r*r*G_i*G_j

    @return Sgb (top of p. 1694 of Labute paper [JCC 29 p. 1693-1698 2008])

    --------------------------------------------------------------------------------------- */

RealOpenMM ReferenceGBVI::Sgb(RealOpenMM t) {

    // ---------------------------------------------------------------------------------------

    // static const char* methodName = "ReferenceGBVI::Sgb";

    static const RealOpenMM zero    = static_cast<RealOpenMM>(0.0);
    static const RealOpenMM one     = static_cast<RealOpenMM>(1.0);
    static const RealOpenMM fourth  = static_cast<RealOpenMM>(0.25);

    // ---------------------------------------------------------------------------------------

    return ((t != zero) ? one/SQRT((one + (fourth*EXP(-t))/t)) : zero);
}

/**---------------------------------------------------------------------------------------

    Get GB/VI energy

    @param atomCoordinates     atomic coordinates
    @param partialCharges      partial charges

    @return energy

    --------------------------------------------------------------------------------------- */

RealOpenMM ReferenceGBVI::computeBornEnergy(const vector<RealVec>& atomCoordinates, const vector<RealOpenMM>& partialCharges) {

    // ---------------------------------------------------------------------------------------

    static const RealOpenMM zero          = static_cast<RealOpenMM>(0.0);
    static const RealOpenMM one           = static_cast<RealOpenMM>(1.0);
    static const RealOpenMM two           = static_cast<RealOpenMM>(2.0);
    static const RealOpenMM three         = static_cast<RealOpenMM>(3.0);
    static const RealOpenMM four          = static_cast<RealOpenMM>(4.0);
    static const RealOpenMM half          = static_cast<RealOpenMM>(0.5);
    static const RealOpenMM fourth        = static_cast<RealOpenMM>(0.25);
    static const RealOpenMM eighth        = static_cast<RealOpenMM>(0.125);

    // ---------------------------------------------------------------------------------------

    const GBVIParameters* gbviParameters       = getGBVIParameters();
    const RealOpenMM preFactor                 = gbviParameters->getElectricConstant();
    const int numberOfAtoms                    = gbviParameters->getNumberOfAtoms();
    const vector<RealOpenMM>& atomicRadii        = gbviParameters->getAtomicRadii();
    const vector<RealOpenMM>& gammaParameters    = gbviParameters->getGammaParameters();

    // compute Born radii

    vector<RealOpenMM> bornRadii(numberOfAtoms);
    computeBornRadii(atomCoordinates, bornRadii);

    // ---------------------------------------------------------------------------------------

    // Eq.2 of Labute paper [JCC 29 p. 1693-1698 2008]
    // to minimze roundoff error sum cavityEnergy separately since in general much
    // smaller than other contributions

    RealOpenMM energy                 = zero;
    RealOpenMM cavityEnergy           = zero;

    for (int atomI = 0; atomI < numberOfAtoms; atomI++) {
 
        RealOpenMM partialChargeI   = partialCharges[atomI];
 
        // self-energy term
 
        RealOpenMM  atomIEnergy     = half*partialChargeI/bornRadii[atomI];
 
        // cavity term
 
        RealOpenMM ratio            = (atomicRadii[atomI]/bornRadii[atomI]);
        cavityEnergy               += gammaParameters[atomI]*ratio*ratio*ratio;
 
        for (int atomJ = atomI + 1; atomJ < numberOfAtoms; atomJ++) {
 
            RealOpenMM deltaR[ReferenceForce::LastDeltaRIndex];
            if (_gbviParameters->getPeriodic())
                ReferenceForce::getDeltaRPeriodic(atomCoordinates[atomI], atomCoordinates[atomJ], _gbviParameters->getPeriodicBox(), deltaR);
            else
                ReferenceForce::getDeltaR(atomCoordinates[atomI], atomCoordinates[atomJ], deltaR);
            if (_gbviParameters->getUseCutoff() && deltaR[ReferenceForce::RIndex] > _gbviParameters->getCutoffDistance())
                continue;
  
            RealOpenMM r2           = deltaR[ReferenceForce::R2Index];
            RealOpenMM t            = fourth*r2/(bornRadii[atomI]*bornRadii[atomJ]);         
            atomIEnergy            += partialCharges[atomJ]*Sgb(t)/deltaR[ReferenceForce::RIndex];
        }
 
        energy += two*partialChargeI*atomIEnergy;
    }
    energy               *= preFactor;
    energy               -= cavityEnergy;

    RealOpenMM conversion = static_cast<RealOpenMM>(gbviParameters->getTau());  
    return (conversion*energy);
 
}

/**---------------------------------------------------------------------------------------

    Get GB/VI forces

    @param atomCoordinates     atomic coordinates
    @param partialCharges      partial charges
    @param forces              forces

    --------------------------------------------------------------------------------------- */


void ReferenceGBVI::computeBornForces(std::vector<RealVec>& atomCoordinates, const vector<RealOpenMM>& partialCharges,
                                 std::vector<OpenMM::RealVec>& inputForces) {

    // ---------------------------------------------------------------------------------------

    static const RealOpenMM zero               = static_cast<RealOpenMM>(0.0);
    static const RealOpenMM one                = static_cast<RealOpenMM>(1.0);
    static const RealOpenMM two                = static_cast<RealOpenMM>(2.0);
    static const RealOpenMM three              = static_cast<RealOpenMM>(3.0);
    static const RealOpenMM four               = static_cast<RealOpenMM>(4.0);
    static const RealOpenMM half               = static_cast<RealOpenMM>(0.5);
    static const RealOpenMM oneThird           = static_cast<RealOpenMM>((1.0/3.0));
    static const RealOpenMM fourth             = static_cast<RealOpenMM>(0.25);
    static const RealOpenMM eighth             = static_cast<RealOpenMM>(0.125);

    // ---------------------------------------------------------------------------------------

    const GBVIParameters* gbviParameters       = getGBVIParameters();
    const int numberOfAtoms                    = gbviParameters->getNumberOfAtoms();
    const vector<RealOpenMM>& atomicRadii        = gbviParameters->getAtomicRadii();
    const vector<RealOpenMM>& gammaParameters    = gbviParameters->getGammaParameters();

    // ---------------------------------------------------------------------------------------

    // constants

    const RealOpenMM preFactor                 = two*gbviParameters->getElectricConstant();

    // ---------------------------------------------------------------------------------------

    // compute Born radii

    vector<RealOpenMM> bornRadii(numberOfAtoms);
    computeBornRadii(atomCoordinates, bornRadii);

    // set energy/forces to zero

    std::vector<OpenMM::RealVec> forces(numberOfAtoms);
    for (int ii = 0; ii < numberOfAtoms; ii++) {
        forces[ii][0] = zero;
        forces[ii][1] = zero;
        forces[ii][2] = zero;
    }

    vector<RealOpenMM> bornForces(numberOfAtoms, 0.0);

    // ---------------------------------------------------------------------------------------

    // first main loop

    for (int atomI = 0; atomI < numberOfAtoms; atomI++) {
 
        // partial of polar term wrt Born radius
        // and (dGpol/dr)(dr/dx)
 
        RealOpenMM partialChargeI = preFactor*partialCharges[atomI];
        for (int atomJ = atomI; atomJ < numberOfAtoms; atomJ++) {
 
            RealOpenMM deltaR[ReferenceForce::LastDeltaRIndex];
            if (_gbviParameters->getPeriodic())
                ReferenceForce::getDeltaRPeriodic(atomCoordinates[atomI], atomCoordinates[atomJ], _gbviParameters->getPeriodicBox(), deltaR);
            else
                ReferenceForce::getDeltaR(atomCoordinates[atomI], atomCoordinates[atomJ], deltaR);
            if (_gbviParameters->getUseCutoff() && deltaR[ReferenceForce::RIndex] > _gbviParameters->getCutoffDistance())
                continue;
  
            RealOpenMM r2                 = deltaR[ReferenceForce::R2Index];
  
            RealOpenMM deltaX             = deltaR[ReferenceForce::XIndex];
            RealOpenMM deltaY             = deltaR[ReferenceForce::YIndex];
            RealOpenMM deltaZ             = deltaR[ReferenceForce::ZIndex];
  
            RealOpenMM alpha2_ij          = bornRadii[atomI]*bornRadii[atomJ];
            RealOpenMM D_ij               = r2/(four*alpha2_ij);
  
            RealOpenMM expTerm            = EXP(-D_ij);
            RealOpenMM denominator2       = r2 + alpha2_ij*expTerm; 
            RealOpenMM denominator        = SQRT(denominator2); 
            
            RealOpenMM Gpol               = (partialChargeI*partialCharges[atomJ])/denominator; 
            RealOpenMM dGpol_dr           = -Gpol*(one - fourth*expTerm)/denominator2;  
  
            RealOpenMM dGpol_dalpha2_ij   = -half*Gpol*expTerm*(one + D_ij)/denominator2;
  
            if (atomI != atomJ) {
  
                bornForces[atomJ] += dGpol_dalpha2_ij*bornRadii[atomI];
   
                deltaX            *= dGpol_dr;
                deltaY            *= dGpol_dr;
                deltaZ            *= dGpol_dr;
   
                forces[atomI][0]  += deltaX;
                forces[atomI][1]  += deltaY;
                forces[atomI][2]  += deltaZ;
  
                forces[atomJ][0]  -= deltaX;
                forces[atomJ][1]  -= deltaY;
                forces[atomJ][2]  -= deltaZ;
   
            }
            bornForces[atomI] += dGpol_dalpha2_ij*bornRadii[atomJ];
 
        }
    }

    // ---------------------------------------------------------------------------------------

    // second main loop: (dGpol/dBornRadius)(dBornRadius/dr)(dr/dx)

    // dGpol/dBornRadius) = bornForces[]
    // dBornRadius/dr     = (1/3)*(bR**4)*(dV/dr)

    const vector<RealOpenMM>& scaledRadii           = gbviParameters->getScaledRadii();
    const vector<RealOpenMM>& switchDeriviative     = getSwitchDeriviative();
    for (int atomI = 0; atomI < numberOfAtoms; atomI++) {
 
        RealOpenMM R        = atomicRadii[atomI];
 
        // partial of cavity term wrt Born radius
       
        RealOpenMM  ratio   = (atomicRadii[atomI]/bornRadii[atomI]);
        bornForces[atomI]  += (three*gammaParameters[atomI]*ratio*ratio*ratio)/bornRadii[atomI]; 
 
        RealOpenMM b2       = bornRadii[atomI]*bornRadii[atomI];
        bornForces[atomI]  *= switchDeriviative[atomI]*oneThird*b2*b2;
 
        for (int atomJ = 0; atomJ < numberOfAtoms; atomJ++) {
 
            if (atomJ != atomI) {
  
                RealOpenMM deltaX             = atomCoordinates[atomJ][0] - atomCoordinates[atomI][0];
                RealOpenMM deltaY             = atomCoordinates[atomJ][1] - atomCoordinates[atomI][1];
                RealOpenMM deltaZ             = atomCoordinates[atomJ][2] - atomCoordinates[atomI][2];
        
                RealOpenMM deltaR[ReferenceForce::LastDeltaRIndex];
                if (_gbviParameters->getPeriodic())
                    ReferenceForce::getDeltaRPeriodic(atomCoordinates[atomI], atomCoordinates[atomJ], _gbviParameters->getPeriodicBox(), deltaR);
                else
                    ReferenceForce::getDeltaR(atomCoordinates[atomI], atomCoordinates[atomJ], deltaR);
                if (_gbviParameters->getUseCutoff() && deltaR[ReferenceForce::RIndex] > _gbviParameters->getCutoffDistance())
                    continue;
       
                RealOpenMM r2                 = deltaR[ReferenceForce::R2Index];
                           deltaX             = deltaR[ReferenceForce::XIndex];
                           deltaY             = deltaR[ReferenceForce::YIndex];
                           deltaZ             = deltaR[ReferenceForce::ZIndex];
   
                RealOpenMM r                  = SQRT(r2);
    
                RealOpenMM S                  = scaledRadii[atomJ];
                RealOpenMM diff               = (S - R);
   
                RealOpenMM de                 = zero;
   
                // find dRb/dr, where Rb is the Born radius
   
                if (FABS(diff) < r) {
                    de = ReferenceGBVI::dL_dr(r, r+S, S) + ReferenceGBVI::dL_dx(r, r+S, S);   
                    if (R > (r - S)) {
                       de -= ReferenceGBVI::dL_dr(r, R, S);  
                    } else {
                       de -= (ReferenceGBVI::dL_dr(r, (r-S), S) + ReferenceGBVI::dL_dx(r, (r-S), S));
                    }
                } else if (r < (S - R)) {
                    de  = ReferenceGBVI::dL_dr(r, r+S, S) + ReferenceGBVI::dL_dx(r, r+S, S);   
                    de -= (ReferenceGBVI::dL_dr(r, r-S, S) + ReferenceGBVI::dL_dx(r, r-S, S));   
                }
   
                 // de = (dG/dRb)(dRb/dr)
   
                de                      *= bornForces[atomI]/r;
   
                deltaX                  *= de;
                deltaY                  *= de;
                deltaZ                  *= de;
       
                forces[atomI][0]        += deltaX;
                forces[atomI][1]        += deltaY;
                forces[atomI][2]        += deltaZ;
     
                forces[atomJ][0]        -= deltaX;
                forces[atomJ][1]        -= deltaY;
                forces[atomJ][2]        -= deltaZ;
   
            }
        }
    }

    // convert from cal to Joule & apply prefactor tau = (1/diel_solute - 1/diel_solvent)

    RealOpenMM conversion = static_cast<RealOpenMM>(gbviParameters->getTau());  
    for (int atomI = 0; atomI < numberOfAtoms; atomI++) {
       inputForces[atomI][0] += conversion*forces[atomI][0];
       inputForces[atomI][1] += conversion*forces[atomI][1];
       inputForces[atomI][2] += conversion*forces[atomI][2];
    }

}

/**---------------------------------------------------------------------------------------

    Use double precision 

    Get volume Eq. 4 of Labute paper [JCC 29 p. 1693-1698 2008])

    @param r                   distance between atoms i & j
    @param R                   atomic radius
    @param S                   scaled atomic radius

    @return volume

    --------------------------------------------------------------------------------------- */

double ReferenceGBVI::getVolumeD(double r, double R, double S) {

    // ---------------------------------------------------------------------------------------

    static const double zero         = 0.0;
    static const double minusThree   = -3.0;

    double              diff    = (S - R);
    if (fabs(diff) < r) {

       double lowerBound = (R > (r - S)) ? R : (r - S);

       return (ReferenceGBVI::getLD(r, (r + S),    S) -
               ReferenceGBVI::getLD(r, lowerBound, S));

    } else if (r < diff) {

       return ReferenceGBVI::getLD(r, (r + S), S) -
              ReferenceGBVI::getLD(r, (r - S), S) + 
              pow(R, minusThree);

    } else {
       return zero;
    }
}

/**---------------------------------------------------------------------------------------

    Use double precision 

    Get L (used in analytical solution for volume integrals) 

    @param r                   distance between atoms i & j
    @param R                   atomic radius
    @param S                   scaled atomic radius

    @return L value (Eq. 4 of Labute paper [JCC 29 p. 1693-1698 2008])

    --------------------------------------------------------------------------------------- */

double ReferenceGBVI::getLD(double r, double x, double S) {

    // ---------------------------------------------------------------------------------------

    static const double one           =  1.0;
    static const double threeHalves   =  1.5;
    static const double third         =  1.0/3.0;
    static const double fourth        =  0.25;
    static const double eighth        =  0.125;

    // ---------------------------------------------------------------------------------------

    double rInv   = one/r;

    double xInv   = one/x;
    double xInv2  = xInv*xInv;
    double xInv3  = xInv2*xInv;

    double diff2  = (r + S)*(r - S);

    return (threeHalves*xInv)*((xInv*fourth*rInv) - (xInv2*third) + (diff2*xInv3*eighth*rInv));
}

/**---------------------------------------------------------------------------------------

    Use double precision 

    Get partial derivative of L wrt r

    @param r                   distance between atoms i & j
    @param R                   atomic radius
    @param S                   scaled atomic radius

    @return partial derivative based on Eq. 4 of Labute paper [JCC 29 p. 1693-1698 2008])

    --------------------------------------------------------------------------------------- */

double ReferenceGBVI::dL_drD(double r, double x, double S) {

    // ---------------------------------------------------------------------------------------

    static const double one           =  1.0;
    static const double threeHalves   =  1.5;
    static const double threeEights   =  0.375;
    static const double third         =  1.0/3.0;
    static const double fourth        =  0.25;
    static const double eighth        =  0.125;

    // ---------------------------------------------------------------------------------------

    double rInv   = one/r;
    double rInv2  = rInv*rInv;

    double xInv   = one/x;
    double xInv2  = xInv*xInv;
    double xInv3  = xInv2*xInv;

    double diff2  = (r + S)*(r - S);

    return ((-threeHalves*xInv2*rInv2)*(fourth + eighth*diff2*xInv2) + threeEights*xInv3*xInv);
}

/**---------------------------------------------------------------------------------------

    Use double precision 

    Get partial derivative of L wrt x

    @param r                   distance between atoms i & j
    @param R                   atomic radius
    @param S                   scaled atomic radius

    @return partial derivative based on Eq. 4 of Labute paper [JCC 29 p. 1693-1698 2008])

    --------------------------------------------------------------------------------------- */

double ReferenceGBVI::dL_dxD(double r, double x, double S) {

    // ---------------------------------------------------------------------------------------

    static const double one           =   1.0;
    static const double half          =   0.5;
    static const double threeHalvesM  =  -1.5;
    static const double third         =   1.0/3.0;

    // ---------------------------------------------------------------------------------------

    double rInv   = one/r;

    double xInv   = one/x;
    double xInv2  = xInv*xInv;
    double xInv3  = xInv2*xInv;

    double diff   = (r + S)*(r - S);

    return (threeHalvesM*xInv3)*((half*rInv) - xInv + (half*diff*xInv2*rInv));
}