#define WARPS_PER_GROUP (THREAD_BLOCK_SIZE/TILE_SIZE)

typedef struct {
    real3 pos, force, torque, inducedDipole, inducedDipolePolar, sphericalDipole;
    real q;
    float thole, damp;
#ifdef INCLUDE_QUADRUPOLES
    real sphericalQuadrupole[5];
#endif
} AtomData;

inline DEVICE AtomData loadAtomData(int atom, GLOBAL const real4* RESTRICT posq, GLOBAL const real* RESTRICT sphericalDipole,
            GLOBAL const real* RESTRICT sphericalQuadrupole, GLOBAL const real* RESTRICT inducedDipole, GLOBAL const real* RESTRICT inducedDipolePolar,
            GLOBAL const float2* RESTRICT dampingAndThole) {
    AtomData data;
    real4 atomPosq = posq[atom];
    data.pos = make_real3(atomPosq.x, atomPosq.y, atomPosq.z);
    data.q = atomPosq.w;
    data.sphericalDipole.x = sphericalDipole[atom*3];
    data.sphericalDipole.y = sphericalDipole[atom*3+1];
    data.sphericalDipole.z = sphericalDipole[atom*3+2];
#ifdef INCLUDE_QUADRUPOLES
    data.sphericalQuadrupole[0] = sphericalQuadrupole[atom*5];
    data.sphericalQuadrupole[1] = sphericalQuadrupole[atom*5+1];
    data.sphericalQuadrupole[2] = sphericalQuadrupole[atom*5+2];
    data.sphericalQuadrupole[3] = sphericalQuadrupole[atom*5+3];
    data.sphericalQuadrupole[4] = sphericalQuadrupole[atom*5+4];
#endif
    data.inducedDipole = make_real3(inducedDipole[3*atom], inducedDipole[3*atom+1], inducedDipole[3*atom+2]);
    data.inducedDipolePolar = make_real3(inducedDipolePolar[3*atom], inducedDipolePolar[3*atom+1], inducedDipolePolar[3*atom+2]);
    float2 temp = dampingAndThole[atom];
    data.damp = temp.x;
    data.thole = temp.y;
    return data;
}

DEVICE real computeDScaleFactor(unsigned int polarizationGroup, int index) {
    return (polarizationGroup & 1<<index ? 0 : 1);
}

DEVICE float computeMScaleFactor(uint2 covalent, int index) {
    int mask = 1<<index;
    bool x = (covalent.x & mask);
    bool y = (covalent.y & mask);
    return (x ? (y ? 0.0f : 0.4f) : (y ? 0.8f : 1.0f));
}

DEVICE float computePScaleFactor(uint2 covalent, unsigned int polarizationGroup, int index) {
    int mask = 1<<index;
    bool x = (covalent.x & mask);
    bool y = (covalent.y & mask);
    bool p = (polarizationGroup & mask);
    return (x && y ? 0.0f : (x && p ? 0.5f : 1.0f));
}

DEVICE void computeOneInteraction(AtomData* atom1, LOCAL_ARG AtomData* atom2, bool hasExclusions, float dScale, float pScale, float mScale, float forceFactor,
                                      mixed* energy, real4 periodicBoxSize, real4 invPeriodicBoxSize, real4 periodicBoxVecX, real4 periodicBoxVecY, real4 periodicBoxVecZ) {
    // Compute the displacement.
    
    real3 delta;
    delta.x = atom2->pos.x - atom1->pos.x;
    delta.y = atom2->pos.y - atom1->pos.y;
    delta.z = atom2->pos.z - atom1->pos.z;
    APPLY_PERIODIC_TO_DELTA(delta)
    real r2 = delta.x*delta.x + delta.y*delta.y + delta.z*delta.z;
    if (r2 > CUTOFF_SQUARED)
        return;

    real rInv = RSQRT(r2);
    real r = r2*rInv;

    // Rotate the various dipoles and quadrupoles.

    real qiRotationMatrix[3][3];
    buildQIRotationMatrix(delta, rInv, qiRotationMatrix);

    
    real3 qiUindI = 0.5f*make_real3(qiRotationMatrix[0][1]*atom1->inducedDipole.x + qiRotationMatrix[0][2]*atom1->inducedDipole.y + qiRotationMatrix[0][0]*atom1->inducedDipole.z,
                                    qiRotationMatrix[1][1]*atom1->inducedDipole.x + qiRotationMatrix[1][2]*atom1->inducedDipole.y + qiRotationMatrix[1][0]*atom1->inducedDipole.z,
                                    qiRotationMatrix[2][1]*atom1->inducedDipole.x + qiRotationMatrix[2][2]*atom1->inducedDipole.y + qiRotationMatrix[2][0]*atom1->inducedDipole.z);
    real3 qiUindJ = 0.5f*make_real3(qiRotationMatrix[0][1]*atom2->inducedDipole.x + qiRotationMatrix[0][2]*atom2->inducedDipole.y + qiRotationMatrix[0][0]*atom2->inducedDipole.z,
                                    qiRotationMatrix[1][1]*atom2->inducedDipole.x + qiRotationMatrix[1][2]*atom2->inducedDipole.y + qiRotationMatrix[1][0]*atom2->inducedDipole.z,
                                    qiRotationMatrix[2][1]*atom2->inducedDipole.x + qiRotationMatrix[2][2]*atom2->inducedDipole.y + qiRotationMatrix[2][0]*atom2->inducedDipole.z);
    real3 qiUinpI = 0.5f*make_real3(qiRotationMatrix[0][1]*atom1->inducedDipolePolar.x + qiRotationMatrix[0][2]*atom1->inducedDipolePolar.y + qiRotationMatrix[0][0]*atom1->inducedDipolePolar.z,
                                    qiRotationMatrix[1][1]*atom1->inducedDipolePolar.x + qiRotationMatrix[1][2]*atom1->inducedDipolePolar.y + qiRotationMatrix[1][0]*atom1->inducedDipolePolar.z,
                                    qiRotationMatrix[2][1]*atom1->inducedDipolePolar.x + qiRotationMatrix[2][2]*atom1->inducedDipolePolar.y + qiRotationMatrix[2][0]*atom1->inducedDipolePolar.z);
    real3 qiUinpJ = 0.5f*make_real3(qiRotationMatrix[0][1]*atom2->inducedDipolePolar.x + qiRotationMatrix[0][2]*atom2->inducedDipolePolar.y + qiRotationMatrix[0][0]*atom2->inducedDipolePolar.z,
                                    qiRotationMatrix[1][1]*atom2->inducedDipolePolar.x + qiRotationMatrix[1][2]*atom2->inducedDipolePolar.y + qiRotationMatrix[1][0]*atom2->inducedDipolePolar.z,
                                    qiRotationMatrix[2][1]*atom2->inducedDipolePolar.x + qiRotationMatrix[2][2]*atom2->inducedDipolePolar.y + qiRotationMatrix[2][0]*atom2->inducedDipolePolar.z);
    
    real3 rotatedDipole1 = rotateDipole(atom1->sphericalDipole, qiRotationMatrix);
    real3 rotatedDipole2 = rotateDipole(atom2->sphericalDipole, qiRotationMatrix);
    real rotatedQuadrupole1[] = {0, 0, 0, 0, 0};
    real rotatedQuadrupole2[] = {0, 0, 0, 0, 0};
#ifdef INCLUDE_QUADRUPOLES
    rotateQuadrupoles(qiRotationMatrix, atom1->sphericalQuadrupole, atom2->sphericalQuadrupole, rotatedQuadrupole1, rotatedQuadrupole2);
#endif    
    
    // The field derivatives at I due to permanent and induced moments on J, and vice-versa.
    // Also, their derivatives w.r.t. R, which are needed for force calculations
    real Vij[9], Vji[9], VjiR[9], VijR[9];
    // The field derivatives at I due to only permanent moments on J, and vice-versa.
    real Vijp[3], Vijd[3], Vjip[3], Vjid[3];
    real rInvVec[7], alphaRVec[8], bVec[5];

    // The rInvVec array is defined such that the ith element is R^-i, with the
    // dieleectric constant folded in, to avoid conversions later.
    rInvVec[1] = rInv;
    for (int i = 2; i < 7; ++i)
        rInvVec[i] = rInvVec[i-1] * rInv;

    // The alpharVec array is defined such that the ith element is (alpha R)^i,
    // where kappa (alpha in OpenMM parlance) is the Ewald attenuation parameter.
    real ralpha = EWALD_ALPHA*r;
    real exp2a = EXP(-(ralpha*ralpha));
#ifdef USE_DOUBLE_PRECISION
    const real erfAlphaR = erf(ralpha);
#else
    // This approximation for erfc is from Abramowitz and Stegun (1964) p. 299.  They cite the following as
    // the original source: C. Hastings, Jr., Approximations for Digital Computers (1955).  It has a maximum
    // error of 1.5e-7.

    const real t = RECIP(1.0f+0.3275911f*ralpha);
    const real erfAlphaR = 1-(0.254829592f+(-0.284496736f+(1.421413741f+(-1.453152027f+1.061405429f*t)*t)*t)*t)*t*exp2a;
#endif
    alphaRVec[1] = ralpha;
    for (int i = 2; i < 8; ++i)
        alphaRVec[i] = alphaRVec[i-1]*ralpha;
    real X = 2*exp2a/SQRT_PI;
    int doubleFactorial = 1, facCount = 1;
    real tmp = alphaRVec[1];
    bVec[1] = -erfAlphaR;
    for (int i = 2; i < 5; ++i) {
        bVec[i] = bVec[i-1] + tmp * X / (real)(doubleFactorial);
        facCount = facCount + 2;
        doubleFactorial = doubleFactorial * facCount;
        tmp *= 2*alphaRVec[2];
    }

    real dmp = atom1->damp*atom2->damp;
    real a = min(atom1->thole, atom2->thole);
    real u = r/dmp;
    real au3 = fabs(dmp) > 1.0e-5f ? a*u*u*u : 0;
    real expau3 = fabs(dmp) > 1.0e-5f ? EXP(-au3) : 0;
    real a2u6 = au3*au3;
    real a3u9 = a2u6*au3;
    // Thole damping factors for energies
    real thole_c  = 1 - expau3;
    real thole_d0 = 1 - expau3*(1 + 1.5f*au3);
    real thole_d1 = 1 - expau3;
    real thole_q0 = 1 - expau3*(1 + au3 + a2u6);
    real thole_q1 = 1 - expau3*(1 + au3);
    // Thole damping factors for derivatives
    real dthole_c  = 1 - expau3*(1 + 1.5f*au3);
    real dthole_d0 = 1 - expau3*(1 + au3 + 1.5f*a2u6);
    real dthole_d1 = 1 - expau3*(1 + au3);
    real dthole_q0 = 1 - expau3*(1 + au3 + 0.25f*a2u6 + 0.75f*a3u9);
    real dthole_q1 = 1 - expau3*(1 + au3 + 0.75f*a2u6);

    // Now we compute the (attenuated) Coulomb operator and its derivatives, contracted with
    // permanent moments and induced dipoles.  Note that the coefficient of the permanent force
    // terms is half of the expected value; this is because we compute the interaction of I with
    // the sum of induced and permanent moments on J, as well as the interaction of J with I's
    // permanent and induced moments; doing so double counts the permanent-permanent interaction.
    real ePermCoef, dPermCoef, eUIndCoef, dUIndCoef, eUInpCoef, dUInpCoef;

    // C-C terms (m=0)
    ePermCoef = rInvVec[1]*(mScale + bVec[2] - alphaRVec[1]*X);
    dPermCoef = -0.5f*(mScale + bVec[2])*rInvVec[2];
    Vij[0]  = ePermCoef*atom2->q;
    Vji[0]  = ePermCoef*atom1->q;
    VijR[0] = dPermCoef*atom2->q;
    VjiR[0] = dPermCoef*atom1->q;

    // C-D and C-Uind terms (m=0)
    ePermCoef = rInvVec[2]*(mScale + bVec[2]);
    eUIndCoef = rInvVec[2]*(pScale*thole_c + bVec[2]);
    eUInpCoef = rInvVec[2]*(dScale*thole_c + bVec[2]);
    dPermCoef = -rInvVec[3]*(mScale + bVec[2] + alphaRVec[3]*X);
    dUIndCoef = -2*rInvVec[3]*(pScale*dthole_c + bVec[2] + alphaRVec[3]*X);
    dUInpCoef = -2*rInvVec[3]*(dScale*dthole_c + bVec[2] + alphaRVec[3]*X);
    Vij[0]  += -(ePermCoef*rotatedDipole2.x + eUIndCoef*qiUindJ.x + eUInpCoef*qiUinpJ.x);
    Vji[1]   = -(ePermCoef*atom1->q);
    VijR[0] += -(dPermCoef*rotatedDipole2.x + dUIndCoef*qiUindJ.x + dUInpCoef*qiUinpJ.x);
    VjiR[1]  = -(dPermCoef*atom1->q);
    Vjip[0]  = -(eUInpCoef*atom1->q);
    Vjid[0]  = -(eUIndCoef*atom1->q);
    // D-C and Uind-C terms (m=0)
    Vij[1]   = ePermCoef*atom2->q;
    Vji[0]  += ePermCoef*rotatedDipole1.x + eUIndCoef*qiUindI.x + eUInpCoef*qiUinpI.x;
    VijR[1]  = dPermCoef*atom2->q;
    VjiR[0] += dPermCoef*rotatedDipole1.x + dUIndCoef*qiUindI.x + dUInpCoef*qiUinpI.x;
    Vijp[0]  = eUInpCoef*atom2->q;
    Vijd[0]  = eUIndCoef*atom2->q;

    // D-D and D-Uind terms (m=0)
    const real twoThirds = (real) 2/3;
    ePermCoef = -twoThirds*rInvVec[3]*(3*(mScale + bVec[3]) + alphaRVec[3]*X);
    eUIndCoef = -twoThirds*rInvVec[3]*(3*(pScale*thole_d0 + bVec[3]) + alphaRVec[3]*X);
    eUInpCoef = -twoThirds*rInvVec[3]*(3*(dScale*thole_d0 + bVec[3]) + alphaRVec[3]*X);
    dPermCoef = rInvVec[4]*(3*(mScale + bVec[3]) + 2*alphaRVec[5]*X);
    dUIndCoef = rInvVec[4]*(6*(pScale*dthole_d0 + bVec[3]) + 4*alphaRVec[5]*X);
    dUInpCoef = rInvVec[4]*(6*(dScale*dthole_d0 + bVec[3]) + 4*alphaRVec[5]*X);
    Vij[1]  += ePermCoef*rotatedDipole2.x + eUIndCoef*qiUindJ.x + eUInpCoef*qiUinpJ.x;
    Vji[1]  += ePermCoef*rotatedDipole1.x + eUIndCoef*qiUindI.x + eUInpCoef*qiUinpI.x;
    VijR[1] += dPermCoef*rotatedDipole2.x + dUIndCoef*qiUindJ.x + dUInpCoef*qiUinpJ.x;
    VjiR[1] += dPermCoef*rotatedDipole1.x + dUIndCoef*qiUindI.x + dUInpCoef*qiUinpI.x;
    Vijp[0] += eUInpCoef*rotatedDipole2.x;
    Vijd[0] += eUIndCoef*rotatedDipole2.x;
    Vjip[0] += eUInpCoef*rotatedDipole1.x;
    Vjid[0] += eUIndCoef*rotatedDipole1.x;
    // D-D and D-Uind terms (m=1)
    ePermCoef = rInvVec[3]*(mScale + bVec[3] - twoThirds*alphaRVec[3]*X);
    eUIndCoef = rInvVec[3]*(pScale*thole_d1 + bVec[3] - twoThirds*alphaRVec[3]*X);
    eUInpCoef = rInvVec[3]*(dScale*thole_d1 + bVec[3] - twoThirds*alphaRVec[3]*X);
    dPermCoef = -1.5f*rInvVec[4]*(mScale + bVec[3]);
    dUIndCoef = -3*rInvVec[4]*(pScale*dthole_d1 + bVec[3]);
    dUInpCoef = -3*rInvVec[4]*(dScale*dthole_d1 + bVec[3]);
    Vij[2]  = ePermCoef*rotatedDipole2.y + eUIndCoef*qiUindJ.y + eUInpCoef*qiUinpJ.y;
    Vji[2]  = ePermCoef*rotatedDipole1.y + eUIndCoef*qiUindI.y + eUInpCoef*qiUinpI.y;
    VijR[2] = dPermCoef*rotatedDipole2.y + dUIndCoef*qiUindJ.y + dUInpCoef*qiUinpJ.y;
    VjiR[2] = dPermCoef*rotatedDipole1.y + dUIndCoef*qiUindI.y + dUInpCoef*qiUinpI.y;
    Vij[3]  = ePermCoef*rotatedDipole2.z + eUIndCoef*qiUindJ.z + eUInpCoef*qiUinpJ.z;
    Vji[3]  = ePermCoef*rotatedDipole1.z + eUIndCoef*qiUindI.z + eUInpCoef*qiUinpI.z;
    VijR[3] = dPermCoef*rotatedDipole2.z + dUIndCoef*qiUindJ.z + dUInpCoef*qiUinpJ.z;
    VjiR[3] = dPermCoef*rotatedDipole1.z + dUIndCoef*qiUindI.z + dUInpCoef*qiUinpI.z;
    Vijp[1] = eUInpCoef*rotatedDipole2.y;
    Vijd[1] = eUIndCoef*rotatedDipole2.y;
    Vjip[1] = eUInpCoef*rotatedDipole1.y;
    Vjid[1] = eUIndCoef*rotatedDipole1.y;
    Vijp[2] = eUInpCoef*rotatedDipole2.z;
    Vijd[2] = eUIndCoef*rotatedDipole2.z;
    Vjip[2] = eUInpCoef*rotatedDipole1.z;
    Vjid[2] = eUIndCoef*rotatedDipole1.z;

    // C-Q terms (m=0)
    ePermCoef = (mScale + bVec[3])*rInvVec[3];
    dPermCoef = -((real) 1/3)*rInvVec[4]*(4.5f*(mScale + bVec[3]) + 2*alphaRVec[5]*X);
    Vij[0]  += ePermCoef*rotatedQuadrupole2[0];
    Vji[4]   = ePermCoef*atom1->q;
    VijR[0] += dPermCoef*rotatedQuadrupole2[0];
    VjiR[4]  = dPermCoef*atom1->q;
    // Q-C terms (m=0)
    Vij[4]   = ePermCoef*atom2->q;
    Vji[0]  += ePermCoef*rotatedQuadrupole1[0];
    VijR[4]  = dPermCoef*atom2->q;
    VjiR[0] += dPermCoef*rotatedQuadrupole1[0];

    // D-Q and Uind-Q terms (m=0)
    const real fourThirds = (real) 4/3;
    ePermCoef = rInvVec[4]*(3*(mScale + bVec[3]) + fourThirds*alphaRVec[5]*X);
    eUIndCoef = rInvVec[4]*(3*(pScale*thole_q0 + bVec[3]) + fourThirds*alphaRVec[5]*X);
    eUInpCoef = rInvVec[4]*(3*(dScale*thole_q0 + bVec[3]) + fourThirds*alphaRVec[5]*X);
    dPermCoef = -fourThirds*rInvVec[5]*(4.5f*(mScale + bVec[3]) + (1 + alphaRVec[2])*alphaRVec[5]*X);
    dUIndCoef = -fourThirds*rInvVec[5]*(9*(pScale*dthole_q0 + bVec[3]) + 2*(1 + alphaRVec[2])*alphaRVec[5]*X);
    dUInpCoef = -fourThirds*rInvVec[5]*(9*(dScale*dthole_q0 + bVec[3]) + 2*(1 + alphaRVec[2])*alphaRVec[5]*X);
    Vij[1]  += ePermCoef*rotatedQuadrupole2[0];
    Vji[4]  += ePermCoef*rotatedDipole1.x + eUIndCoef*qiUindI.x + eUInpCoef*qiUinpI.x;
    VijR[1] += dPermCoef*rotatedQuadrupole2[0];
    VjiR[4] += dPermCoef*rotatedDipole1.x + dUIndCoef*qiUindI.x + dUInpCoef*qiUinpI.x;
    Vijp[0] += eUInpCoef*rotatedQuadrupole2[0];
    Vijd[0] += eUIndCoef*rotatedQuadrupole2[0];
    // Q-D and Q-Uind terms (m=0)
    Vij[4]  += -(ePermCoef*rotatedDipole2.x + eUIndCoef*qiUindJ.x + eUInpCoef*qiUinpJ.x);
    Vji[1]  += -(ePermCoef*rotatedQuadrupole1[0]);
    VijR[4] += -(dPermCoef*rotatedDipole2.x + dUIndCoef*qiUindJ.x + dUInpCoef*qiUinpJ.x);
    VjiR[1] += -(dPermCoef*rotatedQuadrupole1[0]);
    Vjip[0] += -(eUInpCoef*rotatedQuadrupole1[0]);
    Vjid[0] += -(eUIndCoef*rotatedQuadrupole1[0]);

    // D-Q and Uind-Q terms (m=1)
    const real sqrtThree = SQRT((real) 3);
    ePermCoef = -sqrtThree*rInvVec[4]*(mScale + bVec[3]);
    eUIndCoef = -sqrtThree*rInvVec[4]*(pScale*thole_q1 + bVec[3]);
    eUInpCoef = -sqrtThree*rInvVec[4]*(dScale*thole_q1 + bVec[3]);
    const real fourSqrtOneThird = 4/sqrt((real) 3);
    dPermCoef = fourSqrtOneThird*rInvVec[5]*(1.5f*(mScale + bVec[3]) + 0.5f*alphaRVec[5]*X);
    dUIndCoef = fourSqrtOneThird*rInvVec[5]*(3*(pScale*dthole_q1 + bVec[3]) + alphaRVec[5]*X);
    dUInpCoef = fourSqrtOneThird*rInvVec[5]*(3*(dScale*dthole_q1 + bVec[3]) + alphaRVec[5]*X);
    Vij[2]  += ePermCoef*rotatedQuadrupole2[1];
    Vji[5]   = ePermCoef*rotatedDipole1.y + eUIndCoef*qiUindI.y + eUInpCoef*qiUinpI.y;
    VijR[2] += dPermCoef*rotatedQuadrupole2[1];
    VjiR[5]  = dPermCoef*rotatedDipole1.y + dUIndCoef*qiUindI.y + dUInpCoef*qiUinpI.y;
    Vij[3]  += ePermCoef*rotatedQuadrupole2[2];
    Vji[6]   = ePermCoef*rotatedDipole1.z + eUIndCoef*qiUindI.z + eUInpCoef*qiUinpI.z;
    VijR[3] += dPermCoef*rotatedQuadrupole2[2];
    VjiR[6]  = dPermCoef*rotatedDipole1.z + dUIndCoef*qiUindI.z + dUInpCoef*qiUinpI.z;
    Vijp[1] += eUInpCoef*rotatedQuadrupole2[1];
    Vijd[1] += eUIndCoef*rotatedQuadrupole2[1];
    Vijp[2] += eUInpCoef*rotatedQuadrupole2[2];
    Vijd[2] += eUIndCoef*rotatedQuadrupole2[2];
    // D-Q and Uind-Q terms (m=1)
    Vij[5]   = -(ePermCoef*rotatedDipole2.y + eUIndCoef*qiUindJ.y + eUInpCoef*qiUinpJ.y);
    Vji[2]  += -(ePermCoef*rotatedQuadrupole1[1]);
    VijR[5]  = -(dPermCoef*rotatedDipole2.y + dUIndCoef*qiUindJ.y + dUInpCoef*qiUinpJ.y);
    VjiR[2] += -(dPermCoef*rotatedQuadrupole1[1]);
    Vij[6]   = -(ePermCoef*rotatedDipole2.z + eUIndCoef*qiUindJ.z + eUInpCoef*qiUinpJ.z);
    Vji[3]  += -(ePermCoef*rotatedQuadrupole1[2]);
    VijR[6]  = -(dPermCoef*rotatedDipole2.z + dUIndCoef*qiUindJ.z + dUInpCoef*qiUinpJ.z);
    VjiR[3] += -(dPermCoef*rotatedQuadrupole1[2]);
    Vjip[1] += -(eUInpCoef*rotatedQuadrupole1[1]);
    Vjid[1] += -(eUIndCoef*rotatedQuadrupole1[1]);
    Vjip[2] += -(eUInpCoef*rotatedQuadrupole1[2]);
    Vjid[2] += -(eUIndCoef*rotatedQuadrupole1[2]);

    // Q-Q terms (m=0)
    ePermCoef = rInvVec[5]*(6*(mScale + bVec[4]) + ((real) 4/45)*(-3 + 10*alphaRVec[2])*alphaRVec[5]*X);
    dPermCoef = -rInvVec[6]*(135*(mScale + bVec[4]) + 4*(1 + 2*alphaRVec[2])*alphaRVec[7]*X)/9;
    Vij[4]  += ePermCoef*rotatedQuadrupole2[0];
    Vji[4]  += ePermCoef*rotatedQuadrupole1[0];
    VijR[4] += dPermCoef*rotatedQuadrupole2[0];
    VjiR[4] += dPermCoef*rotatedQuadrupole1[0];
    // Q-Q terms (m=1)
    const real fourOverFifteen = (real) 4/15;
    ePermCoef = -fourOverFifteen*rInvVec[5]*(15*(mScale + bVec[4]) + alphaRVec[5]*X);
    dPermCoef = rInvVec[6]*(10*(mScale + bVec[4]) + fourThirds*alphaRVec[7]*X);
    Vij[5]  += ePermCoef*rotatedQuadrupole2[1];
    Vji[5]  += ePermCoef*rotatedQuadrupole1[1];
    VijR[5] += dPermCoef*rotatedQuadrupole2[1];
    VjiR[5] += dPermCoef*rotatedQuadrupole1[1];
    Vij[6]  += ePermCoef*rotatedQuadrupole2[2];
    Vji[6]  += ePermCoef*rotatedQuadrupole1[2];
    VijR[6] += dPermCoef*rotatedQuadrupole2[2];
    VjiR[6] += dPermCoef*rotatedQuadrupole1[2];
    // Q-Q terms (m=2)
    ePermCoef = rInvVec[5]*(mScale + bVec[4] - fourOverFifteen*alphaRVec[5]*X);
    dPermCoef = -2.5f*(mScale + bVec[4])*rInvVec[6];
    Vij[7]  = ePermCoef*rotatedQuadrupole2[3];
    Vji[7]  = ePermCoef*rotatedQuadrupole1[3];
    VijR[7] = dPermCoef*rotatedQuadrupole2[3];
    VjiR[7] = dPermCoef*rotatedQuadrupole1[3];
    Vij[8]  = ePermCoef*rotatedQuadrupole2[4];
    Vji[8]  = ePermCoef*rotatedQuadrupole1[4];
    VijR[8] = dPermCoef*rotatedQuadrupole2[4];
    VjiR[8] = dPermCoef*rotatedQuadrupole1[4];

    // Evaluate the energies, forces and torques due to permanent+induced moments
    // interacting with just the permanent moments.
    *energy += forceFactor*0.5f*(
        atom1->q*Vij[0] + rotatedDipole1.x*Vij[1] + rotatedDipole1.y*Vij[2] + rotatedDipole1.z*Vij[3] + rotatedQuadrupole1[0]*Vij[4] + rotatedQuadrupole1[1]*Vij[5] + rotatedQuadrupole1[2]*Vij[6] + rotatedQuadrupole1[3]*Vij[7] + rotatedQuadrupole1[4]*Vij[8] +
        atom2->q*Vji[0] + rotatedDipole2.x*Vji[1] + rotatedDipole2.y*Vji[2] + rotatedDipole2.z*Vji[3] + rotatedQuadrupole2[0]*Vji[4] + rotatedQuadrupole2[1]*Vji[5] + rotatedQuadrupole2[2]*Vji[6] + rotatedQuadrupole2[3]*Vji[7] + rotatedQuadrupole2[4]*Vji[8]);
    real fIZ = atom1->q*VijR[0] + rotatedDipole1.x*VijR[1] + rotatedDipole1.y*VijR[2] + rotatedDipole1.z*VijR[3] + rotatedQuadrupole1[0]*VijR[4] + rotatedQuadrupole1[1]*VijR[5] + rotatedQuadrupole1[2]*VijR[6] + rotatedQuadrupole1[3]*VijR[7] + rotatedQuadrupole1[4]*VijR[8];
    real fJZ = atom2->q*VjiR[0] + rotatedDipole2.x*VjiR[1] + rotatedDipole2.y*VjiR[2] + rotatedDipole2.z*VjiR[3] + rotatedQuadrupole2[0]*VjiR[4] + rotatedQuadrupole2[1]*VjiR[5] + rotatedQuadrupole2[2]*VjiR[6] + rotatedQuadrupole2[3]*VjiR[7] + rotatedQuadrupole2[4]*VjiR[8];
    real EIX = rotatedDipole1.z*Vij[1] - rotatedDipole1.x*Vij[3] + sqrtThree*rotatedQuadrupole1[2]*Vij[4] + rotatedQuadrupole1[4]*Vij[5] - (sqrtThree*rotatedQuadrupole1[0]+rotatedQuadrupole1[3])*Vij[6] + rotatedQuadrupole1[2]*Vij[7] - rotatedQuadrupole1[1]*Vij[8];
    real EIY = -rotatedDipole1.y*Vij[1] + rotatedDipole1.x*Vij[2] - sqrtThree*rotatedQuadrupole1[1]*Vij[4] + (sqrtThree*rotatedQuadrupole1[0]-rotatedQuadrupole1[3])*Vij[5] - rotatedQuadrupole1[4]*Vij[6] + rotatedQuadrupole1[1]*Vij[7] + rotatedQuadrupole1[2]*Vij[8];
    real EIZ = -rotatedDipole1.z*Vij[2] + rotatedDipole1.y*Vij[3] - rotatedQuadrupole1[2]*Vij[5] + rotatedQuadrupole1[1]*Vij[6] - 2*rotatedQuadrupole1[4]*Vij[7] + 2*rotatedQuadrupole1[3]*Vij[8];
    real EJX = rotatedDipole2.z*Vji[1] - rotatedDipole2.x*Vji[3] + sqrtThree*rotatedQuadrupole2[2]*Vji[4] + rotatedQuadrupole2[4]*Vji[5] - (sqrtThree*rotatedQuadrupole2[0]+rotatedQuadrupole2[3])*Vji[6] + rotatedQuadrupole2[2]*Vji[7] - rotatedQuadrupole2[1]*Vji[8];
    real EJY = -rotatedDipole2.y*Vji[1] + rotatedDipole2.x*Vji[2] - sqrtThree*rotatedQuadrupole2[1]*Vji[4] + (sqrtThree*rotatedQuadrupole2[0]-rotatedQuadrupole2[3])*Vji[5] - rotatedQuadrupole2[4]*Vji[6] + rotatedQuadrupole2[1]*Vji[7] + rotatedQuadrupole2[2]*Vji[8];
    real EJZ = -rotatedDipole2.z*Vji[2] + rotatedDipole2.y*Vji[3] - rotatedQuadrupole2[2]*Vji[5] + rotatedQuadrupole2[1]*Vji[6] - 2*rotatedQuadrupole2[4]*Vji[7] + 2*rotatedQuadrupole2[3]*Vji[8];

    // Define the torque intermediates for the induced dipoles. These are simply the induced dipole torque
    // intermediates dotted with the field due to permanent moments only, at each center. We inline the
    // induced dipole torque intermediates here, for simplicity. N.B. There are no torques on the dipoles
    // themselves, so we accumulate the torque intermediates into separate variables to allow them to be
    // used only in the force calculation.
    //
    // The torque about the x axis (needed to obtain the y force on the induced dipoles, below)
    //    qiUindIx[0] = qiQUindI[2];    qiUindIx[1] = 0;    qiUindIx[2] = -qiQUindI[0]
    real iEIX = qiUinpI.z*Vijp[0] + qiUindI.z*Vijd[0] - qiUinpI.x*Vijp[2] - qiUindI.x*Vijd[2];
    real iEJX = qiUinpJ.z*Vjip[0] + qiUindJ.z*Vjid[0] - qiUinpJ.x*Vjip[2] - qiUindJ.x*Vjid[2];
    // The torque about the y axis (needed to obtain the x force on the induced dipoles, below)
    //    qiUindIy[0] = -qiQUindI[1];   qiUindIy[1] = qiQUindI[0];    qiUindIy[2] = 0
    real iEIY = qiUinpI.x*Vijp[1] + qiUindI.x*Vijd[1] - qiUinpI.y*Vijp[0] - qiUindI.y*Vijd[0];
    real iEJY = qiUinpJ.x*Vjip[1] + qiUindJ.x*Vjid[1] - qiUinpJ.y*Vjip[0] - qiUindJ.y*Vjid[0];

#ifdef MUTUAL_POLARIZATION
    // Uind-Uind terms (m=0)
    real eCoef = -fourThirds*rInvVec[3]*(3*(thole_d0 + bVec[3]) + alphaRVec[3]*X);
    real dCoef = rInvVec[4]*(6*(dthole_d0 + bVec[3]) + 4*alphaRVec[5]*X);
    iEIX += eCoef*(qiUinpI.z*qiUindJ.x + qiUindI.z*qiUinpJ.x);
    iEJX += eCoef*(qiUinpJ.z*qiUindI.x + qiUindJ.z*qiUinpI.x);
    iEIY -= eCoef*(qiUinpI.y*qiUindJ.x + qiUindI.y*qiUinpJ.x);
    iEJY -= eCoef*(qiUinpJ.y*qiUindI.x + qiUindJ.y*qiUinpI.x);
    fIZ += dCoef*(qiUinpI.x*qiUindJ.x + qiUindI.x*qiUinpJ.x);
    fIZ += dCoef*(qiUinpJ.x*qiUindI.x + qiUindJ.x*qiUinpI.x);
    // Uind-Uind terms (m=1)
    eCoef = 2*rInvVec[3]*(thole_d1 + bVec[3] - twoThirds*alphaRVec[3]*X);
    dCoef = -3*rInvVec[4]*(dthole_d1 + bVec[3]);
    iEIX -= eCoef*(qiUinpI.x*qiUindJ.z + qiUindI.x*qiUinpJ.z);
    iEJX -= eCoef*(qiUinpJ.x*qiUindI.z + qiUindJ.x*qiUinpI.z);
    iEIY += eCoef*(qiUinpI.x*qiUindJ.y + qiUindI.x*qiUinpJ.y);
    iEJY += eCoef*(qiUinpJ.x*qiUindI.y + qiUindJ.x*qiUinpI.y);
    fIZ += dCoef*(qiUinpI.y*qiUindJ.y + qiUindI.y*qiUinpJ.y + qiUinpI.z*qiUindJ.z + qiUindI.z*qiUinpJ.z);
    fIZ += dCoef*(qiUinpJ.y*qiUindI.y + qiUindJ.y*qiUinpI.y + qiUinpJ.z*qiUindI.z + qiUindJ.z*qiUinpI.z);
#endif

    // The quasi-internal frame forces and torques.  Note that the induced torque intermediates are
    // used in the force expression, but not in the torques; the induced dipoles are isotropic.
    real qiForce[3] = {rInv*(EIY+EJY+iEIY+iEJY), -rInv*(EIX+EJX+iEIX+iEJX), -(fJZ+fIZ)};
    real qiTorqueI[3] = {-EIX, -EIY, -EIZ};
    real qiTorqueJ[3] = {-EJX, -EJY, -EJZ};


    real3 force = make_real3(qiRotationMatrix[1][1]*qiForce[0] + qiRotationMatrix[2][1]*qiForce[1] + qiRotationMatrix[0][1]*qiForce[2],
                             qiRotationMatrix[1][2]*qiForce[0] + qiRotationMatrix[2][2]*qiForce[1] + qiRotationMatrix[0][2]*qiForce[2],
                             qiRotationMatrix[1][0]*qiForce[0] + qiRotationMatrix[2][0]*qiForce[1] + qiRotationMatrix[0][0]*qiForce[2]);
    atom1->force += force;
    atom1->torque += make_real3(qiRotationMatrix[1][1]*qiTorqueI[0] + qiRotationMatrix[2][1]*qiTorqueI[1] + qiRotationMatrix[0][1]*qiTorqueI[2],
                               qiRotationMatrix[1][2]*qiTorqueI[0] + qiRotationMatrix[2][2]*qiTorqueI[1] + qiRotationMatrix[0][2]*qiTorqueI[2],
                               qiRotationMatrix[1][0]*qiTorqueI[0] + qiRotationMatrix[2][0]*qiTorqueI[1] + qiRotationMatrix[0][0]*qiTorqueI[2]);
    if (forceFactor == 1) {
        atom2->force -= force;
        atom2->torque += make_real3(qiRotationMatrix[1][1]*qiTorqueJ[0] + qiRotationMatrix[2][1]*qiTorqueJ[1] + qiRotationMatrix[0][1]*qiTorqueJ[2],
                                   qiRotationMatrix[1][2]*qiTorqueJ[0] + qiRotationMatrix[2][2]*qiTorqueJ[1] + qiRotationMatrix[0][2]*qiTorqueJ[2],
                                   qiRotationMatrix[1][0]*qiTorqueJ[0] + qiRotationMatrix[2][0]*qiTorqueJ[1] + qiRotationMatrix[0][0]*qiTorqueJ[2]);
    }
}

/**
 * Compute the self energy and self torque.
 */
DEVICE void computeSelfEnergyAndTorque(AtomData* atom1, mixed* energy) {
    real cii = atom1->q*atom1->q;
    real3 dipole = make_real3(atom1->sphericalDipole.y, atom1->sphericalDipole.z, atom1->sphericalDipole.x);
    real dii = dot(dipole, dipole+(atom1->inducedDipole+atom1->inducedDipolePolar)*0.5f);
#ifdef INCLUDE_QUADRUPOLES
    real qii = (atom1->sphericalQuadrupole[0]*atom1->sphericalQuadrupole[0] +
                atom1->sphericalQuadrupole[1]*atom1->sphericalQuadrupole[1] +
                atom1->sphericalQuadrupole[2]*atom1->sphericalQuadrupole[2] +
                atom1->sphericalQuadrupole[3]*atom1->sphericalQuadrupole[3] +
                atom1->sphericalQuadrupole[4]*atom1->sphericalQuadrupole[4]);
#else
    real qii = 0;
#endif
    real prefac = -EWALD_ALPHA/SQRT_PI;
    real a2 = EWALD_ALPHA*EWALD_ALPHA;
    real a4 = a2*a2;
    *energy += prefac*(cii + ((real)2/3)*a2*dii + ((real) 4/15)*a4*qii);

    // self-torque for PME

    real3 ui = atom1->inducedDipole+atom1->inducedDipolePolar;
    atom1->torque += ((2/(real) 3)*(EWALD_ALPHA*EWALD_ALPHA*EWALD_ALPHA)/SQRT_PI)*cross(dipole, ui);
}

/**
 * Compute electrostatic interactions.
 */
KERNEL void computeElectrostatics(
        GLOBAL mm_ulong* RESTRICT forceBuffers, GLOBAL mm_ulong* RESTRICT torqueBuffers, GLOBAL mixed* RESTRICT energyBuffer,
        GLOBAL const real4* RESTRICT posq, GLOBAL const uint2* RESTRICT covalentFlags, GLOBAL const unsigned int* RESTRICT polarizationGroupFlags,
        GLOBAL const int2* RESTRICT exclusionTiles, unsigned int startTileIndex, unsigned int numTileIndices,
#ifdef USE_CUTOFF
        GLOBAL const int* RESTRICT tiles, GLOBAL const unsigned int* RESTRICT interactionCount, real4 periodicBoxSize, real4 invPeriodicBoxSize,
        real4 periodicBoxVecX, real4 periodicBoxVecY, real4 periodicBoxVecZ, unsigned int maxTiles, GLOBAL const real4* RESTRICT blockCenter,
        GLOBAL const unsigned int* RESTRICT interactingAtoms,
#endif
        GLOBAL const real* RESTRICT sphericalDipole, GLOBAL const real* RESTRICT sphericalQuadrupole, GLOBAL const real* RESTRICT inducedDipole,
        GLOBAL const real* RESTRICT inducedDipolePolar, GLOBAL const float2* RESTRICT dampingAndThole) {
    const unsigned int totalWarps = (GLOBAL_SIZE)/TILE_SIZE;
    const unsigned int warp = (GLOBAL_ID)/TILE_SIZE;
    const unsigned int tgx = LOCAL_ID & (TILE_SIZE-1);
    const unsigned int tbx = LOCAL_ID - tgx;
    mixed energy = 0;
    LOCAL AtomData localData[THREAD_BLOCK_SIZE];

    // First loop: process tiles that contain exclusions.
    
    const unsigned int firstExclusionTile = FIRST_EXCLUSION_TILE+warp*(LAST_EXCLUSION_TILE-FIRST_EXCLUSION_TILE)/totalWarps;
    const unsigned int lastExclusionTile = FIRST_EXCLUSION_TILE+(warp+1)*(LAST_EXCLUSION_TILE-FIRST_EXCLUSION_TILE)/totalWarps;
    for (int pos = firstExclusionTile; pos < lastExclusionTile; pos++) {
        const int2 tileIndices = exclusionTiles[pos];
        const unsigned int x = tileIndices.x;
        const unsigned int y = tileIndices.y;
        unsigned int atom1 = x*TILE_SIZE + tgx;
        AtomData data = loadAtomData(atom1, posq, sphericalDipole, sphericalQuadrupole, inducedDipole, inducedDipolePolar, dampingAndThole);
        data.force = make_real3(0);
        data.torque = make_real3(0);
        uint2 covalent = covalentFlags[pos*TILE_SIZE+tgx];
        unsigned int polarizationGroup = polarizationGroupFlags[pos*TILE_SIZE+tgx];
        if (x == y) {
            // This tile is on the diagonal.

            localData[LOCAL_ID].pos = data.pos;
            localData[LOCAL_ID].q = data.q;
            localData[LOCAL_ID].sphericalDipole = data.sphericalDipole;
#ifdef INCLUDE_QUADRUPOLES
            localData[LOCAL_ID].sphericalQuadrupole[0] = data.sphericalQuadrupole[0];
            localData[LOCAL_ID].sphericalQuadrupole[1] = data.sphericalQuadrupole[1];
            localData[LOCAL_ID].sphericalQuadrupole[2] = data.sphericalQuadrupole[2];
            localData[LOCAL_ID].sphericalQuadrupole[3] = data.sphericalQuadrupole[3];
            localData[LOCAL_ID].sphericalQuadrupole[4] = data.sphericalQuadrupole[4];
#endif
            localData[LOCAL_ID].inducedDipole = data.inducedDipole;
            localData[LOCAL_ID].inducedDipolePolar = data.inducedDipolePolar;
            localData[LOCAL_ID].thole = data.thole;
            localData[LOCAL_ID].damp = data.damp;

            // Compute forces.

            SYNC_WARPS;
            for (unsigned int j = 0; j < TILE_SIZE; j++) {
                int atom2 = y*TILE_SIZE+j;
                if (atom1 != atom2 && atom1 < NUM_ATOMS && atom2 < NUM_ATOMS) {
                    float d = computeDScaleFactor(polarizationGroup, j);
                    float p = computePScaleFactor(covalent, polarizationGroup, j);
                    float m = computeMScaleFactor(covalent, j);
                    computeOneInteraction(&data, &localData[tbx+j], true, d, p, m, 0.5f, &energy, periodicBoxSize, invPeriodicBoxSize, periodicBoxVecX, periodicBoxVecY, periodicBoxVecZ);
                }
            }
            if (atom1 < NUM_ATOMS)
                computeSelfEnergyAndTorque(&data, &energy);
            data.force *= -ENERGY_SCALE_FACTOR;
            data.torque *= ENERGY_SCALE_FACTOR;
            ATOMIC_ADD(&forceBuffers[atom1], (mm_ulong) ((mm_long) (data.force.x*0x100000000)));
            ATOMIC_ADD(&forceBuffers[atom1+PADDED_NUM_ATOMS], (mm_ulong) ((mm_long) (data.force.y*0x100000000)));
            ATOMIC_ADD(&forceBuffers[atom1+2*PADDED_NUM_ATOMS], (mm_ulong) ((mm_long) (data.force.z*0x100000000)));
            ATOMIC_ADD(&torqueBuffers[atom1], (mm_ulong) ((mm_long) (data.torque.x*0x100000000)));
            ATOMIC_ADD(&torqueBuffers[atom1+PADDED_NUM_ATOMS], (mm_ulong) ((mm_long) (data.torque.y*0x100000000)));
            ATOMIC_ADD(&torqueBuffers[atom1+2*PADDED_NUM_ATOMS], (mm_ulong) ((mm_long) (data.torque.z*0x100000000)));
            SYNC_WARPS;
        }
        else {
            // This is an off-diagonal tile.

            unsigned int j = y*TILE_SIZE + tgx;
            localData[LOCAL_ID] = loadAtomData(j, posq, sphericalDipole, sphericalQuadrupole, inducedDipole, inducedDipolePolar, dampingAndThole);
            localData[LOCAL_ID].force = make_real3(0);
            localData[LOCAL_ID].torque = make_real3(0);
            unsigned int tj = tgx;
            SYNC_WARPS;
            for (j = 0; j < TILE_SIZE; j++) {
                int atom2 = y*TILE_SIZE+tj;
                if (atom1 < NUM_ATOMS && atom2 < NUM_ATOMS) {
                    float d = computeDScaleFactor(polarizationGroup, tj);
                    float p = computePScaleFactor(covalent, polarizationGroup, tj);
                    float m = computeMScaleFactor(covalent, tj);
                    computeOneInteraction(&data, &localData[tbx+tj], true, d, p, m, 1, &energy, periodicBoxSize, invPeriodicBoxSize, periodicBoxVecX, periodicBoxVecY, periodicBoxVecZ);
                }
                tj = (tj + 1) & (TILE_SIZE - 1);
            }
            SYNC_WARPS;
            data.force *= -ENERGY_SCALE_FACTOR;
            data.torque *= ENERGY_SCALE_FACTOR;
            localData[LOCAL_ID].force *= -ENERGY_SCALE_FACTOR;
            localData[LOCAL_ID].torque *= ENERGY_SCALE_FACTOR;
            unsigned int offset = x*TILE_SIZE + tgx;
            ATOMIC_ADD(&forceBuffers[offset], (mm_ulong) ((mm_long) (data.force.x*0x100000000)));
            ATOMIC_ADD(&forceBuffers[offset+PADDED_NUM_ATOMS], (mm_ulong) ((mm_long) (data.force.y*0x100000000)));
            ATOMIC_ADD(&forceBuffers[offset+2*PADDED_NUM_ATOMS], (mm_ulong) ((mm_long) (data.force.z*0x100000000)));
            ATOMIC_ADD(&torqueBuffers[offset], (mm_ulong) ((mm_long) (data.torque.x*0x100000000)));
            ATOMIC_ADD(&torqueBuffers[offset+PADDED_NUM_ATOMS], (mm_ulong) ((mm_long) (data.torque.y*0x100000000)));
            ATOMIC_ADD(&torqueBuffers[offset+2*PADDED_NUM_ATOMS], (mm_ulong) ((mm_long) (data.torque.z*0x100000000)));
            offset = y*TILE_SIZE + tgx;
            ATOMIC_ADD(&forceBuffers[offset], (mm_ulong) ((mm_long) (localData[LOCAL_ID].force.x*0x100000000)));
            ATOMIC_ADD(&forceBuffers[offset+PADDED_NUM_ATOMS], (mm_ulong) ((mm_long) (localData[LOCAL_ID].force.y*0x100000000)));
            ATOMIC_ADD(&forceBuffers[offset+2*PADDED_NUM_ATOMS], (mm_ulong) ((mm_long) (localData[LOCAL_ID].force.z*0x100000000)));
            ATOMIC_ADD(&torqueBuffers[offset], (mm_ulong) ((mm_long) (localData[LOCAL_ID].torque.x*0x100000000)));
            ATOMIC_ADD(&torqueBuffers[offset+PADDED_NUM_ATOMS], (mm_ulong) ((mm_long) (localData[LOCAL_ID].torque.y*0x100000000)));
            ATOMIC_ADD(&torqueBuffers[offset+2*PADDED_NUM_ATOMS], (mm_ulong) ((mm_long) (localData[LOCAL_ID].torque.z*0x100000000)));
            SYNC_WARPS;
        }
    }

    // Second loop: tiles without exclusions, either from the neighbor list (with cutoff) or just enumerating all
    // of them (no cutoff).

#ifdef USE_CUTOFF
    const unsigned int numTiles = interactionCount[0];
    if (numTiles > maxTiles)
        return; // There wasn't enough memory for the neighbor list.
    int pos = (int) (numTiles > maxTiles ? startTileIndex+warp*(mm_long)numTileIndices/totalWarps : warp*(mm_long)numTiles/totalWarps);
    int end = (int) (numTiles > maxTiles ? startTileIndex+(warp+1)*(mm_long)numTileIndices/totalWarps : (warp+1)*(mm_long)numTiles/totalWarps);
#else
    const unsigned int numTiles = numTileIndices;
    int pos = (int) (startTileIndex+warp*(mm_long)numTiles/totalWarps);
    int end = (int) (startTileIndex+(warp+1)*(mm_long)numTiles/totalWarps);
#endif
    int skipBase = 0;
    int currentSkipIndex = tbx;
    LOCAL int atomIndices[THREAD_BLOCK_SIZE];
    LOCAL volatile int skipTiles[THREAD_BLOCK_SIZE];
    skipTiles[LOCAL_ID] = -1;
    
    while (pos < end) {
        bool includeTile = true;

        // Extract the coordinates of this tile.
        
        int x, y;
#ifdef USE_CUTOFF
        x = tiles[pos];
#else
        y = (int) floor(NUM_BLOCKS+0.5f-SQRT((NUM_BLOCKS+0.5f)*(NUM_BLOCKS+0.5f)-2*pos));
        x = (pos-y*NUM_BLOCKS+y*(y+1)/2);
        if (x < y || x >= NUM_BLOCKS) { // Occasionally happens due to roundoff error.
            y += (x < y ? -1 : 1);
            x = (pos-y*NUM_BLOCKS+y*(y+1)/2);
        }

        // Skip over tiles that have exclusions, since they were already processed.

        SYNC_WARPS;
        while (skipTiles[tbx+TILE_SIZE-1] < pos) {
            SYNC_WARPS;
            if (skipBase+tgx < NUM_TILES_WITH_EXCLUSIONS) {
                int2 tile = exclusionTiles[skipBase+tgx];
                skipTiles[LOCAL_ID] = tile.x + tile.y*NUM_BLOCKS - tile.y*(tile.y+1)/2;
            }
            else
                skipTiles[LOCAL_ID] = end;
            skipBase += TILE_SIZE;            
            currentSkipIndex = tbx;
            SYNC_WARPS;
        }
        while (skipTiles[currentSkipIndex] < pos)
            currentSkipIndex++;
        includeTile = (skipTiles[currentSkipIndex] != pos);
#endif
        if (includeTile) {
            unsigned int atom1 = x*TILE_SIZE + tgx;

            // Load atom data for this tile.

            AtomData data = loadAtomData(atom1, posq, sphericalDipole, sphericalQuadrupole, inducedDipole, inducedDipolePolar, dampingAndThole);
            data.force = make_real3(0);
            data.torque = make_real3(0);
#ifdef USE_CUTOFF
            unsigned int j = interactingAtoms[pos*TILE_SIZE+tgx];
#else
            unsigned int j = y*TILE_SIZE + tgx;
#endif
            atomIndices[LOCAL_ID] = j;
            localData[LOCAL_ID] = loadAtomData(j, posq, sphericalDipole, sphericalQuadrupole, inducedDipole, inducedDipolePolar, dampingAndThole);
            localData[LOCAL_ID].force = make_real3(0);
            localData[LOCAL_ID].torque = make_real3(0);
            SYNC_WARPS;

            // Compute forces.

            unsigned int tj = tgx;
            for (j = 0; j < TILE_SIZE; j++) {
                int atom2 = atomIndices[tbx+tj];
                if (atom1 < NUM_ATOMS && atom2 < NUM_ATOMS) {
                    computeOneInteraction(&data, &localData[tbx+tj], false, 1, 1, 1, 1, &energy, periodicBoxSize, invPeriodicBoxSize, periodicBoxVecX, periodicBoxVecY, periodicBoxVecZ);
                }
                tj = (tj + 1) & (TILE_SIZE - 1);
                SYNC_WARPS;
            }
            data.force *= -ENERGY_SCALE_FACTOR;
            data.torque *= ENERGY_SCALE_FACTOR;
            localData[LOCAL_ID].force *= -ENERGY_SCALE_FACTOR;
            localData[LOCAL_ID].torque *= ENERGY_SCALE_FACTOR;

            // Write results.

            unsigned int offset = x*TILE_SIZE + tgx;
            ATOMIC_ADD(&forceBuffers[offset], (mm_ulong) ((mm_long) (data.force.x*0x100000000)));
            ATOMIC_ADD(&forceBuffers[offset+PADDED_NUM_ATOMS], (mm_ulong) ((mm_long) (data.force.y*0x100000000)));
            ATOMIC_ADD(&forceBuffers[offset+2*PADDED_NUM_ATOMS], (mm_ulong) ((mm_long) (data.force.z*0x100000000)));
            ATOMIC_ADD(&torqueBuffers[offset], (mm_ulong) ((mm_long) (data.torque.x*0x100000000)));
            ATOMIC_ADD(&torqueBuffers[offset+PADDED_NUM_ATOMS], (mm_ulong) ((mm_long) (data.torque.y*0x100000000)));
            ATOMIC_ADD(&torqueBuffers[offset+2*PADDED_NUM_ATOMS], (mm_ulong) ((mm_long) (data.torque.z*0x100000000)));
#ifdef USE_CUTOFF
            offset = atomIndices[LOCAL_ID];
#else
            offset = y*TILE_SIZE + tgx;
#endif
            ATOMIC_ADD(&forceBuffers[offset], (mm_ulong) ((mm_long) (localData[LOCAL_ID].force.x*0x100000000)));
            ATOMIC_ADD(&forceBuffers[offset+PADDED_NUM_ATOMS], (mm_ulong) ((mm_long) (localData[LOCAL_ID].force.y*0x100000000)));
            ATOMIC_ADD(&forceBuffers[offset+2*PADDED_NUM_ATOMS], (mm_ulong) ((mm_long) (localData[LOCAL_ID].force.z*0x100000000)));
            ATOMIC_ADD(&torqueBuffers[offset], (mm_ulong) ((mm_long) (localData[LOCAL_ID].torque.x*0x100000000)));
            ATOMIC_ADD(&torqueBuffers[offset+PADDED_NUM_ATOMS], (mm_ulong) ((mm_long) (localData[LOCAL_ID].torque.y*0x100000000)));
            ATOMIC_ADD(&torqueBuffers[offset+2*PADDED_NUM_ATOMS], (mm_ulong) ((mm_long) (localData[LOCAL_ID].torque.z*0x100000000)));
        }
        pos++;
    }
    energyBuffer[GLOBAL_ID] += energy*ENERGY_SCALE_FACTOR;
}