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====== Stillinger-Weber Potentials ======

----

The Stillinger-Weber potential((F. H. Stillinger and T. A. Weber, //Phys. Rev. B// **31**, 5262 (1985) )) 
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'').