4.8.1. Writing new pair styles — LAMMPS documentation

\(\renewcommand{\AA}{\text{Å}}\)

# 4.8.1. Writing new pair styles

Pair styles are at the core of most simulations with LAMMPS, since they are used to compute the forces (plus energy and virial contributions, if needed) on atoms for pairs or small clusters of atoms within a given cutoff. This is often the dominant computation in LAMMPS, and sometimes even the only one. Pair styles can be grouped into multiple categories:

1. simple pairwise additive interactions of point particles (e.g. Lennard-Jones, Morse, Buckingham)

2. pairwise additive interactions of point particles with added Coulomb interactions or only the Coulomb interactions

3. manybody interactions of point particles (e.g. EAM, Tersoff)

4. complex interactions that include additional per-atom properties (e.g. Discrete Element Models (DEM), Peridynamics, Ellipsoids)

5. special purpose pair styles that may not even compute forces like pair_style zero and pair_style tracker, or are a wrapper for multiple kinds of interactions like pair_style hybrid, pair_style list, and pair_style kim

In the text below, we will discuss aspects of implementing pair styles in LAMMPS by looking at representative case studies. The design of LAMMPS allows developers to focus on the essentials, which is to compute the forces (and energies or virial contributions), enter and manage the global settings as well as the potential parameters, and the pair style specific parts of reading and writing restart and data files. Most of the complex tasks like management of the atom positions, domain decomposition and boundaries, or neighbor list creation are handled transparently by other parts of the LAMMPS code.

As shown on the page for writing or extending pair styles, in order to implement a new pair style, a new class must be written that is either directly or indirectly derived from the `Pair` class. If that class is directly derived from `Pair`, there are three methods that must be re-implemented, since they are “pure” in the base class: `Pair::compute()`, `Pair::settings()`, `Pair::coeff()`. In addition, a custom constructor is needed. All other methods are optional and have default implementations in the base class (most of which do nothing), but they may need to be overridden depending on the requirements of the model.

We are looking at the following cases:

- Case 1: a pairwise additive model

- Case 2: a many-body potential

- Case 3: a potential requiring communication

- Case 4: potentials without a compute() function

# 4.8.2. Case 1: a pairwise additive model

In this section, we will describe the procedure of adding a simple pair style to LAMMPS: an empirical model that can be used to model liquid mercury. The pair style shall be called bond/gauss and the complete implementation can be found in the files `src/EXTRA-PAIR/pair_born_gauss.cpp` and `src/EXTRA-PAIR/pair_born_gauss.h` of the LAMMPS source code.

## Model and general considerations

The functional form of the model according to (Bomont) consists of a repulsive Born-Mayer exponential term and a temperature dependent, attractive Gaussian term.

\[E = A_0 \exp \left( -\alpha r \right) - A_1 \exp\left[ -\beta \left(r - r_0 \right)^2 \right]\]

For the application to mercury, the following parameters are listed:

- \(A_0 = 8.2464 \times 10^{13} \; \textrm{eV}\)

- \(\alpha = 12.48 \; \AA^{-1}\)

- \(\beta = 0.44 \; \AA^{-2}\)

- \(r_0 = 3.56 \; \AA\)

- \(A_1\) is temperature dependent and can be determined from \(A_1 = a_0 + a_1 T + a_2 T^2\) with:

  - \(a_0 = 1.97475 \times 10^{-2} \; \textrm{eV}\)

  - \(a_1 = 8.40841 \times 10^{-5} \; \textrm{eV/K}\)

  - \(a_2 = -2.58717 \times 10^{-8} \; \textrm{eV/K}^{-2}\)

With the optional cutoff, this means we have a total of 5 or 6 parameters for each pair of atom types. Additionally, we need to input a default cutoff value as a global setting.

Because of the combination of Born-Mayer with a Gaussian, the pair style shall be named “born/gauss” and thus the class name would be `PairBornGauss` and the source files `pair_born_gauss.h` and `pair_born_gauss.cpp`. Since this is a rather uncommon potential, it shall be added to the EXTRA-PAIR package.

## Header file

The first segment of any LAMMPS source should be the copyright and license statement. Note the marker in the first line to indicate to editors like emacs that this file is a C++ source, even though the .h extension suggests a C source (this is a convention inherited from the very beginning of the C++ version of LAMMPS).

    /* -*- c++ -*- ----------------------------------------------------------
       LAMMPS - Large-scale Atomic/Molecular Massively Parallel Simulator
       https://www.lammps.org/, Sandia National Laboratories
       LAMMPS development team: developers@lammps.org

       Copyright (2003) Sandia Corporation.  Under the terms of Contract
       DE-AC04-94AL85000 with Sandia Corporation, the U.S. Government retains
       certain rights in this software.  This software is distributed under
       the GNU General Public License.

       See the README file in the top-level LAMMPS directory.
    ------------------------------------------------------------------------- */

Every pair style must be registered in LAMMPS by including the following lines of code in the second part of the header after the copyright message and before the include guards for the class definition:

    #ifdef PAIR_CLASS
    // clang-format off
    PairStyle(born/gauss,PairBornGauss);
    // clang-format on
    #else

    /* the definition of the PairBornGauss class (see below) is inserted here */

    #endif

This block of between `#ifdef PAIR_CLASS` and `#else` will be included by the `Force` class in `force.cpp` to build a map of “factory functions” that will create an instance of these classes and return a pointer to it. The map connects the name of the pair style, “born/gauss”, to the name of the class, `PairBornGauss`. During compilation, LAMMPS constructs a file `style_pair.h` that contains `#include` statements for all “installed” pair styles. Before including `style_pair.h` into `force.cpp`, the `PAIR_CLASS` define is set and the `PairStyle(name,class)` macro defined. The code of the macro adds the installed pair styles to the “factory map” which enables the pair_style command to create the pair style instance.

The list of header files to include is automatically updated by the build system if there are new files, so the presence of the new header file in the `src/EXTRA-PAIR` folder and the enabling of the EXTRA-PAIR package will trigger LAMMPS to include the new pair style when it is (re-)compiled. The “// clang-format” format comments are needed so that running clang-format on the file will not insert unwanted blanks between “born”, “/”, and “gauss” which would break the `PairStyle` macro.

The third part of the header file is the actual class definition of the `PairBornGauss` class. This has the prototypes for all member functions that will be implemented by this pair style. This includes a few required and a number of optional functions. All functions that were labeled in the base class as “virtual” must be given the “override” property, as it is done in the code shown below.

The “override” property helps to detect unexpected mismatches because compilation will stop with an error in case the signature of a function is changed in the base class without also changing it in all derived classes. For example, if this change added an optional argument with a default value, then all existing source code calling the function would not need changes and still compile, but the function in the derived class would no longer override the one in the base class due to the different number of arguments and the behavior of the pair style is thus changed in an unintended way. Using the “override” keyword prevents such issues.

    #ifndef LMP_PAIR_BORN_GAUSS_H
    #define LMP_PAIR_BORN_GAUSS_H

    #include "pair.h"

    namespace LAMMPS_NS {

    class PairBornGauss : public Pair {
     public:
      PairBornGauss(class LAMMPS *);
      ~PairBornGauss() override;

      void compute(int, int) override;
      void settings(int, char **) override;
      void coeff(int, char **) override;
      double init_one(int, int) override;

      void write_restart(FILE *) override;
      void read_restart(FILE *) override;
      void write_restart_settings(FILE *) override;
      void read_restart_settings(FILE *) override;
      void write_data(FILE *) override;
      void write_data_all(FILE *) override;

      double single(int, int, int, int, double, double, double, double &) override;
      void *extract(const char *, int &) override;

Also, variables and arrays for storing global settings and potential parameters are defined. Since these are internal to the class, they are placed after a “protected:” label.

     protected:
      double cut_global;
      double **cut;
      double **biga0, **alpha, **biga1, **beta, **r0;
      double **a0, **a1, **a2;
      double **offset;

      virtual void allocate();
    };
    }    // namespace LAMMPS_NS
    #endif

## Implementation file

We move on to the implementation of the `PairBornGauss` class in the `pair_born_gauss.cpp` file. This file also starts with a LAMMPS copyright and license header. Below that notice is typically the space where comments may be added with additional information about this specific file, the author(s), affiliation(s), and email address(es). This way the contributing author(s) can be easily contacted, when there are questions about the implementation later. Since the file(s) may be around for a long time, it is beneficial to use some kind of “permanent” email address, if possible.

    /* ----------------------------------------------------------------------
       LAMMPS - Large-scale Atomic/Molecular Massively Parallel Simulator
       https://www.lammps.org/, Sandia National Laboratories
       LAMMPS development team: developers@lammps.org

       Copyright (2003) Sandia Corporation.  Under the terms of Contract
       DE-AC04-94AL85000 with Sandia Corporation, the U.S. Government retains
       certain rights in this software.  This software is distributed under
       the GNU General Public License.

       See the README file in the top-level LAMMPS directory.
    ------------------------------------------------------------------------- */

    // Contributing author: Axel Kohlmeyer, Temple University, akohlmey@gmail.com

    #include "pair_born_gauss.h"

    #include "atom.h"
    #include "comm.h"
    #include "error.h"
    #include "fix.h"
    #include "force.h"
    #include "memory.h"
    #include "neigh_list.h"

    #include <cmath>
    #include <cstring>

    using namespace LAMMPS_NS;

The second section of the implementation file has various include statements. The include file for the class header has to come first, then a block of LAMMPS classes (sorted alphabetically) followed by a block of system headers and others, if needed. Note the standardized C++ notation for headers of C-library functions (`cmath` instead of `math.h`). The final statement of this segment imports the `LAMMPS_NS::` namespace globally for this file. This way, all LAMMPS specific functions and classes do not have to be prefixed with `LAMMPS_NS::`.

## Constructor and destructor (required)

The first two functions in the implementation source file are typically the constructor and the destructor.

Pair styles are different from most classes in LAMMPS that define a “style”, as their constructor only uses the LAMMPS class instance pointer as an argument, but not the command line arguments of the pair_style command. Instead, those arguments are processed in the `Pair::settings()` function (or rather the version in the derived class). The constructor is the place where global defaults are set and specifically flags are set indicating which optional features of a pair style are available.

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

    PairBornGauss::PairBornGauss(LAMMPS *lmp) : Pair(lmp)
    {
      writedata = 1;
    }

The writedata = 1; statement indicates that the pair style is capable of writing the current pair coefficient parameters to data files. That is, the class implements specific versions for `Pair::data()` and `Pair::data_all()`. Other statements that could be added here would be single_enable = 1; or respa_enable = 0; to indicate that the `Pair::single()` function is present and the `Pair::compute_(inner|middle|outer)` functions are not, but those are also the default settings and already set in the base class.

In the destructor, we need to delete all memory that was allocated by the pair style, usually to hold force field parameters that were entered with the pair_coeff command. Most of those array pointers will need to be declared in the derived class header, but some (e.g. setflag, cutsq) are already declared in the base class.

    PairBornGauss::~PairBornGauss()
    {
      if (allocated) {
        memory->destroy(setflag);
        memory->destroy(cutsq);
        memory->destroy(cut);
        memory->destroy(biga0);
        memory->destroy(alpha);
        memory->destroy(biga1);
        memory->destroy(beta);
        memory->destroy(r0);
        memory->destroy(offset);
      }
    }

## Settings and coefficients (required)

To enter the global pair style settings and the pair style parameters, the functions `Pair::settings()` and `Pair::coeff()` need to be re-implemented. The arguments to the `settings()` function are the arguments given to the pair_style command. Normally, those would already be processed as part of the constructor, but moving this to a separate function allows users to change global settings like the default cutoff without having to reissue all pair_coeff commands or re-read the `Pair Coeffs` sections from the data file. In the `settings()` function, also the arrays for storing parameters, to define cutoffs, track with pairs of parameters have been explicitly set are allocated and, if needed, initialized. In this case, the memory allocation and initialization is moved to a function `allocate()`.

    /* ----------------------------------------------------------------------
       allocate all arrays
    ------------------------------------------------------------------------- */

    void PairBornGauss::allocate()
    {
      allocated = 1;
      int np1 = atom->ntypes + 1;

      memory->create(setflag, np1, np1, "pair:setflag");
      for (int i = 1; i < np1; i++)
        for (int j = i; j < np1; j++) setflag[i][j] = 0;

      memory->create(cutsq, np1, np1, "pair:cutsq");
      memory->create(cut, np1, np1, "pair:cut");
      memory->create(biga0, np1, np1, "pair:biga0");
      memory->create(alpha, np1, np1, "pair:alpha");
      memory->create(biga1, np1, np1, "pair:biga1");
      memory->create(beta, np1, np1, "pair:beta");
      memory->create(r0, np1, np1, "pair:r0");
      memory->create(offset, np1, np1, "pair:offset");
    }

    /* ----------------------------------------------------------------------
       global settings
    ------------------------------------------------------------------------- */

    void PairBornGauss::settings(int narg, char **arg)
    {
      if (narg != 1) error->all(FLERR, "Pair style bond/gauss must have exactly one argument");
      cut_global = utils::numeric(FLERR, arg[0], false, lmp);

      // reset per-type pair cutoffs that have been explicitly set previously

      if (allocated) {
        for (int i = 1; i <= atom->ntypes; i++)
          for (int j = i; j <= atom->ntypes; j++)
            if (setflag[i][j]) cut[i][j] = cut_global;
      }
    }

The arguments to the `coeff()` function are the arguments to the pair_coeff command. The function is also called when processing the `Pair Coeffs` or `PairIJ Coeffs` sections of data files. In the case of the `Pair Coeffs` section, there is only one atom type per line and thus the first argument is duplicated. Since the atom type arguments of the pair_coeff command may be a range (e.g. *3 for atom types 1, 2, and 3), the corresponding arguments are passed to the `utils::bounds()` function which will then return the low and high end of the range. Note that the `setflag` array is set to 1 for all pairs of atom types processed by this call. This information is later used in the `init_one()` function to determine if any coefficients are missing and, if supported by the potential, generate those missing coefficients from the selected mixing rule.

    /* ----------------------------------------------------------------------
       set coeffs for one or more type pairs
    ------------------------------------------------------------------------- */

    void PairBornGauss::coeff(int narg, char **arg)
    {
      if (narg < 7 || narg > 8) error->all(FLERR, "Incorrect args for pair coefficients");
      if (!allocated) allocate();

      int ilo, ihi, jlo, jhi;
      utils::bounds(FLERR, arg[0], 1, atom->ntypes, ilo, ihi, error);
      utils::bounds(FLERR, arg[1], 1, atom->ntypes, jlo, jhi, error);

      double biga0_one = utils::numeric(FLERR, arg[2], false, lmp);
      double alpha_one = utils::numeric(FLERR, arg[3], false, lmp);
      double biga1_one = utils::numeric(FLERR, arg[4], false, lmp);
      double beta_one = utils::numeric(FLERR, arg[5], false, lmp);
      double r0_one = utils::numeric(FLERR, arg[6], false, lmp);
      double cut_one = cut_global;
      if (narg == 10) cut_one = utils::numeric(FLERR, arg[7], false, lmp);

      int count = 0;
      for (int i = ilo; i <= ihi; i++) {
        for (int j = MAX(jlo, i); j <= jhi; j++) {
          biga0[i][j] = biga0_one;
          alpha[i][j] = alpha_one;
          biga1[i][j] = biga1_one;
          beta[i][j] = beta_one;
          r0[i][j] = r0_one;
          cut[i][j] = cut_one;
          setflag[i][j] = 1;
          count++;
        }
      }

      if (count == 0) error->all(FLERR, "Incorrect args for pair coefficients");
    }

## Initialization

The `init_one()` function is called during the “init” phase of a simulation. This is where potential parameters are checked for completeness, derived parameters computed (e.g. the “offset” of the potential energy at the cutoff distance for use with the pair_modify shift yes command). If a pair style supports generating “mixed” parameters (i.e. where both atoms of a pair have a different atom type) using a “mixing rule” from the parameters of the type with itself, this is the place to compute and store those mixed values. The born/gauss pair style does not support mixing, so we only check for completeness. Another purpose of the `init_one()` function is to symmetrize the potential parameter arrays. The return value of the function is the cutoff for the given pair of atom types. This information is used by the neighbor list code to determine the largest cutoff and then build the neighbor lists accordingly.

    /* ----------------------------------------------------------------------
       init for one type pair i,j and corresponding j,i
    ------------------------------------------------------------------------- */

    double PairBornGauss::init_one(int i, int j)
    {
      if (setflag[i][j] == 0) error->all(FLERR, "All pair coeffs are not set");

      if (offset_flag) {
        double dr = cut[i][j] - r0[i][j];
        offset[i][j] =
            biga0[i][j] * exp(-alpha[i][j] * cut[i][j]) - biga1[i][j] * exp(-beta[i][j] * dr * dr);
      } else
        offset[i][j] = 0.0;

      biga0[j][i] = biga0[i][j];
      alpha[j][i] = alpha[i][j];
      biga1[j][i] = biga1[i][j];
      beta[j][i] = beta[i][j];
      r0[j][i] = r0[i][j];
      offset[j][i] = offset[i][j];

      return cut[i][j];
    }

## Computing forces from the neighbor list (required)

The `compute()` function is the “workhorse” of a pair style. This is where we have the nested loops over all pairs of particles from the neighbor list to compute forces and - if needed - energies and virials.

The first part is to define some variables for later use and store cached copies of data or pointers that we need to access frequently. Also, this is a good place to call `Pair::ev_init()`, which initializes several flags derived from the eflag and vflag parameters signaling whether the energy and virial need to be tallied and whether only globally or also per-atom.

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

    void PairBornGauss::compute(int eflag, int vflag)
    {
      int i, j, ii, jj, inum, jnum, itype, jtype;
      double xtmp, ytmp, ztmp, delx, dely, delz, evdwl, fpair;
      double rsq, r, dr, aexp, bexp, factor_lj;
      int *ilist, *jlist, *numneigh, **firstneigh;

      evdwl = 0.0;
      ev_init(eflag, vflag);

      double **x = atom->x;
      double **f = atom->f;
      int *type = atom->type;
      int nlocal = atom->nlocal;
      double *special_lj = force->special_lj;
      int newton_pair = force->newton_pair;

      inum = list->inum;
      ilist = list->ilist;
      numneigh = list->numneigh;
      firstneigh = list->firstneigh;

The outer loop (index i) is over local atoms of our sub-domain. Typically, the value of inum (the number of neighbor lists) is the same as the number of local atoms (= atoms owned by this sub-domain). But when the pair style is used as a sub-style of a hybrid pair style or neighbor list entries are removed with neigh_modify exclude, this number may be smaller. The array `list->ilist` has the (local) indices of the atoms for which neighbor lists have been created. Then `list->numneigh` is an inum sized array with the number of entries of each list of neighbors, and `list->firstneigh` is a list of pointers to those lists.

For efficiency reasons, cached copies of some properties of the outer loop atoms are also initialized.

    // loop over neighbors of my atoms

    for (ii = 0; ii < inum; ii++) {
      i = ilist[ii];
      xtmp = x[i][0];
      ytmp = x[i][1];
      ztmp = x[i][2];
      itype = type[i];
      jlist = firstneigh[i];
      jnum = numneigh[i];

The inner loop (index j) processes the neighbor lists. The neighbor list code encodes in the upper 2 bits whether a pair is a regular pair of neighbor (= 0) or a pair of 1-2 (= 1), 1-3 (= 2), or 1-4 (= 3) “special” neighbor. The `sbmask()` inline function extracts those bits and converts them into a number. This number is used to look up the corresponding scaling factor for the non-bonded interaction from the `force->special_lj` array and stores it in the factor_lj variable. Due to the additional bits, the value of j would be out of range when accessing data from per-atom arrays, so we apply the NEIGHMASK constant with a bit-wise and operation to mask them out. This step must be done, even if a pair style does not use special bond scaling of forces and energies to avoid segmentation faults.

With the corrected j index, it is now possible to compute the distance of the pair. For efficiency reasons, the square root is only taken after the check for the cutoff (which has been stored as squared cutoff by the `Pair` base class). For some pair styles, like the 12-6 Lennard-Jones potential, computing the square root can be avoided entirely.

    for (jj = 0; jj < jnum; jj++) {
      j = jlist[jj];
      factor_lj = special_lj[sbmask(j)];
      j &= NEIGHMASK;

      delx = xtmp - x[j][0];
      dely = ytmp - x[j][1];
      delz = ztmp - x[j][2];
      rsq = delx * delx + dely * dely + delz * delz;
      jtype = type[j];

The following block of code is the actual application of the model potential to compute the force. Note, that fpair is the pair-wise force divided by the distance, as this simplifies the projection of the x-, y-, and z-components of the force vector by simply multiplying with the respective distances in those directions.

    if (rsq < cutsq[itype][jtype]) {
      r = sqrt(rsq);
      dr = r - r0[itype][jtype];
      aexp = biga0[itype][jtype] * exp(-alpha[itype][jtype] * r);
      bexp = biga1[itype][jtype] * exp(-beta[itype][jtype] * dr * dr);
      fpair = alpha[itype][jtype] * aexp;
      fpair -= 2.0 * beta[itype][jtype] * dr * bexp;
      fpair *= factor_lj / r;

In the next block, the force is added to the per-atom force arrays. This pair style uses a “half” neighbor list (each pair is listed only once) so we take advantage of the fact that \(\vec{F}_{ij} = -\vec{F}_{ji}\), i.e. apply Newton’s third law. The force is always stored when the atom is a “local” atom. Index i atoms are always “local” (i.e. i < nlocal); index j atoms may be “ghost” atoms (j >= nlocal).

Depending on the settings used with the newton command, those pairs are only listed once globally (newton_pair == 1), then forces must be stored even with ghost atoms and after all forces are computed a “reverse communication” is performed to add those ghost atom forces to their corresponding local atoms. If the setting is disabled, then the extra communication is skipped, since for pairs straddling sub-domain boundaries, the forces are computed twice and only stored with the local atoms in the domain that owns it.

    f[i][0] += delx * fpair;
    f[i][1] += dely * fpair;
    f[i][2] += delz * fpair;
    if (newton_pair || j < nlocal) {
      f[j][0] -= delx * fpair;
      f[j][1] -= dely * fpair;
      f[j][2] -= delz * fpair;
    }

The `ev_tally()` function tallies global or per-atom energy and virial. For typical MD simulations, the potential energy is merely a diagnostic and only needed on output. Similarly, the pressure may only be computed for (infrequent) thermodynamic output. For all timesteps where this information is not needed either, eflag or evflag are zero and the computation and call to the tally function skipped. Note that evdwl is initialized to zero at the beginning of the function, so that it still is valid to access it, even if the energy is not computed (e.g. when only the virial is needed).

          if (eflag) evdwl = factor_lj * (aexp - bexp - offset[itype][jtype]);
          if (evflag) ev_tally(i, j, nlocal, newton_pair, evdwl, 0.0, fpair, delx, dely, delz);
        }
      }
    }

If only the global virial is needed and no energy, then calls to `ev_tally()` can be avoided altogether, and the global virial can be computed more efficiently from the dot product of the total per-atom force vector and the position vector of the corresponding atom, \(\vec{F}\cdot\vec{r}\). This has to be done after all pair-wise forces are computed and before the reverse communication to collect data from ghost atoms, since the position has to be the position that was used to compute the force, i.e. not the “local” position if that ghost atom is a periodic copy.

      if (vflag_fdotr) virial_fdotr_compute();
    }

## Computing force and energy for a single pair

Certain features in LAMMPS only require computing interactions between individual pairs of atoms and the (optional) `single()` function is needed to support those features (e.g. for tabulation of force and energy with pair_write). This is a repetition of the force kernel in the `compute()` function, but only for a single pair of atoms, where the (squared) distance is provided as a parameter (so it may not even be an existing distance between two specific atoms). The energy is returned as the return value of the function and the force as the fforce reference. Note, that this is, similar to how fpair is used in the `compute()` function, the magnitude of the force along the vector between the two atoms divided by the distance.

The `single()` function is optional, but it is expected to be implemented for any true pair-wise additive potential. Many-body potentials and special case potentials do not implement it. In a few special cases (EAM, long-range Coulomb), the `single()` function implements the pairwise additive part of the complete force interaction and depends on either pre-computed properties (derivative of embedding term for EAM) or post-computed non-pair-wise force contributions (KSpace style in case of long-range Coulomb).

The member variable single_enable should be set to 0 in the constructor, if it is not implemented (its default value is 1).

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

    double PairBornGauss::single(int /*i*/, int /*j*/, int itype, int jtype, double rsq,
                                 double /*factor_coul*/, double factor_lj, double &fforce)
    {
      double r, dr, aexp, bexp;

      r = sqrt(rsq);
      dr = r - r0[itype][jtype];
      aexp = biga0[itype][jtype] * exp(-alpha[itype][jtype] * r);
      bexp = biga1[itype][jtype] * exp(-beta[itype][jtype] * dr * dr);

      fforce = factor_lj * (alpha[itype][jtype] * aexp - 2.0 * dr * beta[itype][jtype] * bexp) / r;
      return factor_lj * (aexp - bexp - offset[itype][jtype]);
    }

## Reading and writing of restart files

Support for writing and reading binary restart files is provided by the following four functions. Writing is only done by MPI processor rank 0. The output of global (not related to atom types) settings is usually delegated to the `write_restart_settings()` function. This restart facility is commonly only used, if there are small number of per-type parameters. For potentials that use per-element parameters or tabulated data and read these from files, those parameters and the name of the potential file are not written to restart files and the pair_coeff command has to re-issued when restarting. For pair styles like “born/gauss” that do support writing to restart files, this is not required.

Implementing the functions to read and write binary restart files is optional. The member variable restartinfo should be set to 0 in the constructor, if they are not implemented (its default value is 1).

    /* ----------------------------------------------------------------------
       proc 0 writes to restart file
    ------------------------------------------------------------------------- */

    void PairBornGauss::write_restart(FILE *fp)
    {
      write_restart_settings(fp);

      int i, j;
      for (i = 1; i <= atom->ntypes; i++) {
        for (j = i; j <= atom->ntypes; j++) {
          fwrite(&setflag[i][j], sizeof(int), 1, fp);
          if (setflag[i][j]) {
            fwrite(&biga0[i][j], sizeof(double), 1, fp);
            fwrite(&alpha[i][j], sizeof(double), 1, fp);
            fwrite(&biga1[i][j], sizeof(double), 1, fp);
            fwrite(&beta[i][j], sizeof(double), 1, fp);
            fwrite(&r0[i][j], sizeof(double), 1, fp);
            fwrite(&cut[i][j], sizeof(double), 1, fp);
          }
        }
      }
    }

    /* ----------------------------------------------------------------------
       proc 0 writes to restart file
    ------------------------------------------------------------------------- */

    void PairBornGauss::write_restart_settings(FILE *fp)
    {
      fwrite(&cut_global, sizeof(double), 1, fp);
      fwrite(&offset_flag, sizeof(int), 1, fp);
      fwrite(&mix_flag, sizeof(int), 1, fp);
    }

Similarly, on reading, only MPI processor rank 0 has opened the restart file and will read the data. The data is then distributed across all parallel processes using calls to `MPI_Bcast()`. Before reading atom type specific data, the corresponding storage needs to be allocated. Order and number or storage size of items read must be exactly the same as when writing, or else the data will be read incorrectly.

Reading uses the `utils::sfread` utility function to detect read errors and short reads, so that LAMMPS can abort if that happens, e.g. when the restart file is corrupted.

    /* ----------------------------------------------------------------------
       proc 0 reads from restart file, bcasts
    ------------------------------------------------------------------------- */

    void PairBornGauss::read_restart(FILE *fp)
    {
      read_restart_settings(fp);

      allocate();

      int i, j;
      int me = comm->me;
      for (i = 1; i <= atom->ntypes; i++) {
        for (j = i; j <= atom->ntypes; j++) {
          if (me == 0) utils::sfread(FLERR, &setflag[i][j], sizeof(int), 1, fp, nullptr, error);
          MPI_Bcast(&setflag[i][j], 1, MPI_INT, 0, world);
          if (setflag[i][j]) {
            if (me == 0) {
              utils::sfread(FLERR, &biga0[i][j], sizeof(double), 1, fp, nullptr, error);
              utils::sfread(FLERR, &alpha[i][j], sizeof(double), 1, fp, nullptr, error);
              utils::sfread(FLERR, &biga1[i][j], sizeof(double), 1, fp, nullptr, error);
              utils::sfread(FLERR, &beta[i][j], sizeof(double), 1, fp, nullptr, error);
              utils::sfread(FLERR, &r0[i][j], sizeof(double), 1, fp, nullptr, error);
              utils::sfread(FLERR, &cut[i][j], sizeof(double), 1, fp, nullptr, error);
            }
            MPI_Bcast(&biga0[i][j], 1, MPI_DOUBLE, 0, world);
            MPI_Bcast(&alpha[i][j], 1, MPI_DOUBLE, 0, world);
            MPI_Bcast(&biga1[i][j], 1, MPI_DOUBLE, 0, world);
            MPI_Bcast(&beta[i][j], 1, MPI_DOUBLE, 0, world);
            MPI_Bcast(&r0[i][j], 1, MPI_DOUBLE, 0, world);
            MPI_Bcast(&cut[i][j], 1, MPI_DOUBLE, 0, world);
          }
        }
      }
    }

    /* ----------------------------------------------------------------------
       proc 0 reads from restart file, bcasts
    ------------------------------------------------------------------------- */

    void PairBornGauss::read_restart_settings(FILE *fp)
    {
      if (comm->me == 0) {
        utils::sfread(FLERR, &cut_global, sizeof(double), 1, fp, nullptr, error);
        utils::sfread(FLERR, &offset_flag, sizeof(int), 1, fp, nullptr, error);
        utils::sfread(FLERR, &mix_flag, sizeof(int), 1, fp, nullptr, error);
      }
      MPI_Bcast(&cut_global, 1, MPI_DOUBLE, 0, world);
      MPI_Bcast(&offset_flag, 1, MPI_INT, 0, world);
      MPI_Bcast(&mix_flag, 1, MPI_INT, 0, world);
    }

## Writing coefficients to data files

The `write_data()` and `write_data_all()` functions are optional and write out the current state of the pair_coeff settings as “Pair Coeffs” or “PairIJ Coeffs” sections to a data file when using the write_data command. The `write_data()` only writes out the diagonal elements of the pair coefficient matrix, as that is required for the format of the “Pair Coeffs” section. It is called when the “pair” option of the write_data command is “ii” (the default). This is suitable for force fields where all off-diagonal terms of the pair coefficient matrix are generated from mixing. If explicit settings for off-diagonal elements were made, LAMMPS will print a warning, as those would be lost. To avoid this, the “pair ij” option of write_data can be used which will trigger calling the `write_data_all()` function instead, which will write out all settings of the pair coefficient matrix (regardless of whether they were originally created from mixing or not).

These data file output functions are only useful for true pair-wise additive potentials, where the potential parameters can be entered through multiple pair_coeff commands. Pair styles that require a single “pair_coeff * *” command line are not compatible with reading their parameters from data files. For pair styles like born/gauss that do support writing to data files, the potential parameters will be read from the data file, if present and pair_coeff commands may not be needed.

The member variable `writedata` should be set to 1 in the constructor, if these functions are implemented (the default value is 0).

    /* ----------------------------------------------------------------------
       proc 0 writes to data file
    ------------------------------------------------------------------------- */

    void PairBornGauss::write_data(FILE *fp)
    {
      for (int i = 1; i <= atom->ntypes; i++)
        fprintf(fp, "%d %g %g %g %g %g\n", i, biga0[i][i], alpha[i][i], biga1[i][i], beta[i][i],
                r0[i][i]);
    }

    /* ----------------------------------------------------------------------
       proc 0 writes all pairs to data file
    ------------------------------------------------------------------------- */

    void PairBornGauss::write_data_all(FILE *fp)
    {
      for (int i = 1; i <= atom->ntypes; i++)
        for (int j = i; j <= atom->ntypes; j++)
          fprintf(fp, "%d %d %g %g %g %g %g %g\n", i, j, biga0[i][j], alpha[i][j], biga1[i][j],
                  beta[i][j], r0[i][j], cut[i][j]);
    }

## Give access to internal data

The purpose of the `extract()` function is to facilitate access to internal data of the pair style by other parts of LAMMPS. One possible application is to use fix adapt to gradually change potential parameters during a run. Here, we implement access to the pair coefficient matrix parameters.

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

    void *PairBornGauss::extract(const char *str, int &dim)
    {
      dim = 2;
      if (strcmp(str, "biga0") == 0) return (void *) biga0;
      if (strcmp(str, "biga1") == 0) return (void *) biga1;
      if (strcmp(str, "r0") == 0) return (void *) r0;
      return nullptr;
    }

Since the mercury potential, for which we have implemented the born/gauss pair style, has a temperature dependent parameter “biga1”, we can automatically adapt the potential based on the Taylor-MacLaurin expansion for “biga1” when performing a simulation with a temperature ramp. LAMMPS commands for that application are given below:

    variable tlo  index 300.0
    variable thi  index 600.0
    variable temp equal ramp(v_tlo,v_thi)
    variable biga1 equal (-2.58717e-8*v_temp+8.40841e-5)*v_temp+1.97475e-2

    fix             1 all nvt temp ${tlo} ${thi} 0.1
    fix             2 all adapt 1 pair born/gauss biga1 * * v_biga1

# 4.8.3. Case 2: a many-body potential

Since there is a detailed description of the purpose and general layout of a pair style in the previous case, we will focus on where the implementation of a typical many-body potential differs from a pair-wise additive potential. We will use the implementation of the Tersoff potential as pair_style tersoff as an example. The complete implementation can be found in the files `src/MANYBODY/pair_tersoff.cpp` and `src/MANYBODY/pair_tersoff.h` of the LAMMPS source code.

## Constructor

In the constructor, several pair style flags must be set differently for many-body potentials:

- the potential is not pair-wise additive, so the `single()` function cannot be used. This is indicated by setting the single_enable member variable to 0 (default value is 1)

- many-body potentials are usually not written to binary restart files. This is indicated by setting the member variable restartinfo to 0 (default is 1)

- many-body potentials typically read all parameters from a file which stores parameters indexed with a string (e.g. the element). For this, only a single pair_coeff * * command is allowed. This requirement is set and checked for, when the member variable one_coeff is set to 1 (default value is 0)

- many-body potentials can produce incorrect results if pairs of atoms are excluded from the neighbor list, e.g. explicitly by neigh_modify exclude or implicitly through defining bonds, angles, etc. and having a special_bonds setting that is not “special_bonds lj/coul 1.0 1.0 1.0”. LAMMPS will check for this and print a suitable warning, when the member variable manybody_flag is set to 1 (default value is 0).

    PairTersoff::PairTersoff(LAMMPS *lmp) : Pair(lmp)
    {
      single_enable = 0;
      restartinfo = 0;
      one_coeff = 1;
      manybody_flag = 1;

## Neighbor list request

For computing the three-body interactions of the Tersoff potential a “full” neighbor list (both atoms of a pair are listed in each other’s neighbor list) is required. By default a “half” neighbor list is requested (each pair is listed only once). The request is made in the `init_style()` function. A more in-depth discussion of neighbor lists in LAMMPS and how to request them is in this section of the documentation

Also, additional conditions must be met for some global settings which are checked in the `init_style()` function.

    /* ----------------------------------------------------------------------
       init specific to this pair style
    ------------------------------------------------------------------------- */

    void PairTersoff::init_style()
    {
      if (atom->tag_enable == 0)
        error->all(FLERR,"Pair style Tersoff requires atom IDs");
      if (force->newton_pair == 0)
        error->all(FLERR,"Pair style Tersoff requires newton pair on");

      // need a full neighbor list

      neighbor->add_request(this,NeighConst::REQ_FULL);
    }

## Computing forces from the neighbor list

Computing forces for a many-body potential is usually more complex than for a pair-wise additive potential and there are multiple components. For Tersoff, there is a pair-wise additive two-body term (two nested loops over indices i and j) and a three-body term (three nested loops over indices i, j, and k). Since the neighbor list has all neighbors up to the maximum cutoff (for the two-body term), but the three-body interactions have a significantly shorter cutoff, a “short neighbor list” is also constructed at the same time while computing the two-body term and looping over the neighbor list for the first time.

    if (rsq < cutshortsq) {
      neighshort[numshort++] = j;
      if (numshort >= maxshort) {
        maxshort += maxshort/2;
        memory->grow(neighshort,maxshort,"pair:neighshort");
      }
    }

For the two-body term, only a half neighbor list would be needed, even though we have requested a full list (for the three-body loops). Rather than computing all interactions twice, we skip over half of the entries. This is done in a slightly complex way to make certain the same choice is made across all subdomains and so that there is no load imbalance introduced.

    jtag = tag[j];
    if (itag > jtag) {
      if ((itag+jtag) % 2 == 0) continue;
    } else if (itag < jtag) {
      if ((itag+jtag) % 2 == 1) continue;
    } else {
      if (x[j][2] < x[i][2]) continue;
      if (x[j][2] == ztmp && x[j][1] < ytmp) continue;
      if (x[j][2] == ztmp && x[j][1] == ytmp && x[j][0] < xtmp) continue;
    }

For the three-body term, there is one additional nested loop and it uses the “short” neighbor list, accumulated previously.

    // three-body interactions
    // skip immediately if I-J is not within cutoff
    double fjxtmp,fjytmp,fjztmp;

    for (jj = 0; jj < numshort; jj++) {
      j = neighshort[jj];
      jtype = map[type[j]];

      [...]

      for (kk = 0; kk < numshort; kk++) {
        if (jj == kk) continue;
        k = neighshort[kk];
        ktype = map[type[k]];

        [...]
      }
    [...]

## Reading potential parameters

For the Tersoff potential, the parameters are listed in a file and associated with triples of elements. Because we have set the `one_coeff` flag to 1 in the constructor, there may only be a single pair_coeff * * line in the input for this pair style, and as a consequence the `coeff()` function will only be called once. Thus, the `coeff()` function has to do three tasks, each of which is delegated to a function in the `PairTersoff` class:

1. map elements to atom types. Those follow the potential file name in the command line arguments and are processed by the `map_element2type()` function.

2. read and parse the potential parameter file in the `read_file()` function.

3. Build data structures where the original and derived parameters are indexed by all possible triples of atom types and thus can be looked up quickly in the loops for the force computation

    void PairTersoff::coeff(int narg, char **arg)
    {
      if (!allocated) allocate();

      map_element2type(narg-3,arg+3);

      // read potential file and initialize potential parameters

      read_file(arg[2]);
      setup_params();
    }

# 4.8.4. Case 3: a potential requiring communication

For some models, the interactions between atoms depends on properties of their environment which have to be computed before the the forces can be computed. Since LAMMPS is designed to run in parallel using a domain decomposition strategy, not all information of the atoms may be directly available and thus communication steps may be need to collect data from ghost atoms of neighboring subdomains or send data to ghost atoms for application during the pairwise computation.

Specifically, two communication patterns are needed: a “reverse communication” and a “forward communication”. The reverse communication collects data added to “ghost” atoms from neighboring sub-domains and sums it to their corresponding “local” atoms. This communication is only required and thus executed when the `Force::newton_pair` setting is 1 (i.e. newton on, the default). The forward communication is used to copy computed per-atom data from “local” atoms to their corresponding “ghost” atoms in neighboring sub-domains.

For this we will look at how the embedding term of the embedded atom potential EAM is implemented in LAMMPS. The complete implementation of this pair style can be found in the files `src/MANYBODY/pair_eam.cpp` and `src/MANYBODY/pair_eam.h` of the LAMMPS source code.

## Allocating additional per-atom storage

First suitable (local) per-atom arrays (rho, fp, numforce) are allocated. These have to be large enough to include ghost atoms, are not used outside the `compute()` function and are re-initialized to zero once per timestep.

    if (atom->nmax > nmax) {
      memory->destroy(rho);
      memory->destroy(fp);
      memory->destroy(numforce);
      nmax = atom->nmax;
      memory->create(rho,nmax,"pair:rho");
      memory->create(fp,nmax,"pair:fp");
      memory->create(numforce,nmax,"pair:numforce");
    }

## Reverse communication

Then a first loop over all pairs (i and j) is performed, where data is stored in the rho array representing the electron density at the site of i contributed from all neighbors j. Since the EAM pair style uses a half neighbor list (for efficiency reasons), a reverse communication is needed to collect the contributions to rho from ghost atoms (only if newton on is set for pair styles).

    if (newton_pair) comm->reverse_comm(this);

To support the reverse communication, two functions must be defined: `pack_reverse_comm()` that copies relevant data into a buffer for ghost atoms and `unpack_reverse_comm()` that takes the collected data and adds it to the rho array for the corresponding local atoms that match the ghost atoms. In order to allocate sufficiently sized buffers, a flag must be set in the pair style constructor. Since in this case a single double precision number is communicated per atom, the comm_reverse member variable is set to 1 (default is 0 = no reverse communication).

    int PairEAM::pack_reverse_comm(int n, int first, double *buf)
    {
      int i,m,last;

      m = 0;
      last = first + n;
      for (i = first; i < last; i++) buf[m++] = rho[i];
      return m;
    }

    void PairEAM::unpack_reverse_comm(int n, int *list, double *buf)
    {
      int i,j,m;

      m = 0;
      for (i = 0; i < n; i++) {
        j = list[i];
        rho[j] += buf[m++];
      }
    }

## Forward communication

From the density array rho, the derivative of the embedding energy fp is computed. The computation is only done for “local” atoms, but for the force computation, that property also is needed on ghost atoms. For that a forward communication is needed.

    comm->forward_comm(this);

Similar to the reverse communication, this requires implementing a `pack_forward_comm()` and an `unpack_forward_comm()` function. Since there is one double precision number per atom that needs to be communicated, we must set the comm_forward member variable to 1 (default is 0 = no forward communication).

    int PairEAM::pack_forward_comm(int n, int *list, double *buf, int pbc_flag, int *pbc)
    {
      int i,j,m;

      m = 0;
      for (i = 0; i < n; i++) {
        j = list[i];
        buf[m++] = fp[j];
      }
      return m;
    }

    void PairEAM::unpack_forward_comm(int n, int first, double *buf)
    {
      int i,m,last;

      m = 0;
      last = first + n;
      for (i = first; i < last; i++) fp[i] = buf[m++];
    }

# 4.8.5. Case 4: potentials without a compute() function

A small number of pair style classes do not implement a `compute()` function, but instead use that of a different pair style.

## Embedded atom variants “eam/fs” and “eam/alloy”

The pair styles eam/fs and eam/alloy share the same model and potential function as the eam pair style. They differ in the format of the potential files. Pair style eam supports only potential files for single elements. For multi-element systems, the mixed terms are computed from mixed parameters. The eam/fs and eam/alloy pair styles, however, require the use of a single potential file for all elements where the mixed element potential is included in the tabulation. That enables more accurate models for alloys, since the mixed terms can be adjusted for a better representation of material properties compared to terms created from mixing of per-element terms in the `PairEAM` class.

We take a closer at the eam/alloy pair style. The complete implementation is in the files `src/MANYBODY/pair_eam_alloy.cpp` and `src/MANYBODY/pair_eam_alloy.h`.

The `PairEAMAlloy` class is derived from `PairEAM` and not `Pair` and overrides only a small number of functions:

    class PairEAMAlloy : virtual public PairEAM {
     public:
      PairEAMAlloy(class LAMMPS *);
      void coeff(int, char **) override;

     protected:
      void read_file(char *) override;
      void file2array() override;
    };

All other functionality is inherited from the base classes. In the constructor we set the `one_coeff` flag and the `many_body` flag to 1 to indicate the different behavior.

    PairEAMAlloy::PairEAMAlloy(LAMMPS *lmp) : PairEAM(lmp)
    {
      one_coeff = 1;
      manybody_flag = 1;
    }

The `coeff()` function (not shown here) implements the different behavior when processing the pair_coeff command. The `read_file()` and `file2array()` replace the corresponding `PairEAM` class functions to accommodate the different data and file format.

## AIREBO and AIREBO-M potentials

The AIREBO-M potential differs from the better known AIREBO potential in that it use a Morse potential instead of a Lennard-Jones potential for non-bonded interactions. Since this difference is very minimal compared to the entire potential, both potentials are implemented in the `PairAIREBO` class and which non-bonded potential is used is determined by the value of the `morseflag` flag, which would be set to either 0 or 1.

    class PairAIREBOMorse : public PairAIREBO {
     public:
      PairAIREBOMorse(class LAMMPS *);
      void settings(int, char **) override;
    };

The `morseflag` variable defaults to 0 and is set to 1 in the `PairAIREBOMorse::settings()` function which is called by the pair_style command. This function delegates all command line processing and setting of other parameters to the `PairAIREBO::settings()` function of the base class.

    void PairAIREBOMorse::settings(int narg, char **arg)
    {
      PairAIREBO::settings(narg, arg);

      morseflag = 1;
    }

The complete implementation is in the files `src/MANYBODY/pair_airebo.cpp`, `src/MANYBODY/pair_airebo.h`, `src/MANYBODY/pair_airebo_morse.cpp`, `src/MANYBODY/pair_airebo_morse.h`.

(Bomont) Bomont, Bretonnet, J. Chem. Phys. 124, 054504 (2006)