Hard core Yukawa

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The hard core Yukawa potential has two forms, the attractive Yukawa potential:

$\Phi_{12}\left( r \right) = \left\{ \begin{array}{lll} \infty & ; & r < \sigma \\ - \left( \frac{ \epsilon \sigma }{r}\right) \exp \left[- \kappa \left( \frac{r}{\sigma} - 1 \right) \right] & ; & r \ge \sigma \end{array} \right.$

and the repulsive form

$\Phi_{12}\left( r \right) = \left\{ \begin{array}{lll} \infty & ; & r < \sigma \\ \left( \frac{ \epsilon \sigma }{r}\right) \exp \left[- \kappa \left( \frac{r}{\sigma} - 1 \right) \right] & ; & r \ge \sigma \end{array} \right.$

where $\Phi\left( r \right)$ is the intermolecular pair potential, $r := |\mathbf{r}_1 - \mathbf{r}_2|$ is the distance between site 1 and site 2, $σ$ is the hard diameter, $ε$ is the energy well depth ( $ε > 0$ ), and $κ$ is a parameter that controls the interaction range ( $κ > 0$ ).