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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:

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

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).

F. H. Stillinger and T. A. Weber, Phys. Rev. B 31, 5262 (1985)
interactions/stiweb.txt ยท Last modified: 2018/01/06 10:39 by daniel