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).