Hard core Yukawa potential

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

\Phi \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\geq \sigma \end{array}}\right.

and the repulsive form

\Phi \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\geq \sigma \end{array}}\right.

where $\Phi \left(r\right)$ is the intermolecular pair potential, $r$ is the distance, $\sigma$ is the hard diameter, $\epsilon$ is the energy well depth ( $\epsilon >0$ ), and $\kappa$ is a parameter that controls the interaction range ( $\kappa >0$ ).