Improve Verlet buffer constraint estimate

[gromacs.git] / src / gromacs / mdlib / calc_verletbuf.cpp

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#include "gmxpre.h"

#include "calc_verletbuf.h"

#include <assert.h>
#include <stdlib.h>

#include <cmath>

#include <algorithm>

#include "gromacs/math/calculate-ewald-splitting-coefficient.h"
#include "gromacs/math/functions.h"
#include "gromacs/math/units.h"
#include "gromacs/math/vec.h"
#include "gromacs/mdlib/nb_verlet.h"
#include "gromacs/mdlib/nbnxn_simd.h"
#include "gromacs/mdlib/nbnxn_util.h"
#include "gromacs/mdtypes/inputrec.h"
#include "gromacs/mdtypes/md_enums.h"
#include "gromacs/topology/ifunc.h"
#include "gromacs/topology/topology.h"
#include "gromacs/utility/fatalerror.h"
#include "gromacs/utility/smalloc.h"

/* The code in this file estimates a pairlist buffer length
 * given a target energy drift per atom per picosecond.
 * This is done by estimating the drift given a buffer length.
 * Ideally we would like to have a tight overestimate of the drift,
 * but that can be difficult to achieve.
 *
 * Significant approximations used:
 *
 * Uniform particle density. UNDERESTIMATES the drift by rho_global/rho_local.
 *
 * Interactions don't affect particle motion. OVERESTIMATES the drift on longer
 * time scales. This approximation probably introduces the largest errors.
 *
 * Only take one constraint per particle into account: OVERESTIMATES the drift.
 *
 * For rotating constraints assume the same functional shape for time scales
 * where the constraints rotate significantly as the exact expression for
 * short time scales. OVERESTIMATES the drift on long time scales.
 *
 * For non-linear virtual sites use the mass of the lightest constructing atom
 * to determine the displacement. OVER/UNDERESTIMATES the drift, depending on
 * the geometry and masses of constructing atoms.
 *
 * Note that the formulas for normal atoms and linear virtual sites are exact,
 * apart from the first two approximations.
 *
 * Note that apart from the effect of the above approximations, the actual
 * drift of the total energy of a system can be order of magnitude smaller
 * due to cancellation of positive and negative drift for different pairs.
 */


/* Struct for unique atom type for calculating the energy drift.
 * The atom displacement depends on mass and constraints.
 * The energy jump for given distance depend on LJ type and q.
 */
typedef struct
{
    atom_nonbonded_kinetic_prop_t prop; /* non-bonded and kinetic atom prop. */
    int                           n;    /* #atoms of this type in the system */
} verletbuf_atomtype_t;

// Struct for derivatives of a non-bonded interaction potential
typedef struct
{
    real  md1; // -V' at the cutoff
    real  d2;  //  V'' at the cutoff
    real  md3; // -V''' at the cutoff
} pot_derivatives_t;

void verletbuf_get_list_setup(gmx_bool gmx_unused     bSIMD,
                              gmx_bool                bGPU,
                              verletbuf_list_setup_t *list_setup)
{
    /* When calling this function we often don't know which kernel type we
     * are going to use. W choose the kernel type with the smallest possible
     * i- and j-cluster sizes, so we potentially overestimate, but never
     * underestimate, the buffer drift.
     * Note that the current buffer estimation code only handles clusters
     * of size 1, 2 or 4, so for 4x8 or 8x8 we use the estimate for 4x4.
     */

    if (bGPU)
    {
        /* The CUDA kernels split the j-clusters in two halves */
        list_setup->cluster_size_i = nbnxn_kernel_to_cluster_i_size(nbnxnk8x8x8_GPU);
        list_setup->cluster_size_j = nbnxn_kernel_to_cluster_j_size(nbnxnk8x8x8_GPU)/2;
    }
    else
    {
        int kernel_type;

        kernel_type = nbnxnk4x4_PlainC;

#if GMX_SIMD
        if (bSIMD)
        {
#ifdef GMX_NBNXN_SIMD_2XNN
            /* We use the smallest cluster size to be on the safe side */
            kernel_type = nbnxnk4xN_SIMD_2xNN;
#else
            kernel_type = nbnxnk4xN_SIMD_4xN;
#endif
        }
#endif

        list_setup->cluster_size_i = nbnxn_kernel_to_cluster_i_size(kernel_type);
        list_setup->cluster_size_j = nbnxn_kernel_to_cluster_j_size(kernel_type);
    }
}

static gmx_bool
atom_nonbonded_kinetic_prop_equal(const atom_nonbonded_kinetic_prop_t *prop1,
                                  const atom_nonbonded_kinetic_prop_t *prop2)
{
    return (prop1->mass     == prop2->mass &&
            prop1->type     == prop2->type &&
            prop1->q        == prop2->q &&
            prop1->bConstr  == prop2->bConstr &&
            prop1->con_mass == prop2->con_mass &&
            prop1->con_len  == prop2->con_len);
}

static void add_at(verletbuf_atomtype_t **att_p, int *natt_p,
                   const atom_nonbonded_kinetic_prop_t *prop,
                   int nmol)
{
    verletbuf_atomtype_t   *att;
    int                     natt, i;

    if (prop->mass == 0)
    {
        /* Ignore massless particles */
        return;
    }

    att  = *att_p;
    natt = *natt_p;

    i = 0;
    while (i < natt && !atom_nonbonded_kinetic_prop_equal(prop, &att[i].prop))
    {
        i++;
    }

    if (i < natt)
    {
        att[i].n += nmol;
    }
    else
    {
        (*natt_p)++;
        srenew(*att_p, *natt_p);
        (*att_p)[i].prop = *prop;
        (*att_p)[i].n    = nmol;
    }
}

static void get_vsite_masses(const gmx_moltype_t  *moltype,
                             const gmx_ffparams_t *ffparams,
                             real                 *vsite_m,
                             int                  *n_nonlin_vsite)
{
    int            ft, i;
    const t_ilist *il;

    *n_nonlin_vsite = 0;

    /* Check for virtual sites, determine mass from constructing atoms */
    for (ft = 0; ft < F_NRE; ft++)
    {
        if (IS_VSITE(ft))
        {
            il = &moltype->ilist[ft];

            for (i = 0; i < il->nr; i += 1+NRAL(ft))
            {
                const t_iparams *ip;
                real             inv_mass, coeff, m_aj;
                int              a1, aj;

                ip = &ffparams->iparams[il->iatoms[i]];

                a1 = il->iatoms[i+1];

                if (ft != F_VSITEN)
                {
                    /* Only vsiten can have more than four
                       constructing atoms, so NRAL(ft) <= 5 */
                    int        j;
                    real      *cam;
                    const int  maxj = NRAL(ft);

                    snew(cam, maxj);
                    assert(maxj <= 5);
                    for (j = 1; j < maxj; j++)
                    {
                        cam[j] = moltype->atoms.atom[il->iatoms[i+1+j]].m;
                        if (cam[j] == 0)
                        {
                            cam[j] = vsite_m[il->iatoms[i+1+j]];
                        }
                        if (cam[j] == 0)
                        {
                            gmx_fatal(FARGS, "In molecule type '%s' %s construction involves atom %d, which is a virtual site of equal or high complexity. This is not supported.",
                                      *moltype->name,
                                      interaction_function[ft].longname,
                                      il->iatoms[i+1+j]+1);
                        }
                    }

                    switch (ft)
                    {
                        case F_VSITE2:
                            /* Exact */
                            vsite_m[a1] = (cam[1]*cam[2])/(cam[2]*gmx::square(1-ip->vsite.a) + cam[1]*gmx::square(ip->vsite.a));
                            break;
                        case F_VSITE3:
                            /* Exact */
                            vsite_m[a1] = (cam[1]*cam[2]*cam[3])/(cam[2]*cam[3]*gmx::square(1-ip->vsite.a-ip->vsite.b) + cam[1]*cam[3]*gmx::square(ip->vsite.a) + cam[1]*cam[2]*gmx::square(ip->vsite.b));
                            break;
                        case F_VSITEN:
                            gmx_incons("Invalid vsite type");
                            break;
                        default:
                            /* Use the mass of the lightest constructing atom.
                             * This is an approximation.
                             * If the distance of the virtual site to the
                             * constructing atom is less than all distances
                             * between constructing atoms, this is a safe
                             * over-estimate of the displacement of the vsite.
                             * This condition holds for all H mass replacement
                             * vsite constructions, except for SP2/3 groups.
                             * In SP3 groups one H will have a F_VSITE3
                             * construction, so even there the total drift
                             * estimate shouldn't be far off.
                             */
                            vsite_m[a1] = cam[1];
                            for (j = 2; j < maxj; j++)
                            {
                                vsite_m[a1] = std::min(vsite_m[a1], cam[j]);
                            }
                            (*n_nonlin_vsite)++;
                            break;
                    }
                    sfree(cam);
                }
                else
                {
                    int j;

                    /* Exact */
                    inv_mass = 0;
                    for (j = 0; j < 3*ffparams->iparams[il->iatoms[i]].vsiten.n; j += 3)
                    {
                        aj    = il->iatoms[i+j+2];
                        coeff = ffparams->iparams[il->iatoms[i+j]].vsiten.a;
                        if (moltype->atoms.atom[aj].ptype == eptVSite)
                        {
                            m_aj = vsite_m[aj];
                        }
                        else
                        {
                            m_aj = moltype->atoms.atom[aj].m;
                        }
                        if (m_aj <= 0)
                        {
                            gmx_incons("The mass of a vsiten constructing atom is <= 0");
                        }
                        inv_mass += coeff*coeff/m_aj;
                    }
                    vsite_m[a1] = 1/inv_mass;
                    /* Correct for loop increment of i */
                    i += j - 1 - NRAL(ft);
                }
                if (gmx_debug_at)
                {
                    fprintf(debug, "atom %4d %-20s mass %6.3f\n",
                            a1, interaction_function[ft].longname, vsite_m[a1]);
                }
            }
        }
    }
}

static void get_verlet_buffer_atomtypes(const gmx_mtop_t      *mtop,
                                        verletbuf_atomtype_t **att_p,
                                        int                   *natt_p,
                                        int                   *n_nonlin_vsite)
{
    verletbuf_atomtype_t          *att;
    int                            natt;
    int                            mb, nmol, ft, i, a1, a2, a3, a;
    const t_atoms                 *atoms;
    const t_ilist                 *il;
    const t_iparams               *ip;
    atom_nonbonded_kinetic_prop_t *prop;
    real                          *vsite_m;
    int                            n_nonlin_vsite_mol;

    att  = nullptr;
    natt = 0;

    if (n_nonlin_vsite != nullptr)
    {
        *n_nonlin_vsite = 0;
    }

    for (mb = 0; mb < mtop->nmolblock; mb++)
    {
        nmol = mtop->molblock[mb].nmol;

        atoms = &mtop->moltype[mtop->molblock[mb].type].atoms;

        /* Check for constraints, as they affect the kinetic energy.
         * For virtual sites we need the masses and geometry of
         * the constructing atoms to determine their velocity distribution.
         */
        snew(prop, atoms->nr);
        snew(vsite_m, atoms->nr);

        for (ft = F_CONSTR; ft <= F_CONSTRNC; ft++)
        {
            il = &mtop->moltype[mtop->molblock[mb].type].ilist[ft];

            for (i = 0; i < il->nr; i += 1+NRAL(ft))
            {
                ip         = &mtop->ffparams.iparams[il->iatoms[i]];
                a1         = il->iatoms[i+1];
                a2         = il->iatoms[i+2];
                if (atoms->atom[a2].m > prop[a1].con_mass)
                {
                    prop[a1].con_mass = atoms->atom[a2].m;
                    prop[a1].con_len  = ip->constr.dA;
                }
                if (atoms->atom[a1].m > prop[a2].con_mass)
                {
                    prop[a2].con_mass = atoms->atom[a1].m;
                    prop[a2].con_len  = ip->constr.dA;
                }
            }
        }

        il = &mtop->moltype[mtop->molblock[mb].type].ilist[F_SETTLE];

        for (i = 0; i < il->nr; i += 1+NRAL(F_SETTLE))
        {
            ip         = &mtop->ffparams.iparams[il->iatoms[i]];
            a1         = il->iatoms[i+1];
            a2         = il->iatoms[i+2];
            a3         = il->iatoms[i+3];
            /* Usually the mass of a1 (usually oxygen) is larger than a2/a3.
             * If this is not the case, we overestimate the displacement,
             * which leads to a larger buffer (ok since this is an exotic case).
             */
            prop[a1].con_mass = atoms->atom[a2].m;
            prop[a1].con_len  = ip->settle.doh;

            prop[a2].con_mass = atoms->atom[a1].m;
            prop[a2].con_len  = ip->settle.doh;

            prop[a3].con_mass = atoms->atom[a1].m;
            prop[a3].con_len  = ip->settle.doh;
        }

        get_vsite_masses(&mtop->moltype[mtop->molblock[mb].type],
                         &mtop->ffparams,
                         vsite_m,
                         &n_nonlin_vsite_mol);
        if (n_nonlin_vsite != nullptr)
        {
            *n_nonlin_vsite += nmol*n_nonlin_vsite_mol;
        }

        for (a = 0; a < atoms->nr; a++)
        {
            if (atoms->atom[a].ptype == eptVSite)
            {
                prop[a].mass = vsite_m[a];
            }
            else
            {
                prop[a].mass = atoms->atom[a].m;
            }
            prop[a].type     = atoms->atom[a].type;
            prop[a].q        = atoms->atom[a].q;
            /* We consider an atom constrained, #DOF=2, when it is
             * connected with constraints to (at least one) atom with
             * a mass of more than 0.4x its own mass. This is not a critical
             * parameter, since with roughly equal masses the unconstrained
             * and constrained displacement will not differ much (and both
             * overestimate the displacement).
             */
            prop[a].bConstr = (prop[a].con_mass > 0.4*prop[a].mass);

            add_at(&att, &natt, &prop[a], nmol);
        }

        sfree(vsite_m);
        sfree(prop);
    }

    if (gmx_debug_at)
    {
        for (a = 0; a < natt; a++)
        {
            fprintf(debug, "type %d: m %5.2f t %d q %6.3f con %d con_m %5.3f con_l %5.3f n %d\n",
                    a, att[a].prop.mass, att[a].prop.type, att[a].prop.q,
                    att[a].prop.bConstr, att[a].prop.con_mass, att[a].prop.con_len,
                    att[a].n);
        }
    }

    *att_p  = att;
    *natt_p = natt;
}

/* This function computes two components of the estimate of the variance
 * in the displacement of one atom in a system of two constrained atoms.
 * Returns in sigma2_2d the variance due to rotation of the constrained
 * atom around the atom to which it constrained.
 * Returns in sigma2_3d the variance due to displacement of the COM
 * of the whole system of the two constrained atoms.
 *
 * Note that we only take a single constraint (the one to the heaviest atom)
 * into account. If an atom has multiple constraints, this will result in
 * an overestimate of the displacement, which gives a larger drift and buffer.
 */
void constrained_atom_sigma2(real                                 kT_fac,
                             const atom_nonbonded_kinetic_prop_t *prop,
                             real                                *sigma2_2d,
                             real                                *sigma2_3d)
{
    /* Here we decompose the motion of a constrained atom into two
     * components: rotation around the COM and translation of the COM.
     */

    /* Determine the variance of the arc length for the two rotational DOFs */
    real massFraction = prop->con_mass/(prop->mass + prop->con_mass);
    real sigma2_rot   = kT_fac*massFraction/prop->mass;

    /* The distance from the atom to the COM, i.e. the rotational arm */
    real comDistance  = prop->con_len*massFraction;

    /* The variance relative to the arm */
    real sigma2_rel   = sigma2_rot/gmx::square(comDistance);

    /* For sigma2_rel << 1 we don't notice the rotational effect and
     * we have a normal, Gaussian displacement distribution.
     * For larger sigma2_rel the displacement is much less, in fact it can
     * not exceed 2*comDistance. We can calculate MSD/arm^2 as:
     *   integral_x=0-inf distance2(x) x/sigma2_rel exp(-x^2/(2 sigma2_rel)) dx
     * where x is angular displacement and distance2(x) is the distance^2
     * between points at angle 0 and x:
     *   distance2(x) = (sin(x) - sin(0))^2 + (cos(x) - cos(0))^2
     * The limiting value of this MSD is 2, which is also the value for
     * a uniform rotation distribution that would be reached at long time.
     * The maximum is 2.5695 at sigma2_rel = 4.5119.
     * We approximate this integral with a rational polynomial with
     * coefficients from a Taylor expansion. This approximation is an
     * overestimate for all values of sigma2_rel. Its maximum value
     * of 2.6491 is reached at sigma2_rel = sqrt(45/2) = 4.7434.
     * We keep the approximation constant after that.
     * We use this approximate MSD as the variance for a Gaussian distribution.
     *
     * NOTE: For any sensible buffer tolerance this will result in a (large)
     * overestimate of the buffer size, since the Gaussian has a long tail,
     * whereas the actual distribution can not reach values larger than 2.
     */
    /* Coeffients obtained from a Taylor expansion */
    const real a = 1.0/3.0;
    const real b = 2.0/45.0;

    /* Our approximation is constant after sigma2_rel = 1/sqrt(b) */
    sigma2_rel   = std::min(sigma2_rel, 1/std::sqrt(b));

    /* Compute the approximate sigma^2 for 2D motion due to the rotation */
    *sigma2_2d   = gmx::square(comDistance)*
        sigma2_rel/(1 + a*sigma2_rel + b*gmx::square(sigma2_rel));

    /* The constrained atom also moves (in 3D) with the COM of both atoms */
    *sigma2_3d   = kT_fac/(prop->mass + prop->con_mass);
}

static void get_atom_sigma2(real                                 kT_fac,
                            const atom_nonbonded_kinetic_prop_t *prop,
                            real                                *sigma2_2d,
                            real                                *sigma2_3d)
{
    if (prop->bConstr)
    {
        /* Complicated constraint calculation in a separate function */
        constrained_atom_sigma2(kT_fac, prop, sigma2_2d, sigma2_3d);
    }
    else
    {
        /* Unconstrained atom: trivial */
        *sigma2_2d = 0;
        *sigma2_3d = kT_fac/prop->mass;
    }
}

static void approx_2dof(real s2, real x, real *shift, real *scale)
{
    /* A particle with 1 DOF constrained has 2 DOFs instead of 3.
     * This code is also used for particles with multiple constraints,
     * in which case we overestimate the displacement.
     * The 2DOF distribution is sqrt(pi/2)*erfc(r/(sqrt(2)*s))/(2*s).
     * We approximate this with scale*Gaussian(s,r+shift),
     * by matching the distribution value and derivative at x.
     * This is a tight overestimate for all r>=0 at any s and x.
     */
    real ex, er;

    ex = std::exp(-x*x/(2*s2));
    er = std::erfc(x/std::sqrt(2*s2));

    *shift = -x + std::sqrt(2*s2/M_PI)*ex/er;
    *scale = 0.5*M_PI*std::exp(ex*ex/(M_PI*er*er))*er;
}

// Returns an (over)estimate of the energy drift for a single atom pair,
// given the kinetic properties, displacement variances and list buffer.
static real energyDriftAtomPair(const atom_nonbonded_kinetic_prop_t *prop_i,
                                const atom_nonbonded_kinetic_prop_t *prop_j,
                                real s2, real s2i_2d, real s2j_2d,
                                real r_buffer,
                                const pot_derivatives_t *der)
{
    // For relatively small arguments erfc() is so small that if will be 0.0
    // when stored in a float. We set an argument limit of 8 (Erfc(8)=1e-29),
    // such that we can divide by erfc and have some space left for arithmetic.
    const real erfc_arg_max = 8.0;

    real       rsh    = r_buffer;
    real       sc_fac = 1.0;

    real       c_exp, c_erfc;

    if (rsh*rsh > 2*s2*erfc_arg_max*erfc_arg_max)
    {
        // Below we calculate c_erfc = 0.5*erfc(rsh/sqrt(2*s2))
        // When rsh/sqrt(2*s2) increases, this erfc will be the first
        // result that underflows and becomes 0.0. To avoid this,
        // we set c_exp=0 and c_erfc=0 for large arguments.
        // This also avoids NaN in approx_2dof().
        // In any relevant case this has no effect on the results,
        // since c_exp < 6e-29, so the displacement is completely
        // negligible for such atom pairs (and an overestimate).
        // In nearly all use cases, there will be other atom pairs
        // that contribute much more to the total, so zeroing
        // this particular contribution has no effect at all.
        c_exp  = 0;
        c_erfc = 0;
    }
    else
    {
        /* For constraints: adapt r and scaling for the Gaussian */
        if (prop_i->bConstr)
        {
            real sh, sc;

            approx_2dof(s2i_2d, r_buffer*s2i_2d/s2, &sh, &sc);
            rsh    += sh;
            sc_fac *= sc;
        }
        if (prop_j->bConstr)
        {
            real sh, sc;

            approx_2dof(s2j_2d, r_buffer*s2j_2d/s2, &sh, &sc);
            rsh    += sh;
            sc_fac *= sc;
        }

        /* Exact contribution of an atom pair with Gaussian displacement
         * with sigma s to the energy drift for a potential with
         * derivative -md and second derivative dd at the cut-off.
         * The only catch is that for potentials that change sign
         * near the cut-off there could be an unlucky compensation
         * of positive and negative energy drift.
         * Such potentials are extremely rare though.
         *
         * Note that pot has unit energy*length, as the linear
         * atom density still needs to be put in.
         */
        c_exp  = std::exp(-rsh*rsh/(2*s2))/std::sqrt(2*M_PI);
        c_erfc = 0.5*std::erfc(rsh/(std::sqrt(2*s2)));
    }
    real s    = std::sqrt(s2);
    real rsh2 = rsh*rsh;

    real pot1 = sc_fac*
        der->md1/2*((rsh2 + s2)*c_erfc - rsh*s*c_exp);
    real pot2 = sc_fac*
        der->d2/6*(s*(rsh2 + 2*s2)*c_exp - rsh*(rsh2 + 3*s2)*c_erfc);
    real pot3 = sc_fac*
        der->md3/24*((rsh2*rsh2 + 6*rsh2*s2 + 3*s2*s2)*c_erfc - rsh*s*(rsh2 + 5*s2)*c_exp);

    return pot1 + pot2 + pot3;
}

static real energyDrift(const verletbuf_atomtype_t *att, int natt,
                        const gmx_ffparams_t *ffp,
                        real kT_fac,
                        const pot_derivatives_t *ljDisp,
                        const pot_derivatives_t *ljRep,
                        const pot_derivatives_t *elec,
                        real rlj, real rcoulomb,
                        real rlist, real boxvol)
{
    double drift_tot = 0;

    if (kT_fac == 0)
    {
        /* No atom displacements: no drift, avoid division by 0 */
        return drift_tot;
    }

    // Here add up the contribution of all atom pairs in the system to
    // (estimated) energy drift by looping over all atom type pairs.
    for (int i = 0; i < natt; i++)
    {
        // Get the thermal displacement variance for the i-atom type
        const atom_nonbonded_kinetic_prop_t *prop_i = &att[i].prop;
        real                                 s2i_2d, s2i_3d;
        get_atom_sigma2(kT_fac, prop_i, &s2i_2d, &s2i_3d);

        for (int j = i; j < natt; j++)
        {
            // Get the thermal displacement variance for the j-atom type
            const atom_nonbonded_kinetic_prop_t *prop_j = &att[j].prop;
            real                                 s2j_2d, s2j_3d;
            get_atom_sigma2(kT_fac, prop_j, &s2j_2d, &s2j_3d);

            /* Add up the up to four independent variances */
            real s2 = s2i_2d + s2i_3d + s2j_2d + s2j_3d;

            // Set -V', V'' and -V''' at the cut-off for LJ */
            real              c6  = ffp->iparams[prop_i->type*ffp->atnr + prop_j->type].lj.c6;
            real              c12 = ffp->iparams[prop_i->type*ffp->atnr + prop_j->type].lj.c12;
            pot_derivatives_t lj;
            lj.md1 = c6*ljDisp->md1 + c12*ljRep->md1;
            lj.d2  = c6*ljDisp->d2  + c12*ljRep->d2;
            lj.md3 = c6*ljDisp->md3 + c12*ljRep->md3;

            real pot_lj = energyDriftAtomPair(prop_i, prop_j,
                                              s2, s2i_2d, s2j_2d,
                                              rlist - rlj,
                                              &lj);

            // Set -V' and V'' at the cut-off for Coulomb
            pot_derivatives_t elec_qq;
            elec_qq.md1 = elec->md1*prop_i->q*prop_j->q;
            elec_qq.d2  = elec->d2 *prop_i->q*prop_j->q;
            elec_qq.md3 = 0;

            real pot_q  = energyDriftAtomPair(prop_i, prop_j,
                                              s2, s2i_2d, s2j_2d,
                                              rlist - rcoulomb,
                                              &elec_qq);

            // Note that attractive and repulsive potentials for individual
            // pairs can partially cancel.
            real pot = pot_lj + pot_q;

            /* Multiply by the number of atom pairs */
            if (j == i)
            {
                pot *= (double)att[i].n*(att[i].n - 1)/2;
            }
            else
            {
                pot *= (double)att[i].n*att[j].n;
            }
            /* We need the line density to get the energy drift of the system.
             * The effective average r^2 is close to (rlist+sigma)^2.
             */
            pot *= 4*M_PI*gmx::square(rlist + std::sqrt(s2))/boxvol;

            /* Add the unsigned drift to avoid cancellation of errors */
            drift_tot += std::abs(pot);
        }
    }

    return drift_tot;
}

static real surface_frac(int cluster_size, real particle_distance, real rlist)
{
    real d, area_rel;

    if (rlist < 0.5*particle_distance)
    {
        /* We have non overlapping spheres */
        return 1.0;
    }

    /* Half the inter-particle distance relative to rlist */
    d = 0.5*particle_distance/rlist;

    /* Determine the area of the surface at distance rlist to the closest
     * particle, relative to surface of a sphere of radius rlist.
     * The formulas below assume close to cubic cells for the pair search grid,
     * which the pair search code tries to achieve.
     * Note that in practice particle distances will not be delta distributed,
     * but have some spread, often involving shorter distances,
     * as e.g. O-H bonds in a water molecule. Thus the estimates below will
     * usually be slightly too high and thus conservative.
     */
    switch (cluster_size)
    {
        case 1:
            /* One particle: trivial */
            area_rel = 1.0;
            break;
        case 2:
            /* Two particles: two spheres at fractional distance 2*a */
            area_rel = 1.0 + d;
            break;
        case 4:
            /* We assume a perfect, symmetric tetrahedron geometry.
             * The surface around a tetrahedron is too complex for a full
             * analytical solution, so we use a Taylor expansion.
             */
            area_rel = (1.0 + 1/M_PI*(6*std::acos(1/std::sqrt(3))*d +
                                      std::sqrt(3)*d*d*(1.0 +
                                                        5.0/18.0*d*d +
                                                        7.0/45.0*d*d*d*d +
                                                        83.0/756.0*d*d*d*d*d*d)));
            break;
        default:
            gmx_incons("surface_frac called with unsupported cluster_size");
            area_rel = 1.0;
    }

    return area_rel/cluster_size;
}

/* Returns the negative of the third derivative of a potential r^-p
 * with a force-switch function, evaluated at the cut-off rc.
 */
static real md3_force_switch(real p, real rswitch, real rc)
{
    /* The switched force function is:
     * p*r^-(p+1) + a*(r - rswitch)^2 + b*(r - rswitch)^3
     */
    real a, b;
    real md3_pot, md3_sw;

    a = -((p + 4)*rc - (p + 1)*rswitch)/(pow(rc, p+2)*gmx::square(rc-rswitch));
    b =  ((p + 3)*rc - (p + 1)*rswitch)/(pow(rc, p+2)*gmx::power3(rc-rswitch));

    md3_pot = (p + 2)*(p + 1)*p*pow(rc, p+3);
    md3_sw  = 2*a + 6*b*(rc - rswitch);

    return md3_pot + md3_sw;
}

void calc_verlet_buffer_size(const gmx_mtop_t *mtop, real boxvol,
                             const t_inputrec *ir,
                             real reference_temperature,
                             const verletbuf_list_setup_t *list_setup,
                             int *n_nonlin_vsite,
                             real *rlist)
{
    double                resolution;
    char                 *env;

    real                  particle_distance;
    real                  nb_clust_frac_pairs_not_in_list_at_cutoff;

    verletbuf_atomtype_t *att  = nullptr;
    int                   natt = -1, i;
    real                  elfac;
    real                  kT_fac, mass_min;
    int                   ib0, ib1, ib;
    real                  rb, rl;
    real                  drift;

    if (!EI_DYNAMICS(ir->eI))
    {
        gmx_incons("Can only determine the Verlet buffer size for integrators that perform dynamics");
    }
    if (ir->verletbuf_tol <= 0)
    {
        gmx_incons("The Verlet buffer tolerance needs to be larger than zero");
    }

    if (reference_temperature < 0)
    {
        if (EI_MD(ir->eI) && ir->etc == etcNO)
        {
            /* This case should be handled outside calc_verlet_buffer_size */
            gmx_incons("calc_verlet_buffer_size called with an NVE ensemble and reference_temperature < 0");
        }

        /* We use the maximum temperature with multiple T-coupl groups.
         * We could use a per particle temperature, but since particles
         * interact, this might underestimate the buffer size.
         */
        reference_temperature = 0;
        for (i = 0; i < ir->opts.ngtc; i++)
        {
            if (ir->opts.tau_t[i] >= 0)
            {
                reference_temperature = std::max(reference_temperature,
                                                 ir->opts.ref_t[i]);
            }
        }
    }

    /* Resolution of the buffer size */
    resolution = 0.001;

    env = getenv("GMX_VERLET_BUFFER_RES");
    if (env != nullptr)
    {
        sscanf(env, "%lf", &resolution);
    }

    /* In an atom wise pair-list there would be no pairs in the list
     * beyond the pair-list cut-off.
     * However, we use a pair-list of groups vs groups of atoms.
     * For groups of 4 atoms, the parallelism of SSE instructions, only
     * 10% of the atoms pairs are not in the list just beyond the cut-off.
     * As this percentage increases slowly compared to the decrease of the
     * Gaussian displacement distribution over this range, we can simply
     * reduce the drift by this fraction.
     * For larger groups, e.g. of 8 atoms, this fraction will be lower,
     * so then buffer size will be on the conservative (large) side.
     *
     * Note that the formulas used here do not take into account
     * cancellation of errors which could occur by missing both
     * attractive and repulsive interactions.
     *
     * The only major assumption is homogeneous particle distribution.
     * For an inhomogeneous system, such as a liquid-vapor system,
     * the buffer will be underestimated. The actual energy drift
     * will be higher by the factor: local/homogeneous particle density.
     *
     * The results of this estimate have been checked againt simulations.
     * In most cases the real drift differs by less than a factor 2.
     */

    /* Worst case assumption: HCP packing of particles gives largest distance */
    particle_distance = std::cbrt(boxvol*std::sqrt(2)/mtop->natoms);

    get_verlet_buffer_atomtypes(mtop, &att, &natt, n_nonlin_vsite);
    assert(att != NULL && natt >= 0);

    if (debug)
    {
        fprintf(debug, "particle distance assuming HCP packing: %f nm\n",
                particle_distance);
        fprintf(debug, "energy drift atom types: %d\n", natt);
    }

    pot_derivatives_t ljDisp = { 0, 0, 0 };
    pot_derivatives_t ljRep  = { 0, 0, 0 };
    real              repPow = mtop->ffparams.reppow;

    if (ir->vdwtype == evdwCUT)
    {
        real sw_range, md3_pswf;

        switch (ir->vdw_modifier)
        {
            case eintmodNONE:
            case eintmodPOTSHIFT:
                /* -dV/dr of -r^-6 and r^-reppow */
                ljDisp.md1 =     -6*std::pow(ir->rvdw, -7.0);
                ljRep.md1  = repPow*std::pow(ir->rvdw, -(repPow + 1));
                /* The contribution of the higher derivatives is negligible */
                break;
            case eintmodFORCESWITCH:
                /* At the cut-off: V=V'=V''=0, so we use only V''' */
                ljDisp.md3 = -md3_force_switch(6.0,    ir->rvdw_switch, ir->rvdw);
                ljRep.md3  =  md3_force_switch(repPow, ir->rvdw_switch, ir->rvdw);
                break;
            case eintmodPOTSWITCH:
                /* At the cut-off: V=V'=V''=0.
                 * V''' is given by the original potential times
                 * the third derivative of the switch function.
                 */
                sw_range   = ir->rvdw - ir->rvdw_switch;
                md3_pswf   = 60.0/gmx::power3(sw_range);

                ljDisp.md3 = -std::pow(ir->rvdw, -6.0   )*md3_pswf;
                ljRep.md3  =  std::pow(ir->rvdw, -repPow)*md3_pswf;
                break;
            default:
                gmx_incons("Unimplemented VdW modifier");
        }
    }
    else if (EVDW_PME(ir->vdwtype))
    {
        real b     = calc_ewaldcoeff_lj(ir->rvdw, ir->ewald_rtol_lj);
        real r     = ir->rvdw;
        real br    = b*r;
        real br2   = br*br;
        real br4   = br2*br2;
        real br6   = br4*br2;
        // -dV/dr of g(br)*r^-6 [where g(x) = exp(-x^2)(1+x^2+x^4/2),
        // see LJ-PME equations in manual] and r^-reppow
        ljDisp.md1 = -std::exp(-br2)*(br6 + 3.0*br4 + 6.0*br2 + 6.0)*std::pow(r, -7.0);
        ljRep.md1  = repPow*pow(r, -(repPow + 1));
        // The contribution of the higher derivatives is negligible
    }
    else
    {
        gmx_fatal(FARGS, "Energy drift calculation is only implemented for plain cut-off Lennard-Jones interactions");
    }

    elfac = ONE_4PI_EPS0/ir->epsilon_r;

    // Determine the 1st and 2nd derivative for the electostatics
    pot_derivatives_t elec = { 0, 0, 0 };

    if (ir->coulombtype == eelCUT || EEL_RF(ir->coulombtype))
    {
        real eps_rf, k_rf;

        if (ir->coulombtype == eelCUT)
        {
            eps_rf = 1;
            k_rf   = 0;
        }
        else
        {
            eps_rf = ir->epsilon_rf/ir->epsilon_r;
            if (eps_rf != 0)
            {
                k_rf = (eps_rf - ir->epsilon_r)/( gmx::power3(ir->rcoulomb) * (2*eps_rf + ir->epsilon_r) );
            }
            else
            {
                /* epsilon_rf = infinity */
                k_rf = 0.5/gmx::power3(ir->rcoulomb);
            }
        }

        if (eps_rf > 0)
        {
            elec.md1 = elfac*(1.0/gmx::square(ir->rcoulomb) - 2*k_rf*ir->rcoulomb);
        }
        elec.d2      = elfac*(2.0/gmx::power3(ir->rcoulomb) + 2*k_rf);
    }
    else if (EEL_PME(ir->coulombtype) || ir->coulombtype == eelEWALD)
    {
        real b, rc, br;

        b        = calc_ewaldcoeff_q(ir->rcoulomb, ir->ewald_rtol);
        rc       = ir->rcoulomb;
        br       = b*rc;
        elec.md1 = elfac*(b*std::exp(-br*br)*M_2_SQRTPI/rc + std::erfc(br)/(rc*rc));
        elec.d2  = elfac/(rc*rc)*(2*b*(1 + br*br)*std::exp(-br*br)*M_2_SQRTPI + 2*std::erfc(br)/rc);
    }
    else
    {
        gmx_fatal(FARGS, "Energy drift calculation is only implemented for Reaction-Field and Ewald electrostatics");
    }

    /* Determine the variance of the atomic displacement
     * over nstlist-1 steps: kT_fac
     * For inertial dynamics (not Brownian dynamics) the mass factor
     * is not included in kT_fac, it is added later.
     */
    if (ir->eI == eiBD)
    {
        /* Get the displacement distribution from the random component only.
         * With accurate integration the systematic (force) displacement
         * should be negligible (unless nstlist is extremely large, which
         * you wouldn't do anyhow).
         */
        kT_fac = 2*BOLTZ*reference_temperature*(ir->nstlist-1)*ir->delta_t;
        if (ir->bd_fric > 0)
        {
            /* This is directly sigma^2 of the displacement */
            kT_fac /= ir->bd_fric;

            /* Set the masses to 1 as kT_fac is the full sigma^2,
             * but we divide by m in ener_drift().
             */
            for (i = 0; i < natt; i++)
            {
                att[i].prop.mass = 1;
            }
        }
        else
        {
            real tau_t;

            /* Per group tau_t is not implemented yet, use the maximum */
            tau_t = ir->opts.tau_t[0];
            for (i = 1; i < ir->opts.ngtc; i++)
            {
                tau_t = std::max(tau_t, ir->opts.tau_t[i]);
            }

            kT_fac *= tau_t;
            /* This kT_fac needs to be divided by the mass to get sigma^2 */
        }
    }
    else
    {
        kT_fac = BOLTZ*reference_temperature*gmx::square((ir->nstlist-1)*ir->delta_t);
    }

    mass_min = att[0].prop.mass;
    for (i = 1; i < natt; i++)
    {
        mass_min = std::min(mass_min, att[i].prop.mass);
    }

    if (debug)
    {
        fprintf(debug, "Derivatives of non-bonded potentials at the cut-off:\n");
        fprintf(debug, "LJ disp. -V' %9.2e V'' %9.2e -V''' %9.2e\n", ljDisp.md1, ljDisp.d2, ljDisp.md3);
        fprintf(debug, "LJ rep.  -V' %9.2e V'' %9.2e -V''' %9.2e\n", ljRep.md1, ljRep.d2, ljRep.md3);
        fprintf(debug, "Electro. -V' %9.2e V'' %9.2e\n", elec.md1, elec.d2);
        fprintf(debug, "sqrt(kT_fac) %f\n", std::sqrt(kT_fac));
        fprintf(debug, "mass_min %f\n", mass_min);
    }

    /* Search using bisection */
    ib0 = -1;
    /* The drift will be neglible at 5 times the max sigma */
    ib1 = (int)(5*2*std::sqrt(kT_fac/mass_min)/resolution) + 1;
    while (ib1 - ib0 > 1)
    {
        ib = (ib0 + ib1)/2;
        rb = ib*resolution;
        rl = std::max(ir->rvdw, ir->rcoulomb) + rb;

        /* Calculate the average energy drift at the last step
         * of the nstlist steps at which the pair-list is used.
         */
        drift = energyDrift(att, natt, &mtop->ffparams,
                            kT_fac,
                            &ljDisp, &ljRep, &elec,
                            ir->rvdw, ir->rcoulomb,
                            rl, boxvol);

        /* Correct for the fact that we are using a Ni x Nj particle pair list
         * and not a 1 x 1 particle pair list. This reduces the drift.
         */
        /* We don't have a formula for 8 (yet), use 4 which is conservative */
        nb_clust_frac_pairs_not_in_list_at_cutoff =
            surface_frac(std::min(list_setup->cluster_size_i, 4),
                         particle_distance, rl)*
            surface_frac(std::min(list_setup->cluster_size_j, 4),
                         particle_distance, rl);
        drift *= nb_clust_frac_pairs_not_in_list_at_cutoff;

        /* Convert the drift to drift per unit time per atom */
        drift /= ir->nstlist*ir->delta_t*mtop->natoms;

        if (debug)
        {
            fprintf(debug, "ib %3d %3d %3d rb %.3f %dx%d fac %.3f drift %.1e\n",
                    ib0, ib, ib1, rb,
                    list_setup->cluster_size_i, list_setup->cluster_size_j,
                    nb_clust_frac_pairs_not_in_list_at_cutoff,
                    drift);
        }

        if (std::abs(drift) > ir->verletbuf_tol)
        {
            ib0 = ib;
        }
        else
        {
            ib1 = ib;
        }
    }

    sfree(att);

    *rlist = std::max(ir->rvdw, ir->rcoulomb) + ib1*resolution;
}