Description

This Model Driver implements the Stillinger-Weber three-body potential

Parameters

Symbols:

\[a, A, B, p, q, \sigma, \lambda, \gamma\]

Corresponding variables in code:
a, A, B,p, q, sigma, lamda, gamma

Details

The total potential energy function \(\phi\) for Si crystal is approximated by a combination of a pair and three-body potential \(v_2\) and \(v_3\) respectively. It takes the following form:

\[\phi = \sum_{i,j; i<j} v_2(i,j) + \sum_{i,j,k; i<j<k} v_3(i,j,k)\]

Where

\[v_2(r_{ij}) = \epsilon f_2(r_{ij}/\sigma)\] \[v_3(\mathbf{r}_i ,\mathbf{r}_j, \mathbf{r}_k) = \epsilon f_3(\mathbf{r}_i/\sigma, \mathbf{r}_j/\sigma, \mathbf{r}_k/\sigma)\]

The function \(f_2\) has the following form:

\[f_2(r) = \begin{cases} A(Br^{-p} - r^{-q})\exp[(r-a)^{-1}] & \text{if } r < a\\ 0 & \text{if }r >=a\\ \end{cases}\]

The function \(f_3\) takes the form:

\[f_3(\mathbf{r}_i, \mathbf{r}_j, \mathbf{r}_k) = h(r_{ij},r_{ik},\theta_{jik}) + h(r_{ji},r_{jk},\theta_{ijk}) + h(r_{ki},r_{kj},\theta_{ikj})\]

where

\[h(r_{ij},r_{ik},\theta_{jik}) = \lambda \exp[\gamma(r_{ij}-a)^{-1} + \gamma(r_{ik}-a)^{-1}](\cos\theta_{jik} + 1/3)^2\]

The term \((\cos\theta_{jik} + 1/3)^2\) is a penalty term which makes \(h\) vanish for the ideal tetrahedral angle i.e. \(\cos\theta = -1/3\).