\(\renewcommand{\AA}{\text{Å}}\)

# pair_style sw command

Accelerator Variants: sw/gpu, sw/intel, sw/kk, sw/omp

# pair_style sw/mod command

Accelerator Variants: sw/mod/omp

## Syntax

    pair_style style keyword values

- style = sw or sw/mod

- keyword = maxdelcs or threebody

      maxdelcs value = delta1 delta2 (optional, sw/mod only)
        delta1 = The minimum thershold for the variation of cosine of three-body angle
        delta2 = The maximum threshold for the variation of cosine of three-body angle
      threebody value = on or off (optional, sw only)
        on (default) = Compute both the three-body and two-body terms of the potential
        off = Compute only the two-body term of the potential

## Examples

    pair_style sw
    pair_coeff * * si.sw Si
    pair_coeff * * GaN.sw Ga N Ga

    pair_style sw/mod maxdelcs 0.25 0.35
    pair_coeff * * tmd.sw.mod Mo S S

    pair_style hybrid sw threebody on sw threebody off
    pair_coeff * * sw 1 mW_xL.sw mW NULL
    pair_coeff 1 2 sw 2 mW_xL.sw mW xL
    pair_coeff 2 2 sw 2 mW_xL.sw mW xL

## Description

The sw style computes a 3-body Stillinger-Weber potential for the energy E of a system of atoms as

\[\begin{split} E & = \sum_i \sum_{j > i} \phi_2 (r_{ij}) + \sum_i \sum_{j \neq i} \sum_{k > j} \phi_3 (r_{ij}, r_{ik}, \theta_{ijk}) \\ \phi_2(r_{ij}) & = A_{ij} \epsilon_{ij} \left[ B_{ij} (\frac{\sigma_{ij}}{r_{ij}})^{p_{ij}} - (\frac{\sigma_{ij}}{r_{ij}})^{q_{ij}} \right] \exp \left( \frac{\sigma_{ij}}{r_{ij} - a_{ij} \sigma_{ij}} \right) \\ \phi_3(r_{ij},r_{ik},\theta_{ijk}) & = \lambda_{ijk} \epsilon_{ijk} \left[ \cos \theta_{ijk} - \cos \theta_{0ijk} \right]^2 \exp \left( \frac{\gamma_{ij} \sigma_{ij}}{r_{ij} - a_{ij} \sigma_{ij}} \right) \exp \left( \frac{\gamma_{ik} \sigma_{ik}}{r_{ik} - a_{ik} \sigma_{ik}} \right)\end{split}\]

where \(\phi_2\) is a two-body term and \(\phi_3\) is a three-body term. The summations in the formula are over all neighbors J and K of atom I within a cutoff distance \(a `\sigma\).

Added in version 14Dec2021.

The sw/mod style is designed for simulations of materials when distinguishing three-body angles are necessary, such as borophene and transition metal dichalcogenides, which cannot be described by the original code for the Stillinger-Weber potential. For instance, there are several types of angles around each Mo atom in MoS_2, and some unnecessary angle types should be excluded in the three-body interaction. Such exclusion may be realized by selecting proper angle types directly. The exclusion of unnecessary angles is achieved here by the cut-off function (f_C(delta)), which induces only minimum modifications for LAMMPS.

Validation, benchmark tests, and applications of the sw/mod style can be found in (Jiang2) and (Jiang3).

The sw/mod style computes the energy E of a system of atoms, whose potential function is mostly the same as the Stillinger-Weber potential. The only modification is in the three-body term, where the value of \(\delta = \cos \theta_{ijk} - \cos \theta_{0ijk}\) used in the original energy and force expression is scaled by a switching factor \(f_C(\delta)\):

\[\begin{split}f_C(\delta) & = \left\{ \begin{array} {r@{\quad:\quad}l} 1 & \left| \delta \right| < \delta_1 \\ \frac{1}{2} + \frac{1}{2} \cos \left( \pi \frac{\left| \delta \right| - \delta_1}{\delta_2 - \delta_1} \right) & \delta_1 < \left| \delta \right| < \delta_2 \\ 0 & \left| \delta \right| > \delta_2 \end{array} \right. \\\end{split}\]

This cut-off function decreases smoothly from 1 to 0 over the range \([\delta_1, \delta_2]\). This smoothly turns off the energy and force contributions for \(\left| \delta \right| > \delta_2\). It is suggested that \(\delta 1\) and \(\delta_2\) to be the value around \(0.5 \left| \cos \theta_1 - \cos \theta_2 \right|\), with \(\theta_1\) and \(\theta_2\) as the different types of angles around an atom. For borophene and transition metal dichalcogenides, \(\delta_1 = 0.25\) and \(\delta_2 = 0.35\). This value enables the cut-off function to exclude unnecessary angles in the three-body SW terms.

Note

The cut-off function is just to be used as a technique to exclude some unnecessary angles, and it has no physical meaning. It should be noted that the force and potential are inconsistent with each other in the decaying range of the cut-off function, as the angle dependence for the cut-off function is not implemented in the force (first derivation of potential). However, the angle variation is much smaller than the given threshold value for actual simulations, so the inconsistency between potential and force can be neglected in actual simulations.

Added in version 3Aug2022.

The threebody keyword is optional and determines whether or not the three-body term of the potential is calculated. The default value is “on” and it is only available for the plain sw pair style variants, but not available for the sw/mod and sw/angle/table pair style variants. To turn off the threebody contributions all \(\lambda_{ijk}\) parameters from the potential file are forcibly set to 0. In addition the pair style implementation may employ code optimizations for the threebody off setting that can result in significant speedups versus the default. These code optimizations are currently only available for the MANYBODY and OPENMP packages.

Only a single pair_coeff command is used with the sw and sw/mod styles which specifies a Stillinger-Weber potential file with parameters for all needed elements, except for when the threebody off setting is used (see note below). These are mapped to LAMMPS atom types by specifying N additional arguments after the filename in the pair_coeff command, where N is the number of LAMMPS atom types:

- filename

- N element names = mapping of SW elements to atom types

See the pair_coeff page for alternate ways to specify the path for the potential file.

As an example, imagine a file SiC.sw has Stillinger-Weber values for Si and C. If your LAMMPS simulation has 4 atoms types and you want the first 3 to be Si, and the fourth to be C, you would use the following pair_coeff command:

    pair_style sw
    pair_coeff * * SiC.sw Si Si Si C

The first 2 arguments must be * * so as to span all LAMMPS atom types. The first three Si arguments map LAMMPS atom types 1, 2, and 3 to the Si element in the SW file. The final C argument maps LAMMPS atom type 4 to the C element in the SW file. If an argument value is specified as NULL, the mapping is not performed. This can be used when an sw potential is used as part of the hybrid pair style. The NULL values are placeholders for atom types that will be used with other potentials.

Note

When the threebody off keyword is used, multiple pair_coeff commands may be used to specific the pairs of atoms which don’t require three-body term. In these cases, the first 2 arguments are not required to be * *, the potential parameter file is only read by the first pair_coeff command and the element to atom type mappings must be consistent across all pair_coeff statements. If not LAMMPS will abort with an error.

Stillinger-Weber files in the potentials directory of the LAMMPS distribution have a “.sw” suffix. Lines that are not blank or comments (starting with #) define parameters for a triplet of elements. The parameters in a single entry correspond to the two-body and three-body coefficients in the formula above:

- element 1 (the center atom in a 3-body interaction)

- element 2

- element 3

- \(\epsilon\) (energy units)

- \(\sigma\) (distance units)

- a

- \(\lambda\)

- \(\gamma\)

- \(\cos\theta_0\)

- A

- B

- p

- q

- tol

The A, B, p, and q parameters are used only for two-body interactions. The \(\lambda\) and \(\cos\theta_0\) parameters are used only for three-body interactions. The \(\epsilon\), \(\sigma\) and a parameters are used for both two-body and three-body interactions. \(\gamma\) is used only in the three-body interactions, but is defined for pairs of atoms. The non-annotated parameters are unitless.

LAMMPS introduces an additional performance-optimization parameter tol that is used for both two-body and three-body interactions. In the Stillinger-Weber potential, the interaction energies become negligibly small at atomic separations substantially less than the theoretical cutoff distances. LAMMPS therefore defines a virtual cutoff distance based on a user defined tolerance tol. The use of the virtual cutoff distance in constructing atom neighbor lists can significantly reduce the neighbor list sizes and therefore the computational cost. LAMMPS provides a tol value for each of the three-body entries so that they can be separately controlled. If tol = 0.0, then the standard Stillinger-Weber cutoff is used.

The Stillinger-Weber potential file must contain entries for all the elements listed in the pair_coeff command. It can also contain entries for additional elements not being used in a particular simulation; LAMMPS ignores those entries.

For a single-element simulation, only a single entry is required (e.g. SiSiSi). For a two-element simulation, the file must contain 8 entries (for SiSiSi, SiSiC, SiCSi, SiCC, CSiSi, CSiC, CCSi, CCC), that specify SW parameters for all permutations of the two elements interacting in three-body configurations. Thus for 3 elements, 27 entries would be required, etc.

As annotated above, the first element in the entry is the center atom in a three-body interaction. Thus an entry for SiCC means a Si atom with 2 C atoms as neighbors. The parameter values used for the two-body interaction come from the entry where the second and third elements are the same. Thus the two-body parameters for Si interacting with C, comes from the SiCC entry. The three-body parameters can in principle be specific to the three elements of the configuration. In the literature, however, the three-body parameters are usually defined by simple formulas involving two sets of pairwise parameters, corresponding to the ij and ik pairs, where i is the center atom. The user must ensure that the correct combining rule is used to calculate the values of the three-body parameters for alloys. Note also that the function \(\phi_3\) contains two exponential screening factors with parameter values from the ij pair and ik pairs. So \(\phi_3\) for a C atom bonded to a Si atom and a second C atom will depend on the three-body parameters for the CSiC entry, and also on the two-body parameters for the CCC and CSiSi entries. Since the order of the two neighbors is arbitrary, the three-body parameters for entries CSiC and CCSi should be the same. Similarly, the two-body parameters for entries SiCC and CSiSi should also be the same. The parameters used only for two-body interactions (A, B, p, and q) in entries whose second and third element are different (e.g. SiCSi) are not used for anything and can be set to 0.0 if desired. This is also true for the parameters in \(\phi_3\) that are taken from the ij and ik pairs (\(\sigma\), a, \(\gamma\))

Styles with a gpu, intel, kk, omp, or opt suffix are functionally the same as the corresponding style without the suffix. They have been optimized to run faster, depending on your available hardware, as discussed on the Accelerator packages page. The accelerated styles take the same arguments and should produce the same results, except for round-off and precision issues.

These accelerated styles are part of the GPU, INTEL, KOKKOS, OPENMP, and OPT packages, respectively. They are only enabled if LAMMPS was built with those packages. See the Build package page for more info.

You can specify the accelerated styles explicitly in your input script by including their suffix, or you can use the -suffix command-line switch when you invoke LAMMPS, or you can use the suffix command in your input script.

See the Accelerator packages page for more instructions on how to use the accelerated styles effectively.

Note

When using the INTEL package with this style, there is an additional 5 to 10 percent performance improvement when the Stillinger-Weber parameters p and q are set to 4 and 0 respectively. These parameters are common for modeling silicon and water.

## Mixing, shift, table, tail correction, restart, rRESPA info

For atom type pairs I,J and I != J, where types I and J correspond to two different element types, mixing is performed by LAMMPS as described above from values in the potential file.

This pair style does not support the pair_modify shift, table, and tail options.

This pair style does not write its information to binary restart files, since it is stored in potential files. Thus, you need to re-specify the pair_style and pair_coeff commands in an input script that reads a restart file.

This pair style can only be used via the pair keyword of the run_style respa command. It does not support the inner, middle, outer keywords.

The single() function of the sw pair style is only enabled and supported for the case of the threebody off setting.

## Restrictions

This pair style is part of the MANYBODY package. It is only enabled if LAMMPS was built with that package. See the Build package page for more info.

This pair style requires the newton setting to be “on” for pair interactions.

The Stillinger-Weber potential files provided with LAMMPS (see the potentials directory) are parameterized for metal units. You can use the sw or sw/mod pair styles with any LAMMPS units, but you would need to create your own SW potential file with coefficients listed in the appropriate units if your simulation does not use “metal” units. If the potential file contains a ‘UNITS:’ metadata tag in the first line of the potential file, then LAMMPS can convert it transparently between “metal” and “real” units.

## Related commands

pair_coeff

## Default

The default value for the threebody setting of the “sw” pair style is “on”, the default values for the “maxdelcs setting of the sw/mod pair style are delta1 = 0.25 and delta2 = 0.35`.

(Stillinger) Stillinger and Weber, Phys Rev B, 31, 5262 (1985).

(Jiang2) J.-W. Jiang, Nanotechnology 26, 315706 (2015).

(Jiang3) J.-W. Jiang, Acta Mech. Solida. Sin 32, 17 (2019).