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# Stillinger-Weber Potentials

The Stillinger-Weber potential1) is available when potfit is compiled with the stiweb flag, which also implies the apot flag for analytic potentials.

## Basic Theory

The total potential energy is defined as

$$E_{\text{total}}=\sum_{i<j}V_2(r_{ij}) + \sum_{\substack{i\neq j,k\\j<k}}V_3(r_{ij},r_{ik},r_{jk}),$$

where $V_2$ and $V_3$ are given by:

$$V_2(r_{ij}) = \left(A_{ij}r_{ij}^{-p_{ij}}-B_{ij}r_{ij}^{-q_{ij}}\right)\exp\left(\frac{\delta_{ij}}{r_{ij}-a_{ij}}\right)$$

$$V_3(r_{ij},r_{ik},r_{jk}) = \lambda_{ijk} \exp\left(\frac{\gamma_{ij}}{r_{ij}-b_{ij}} + \frac{\gamma_{ik}}{r_{ik}-b_{ik}}\right)\left(\cos\theta_{ijk}+\frac{1}{3}\right)^2.$$

$\theta_{ijk}$ is the angle formed by the atoms $i,j$ and $k$, with $i$ being the central atom.

### Analytic functions

To defined an analytic Stillinger-Weber potential in potfit, there are 3 functional forms required. They are available as the stiweb_2, stiweb_3 and stiweb_lambda function types. The stiweb_2 contains the parameters for the $V_2$ part, the stiweb_3 the pair parameters for the $V_3$ part and stiweb_lambda the $\lambda_{ijk}$ for all possible combinations. For more details please take a look at the examples.

## Number of potential functions

To describe a system with $N$ atom types you need $N^2+N+1$ potentials.

# atom types stiweb_2 stiweb_3 stiweb_lambda Total # potentials
$N$ $N(N+1)/2$ $N(N+1)/2$ 1 $N^2+N+1$
1 1 1 1 3
2 3 3 1 7
3 6 6 1 13
4 10 10 1 21

## Order of potential functions

The potential table is assumed to be symmetric, i.e. the potential for the atom types 1-0 is the same as the potential 0-1.

The order of the potentials in the potential file for $N$ atom types is:

$S2_{00}, \ldots, S2_{0N}, S2_{11}, \ldots, S2_{1N}, \ldots, S2_{NN}$
$S3_{00}, \ldots, S3_{0N}, S3_{11}, \ldots, S3_{1N}, \ldots, S3_{NN}$
$S\lambda$

where $S2$ stands for stiweb_2, $S3$ for stiweb_3 and $S\lambda$ for stiweb_lambda potentials.

### IMD output

An analytic IMD potential which can be used with the stiweb option of IMD is written to *.imd.sw.pot. This file, however, may not be used as a potential file for IMD. Instead, its contents need to be copied INTO the IMD parameter file.

### LAMMPS output

The potential can also be written in LAMMPS format. The name of the output file is *.lammps.sw. As LAMMPS uses a slightly different parametrization of the Stillinger-Weber potential, there is a gauge degree of freedom when creating the LAMMPS potential. The scaling factor for the energy is set to 1.0, this can be adjusted in the potfit source file potential_output.c (search for energy scaling factor).

1)
F. H. Stillinger and T. A. Weber, Phys. Rev. B 31, 5262 (1985)