Modulated patchy Lennard-Jones model

The modulated patchy Lennard-Jones model is given by

$u_{\mathrm {patchy} }({\mathbf {r} }_{ij},{\mathbf {\Omega } }_{i},{\mathbf {\Omega } }_{j})=\left\{{\begin{array}{lll}u_{\mathrm {LJ} }(r_{ij})&;&r_{ij}<\sigma _{\mathrm {LJ} }\\u_{\mathrm {LJ} }(r_{ij})\exp \left(-{\frac {\theta _{k_{min},ij}^{2}}{2\sigma ^{2}}}\right)\exp \left(-{\frac {\theta _{l_{min},ji}^{2}}{2\sigma ^{2}}}\right)&;&r_{ij}\geq \sigma _{\mathrm {LJ} }\end{array}}\right.$

where $u_{\mathrm {LJ} }(r_{ij})$ is the Lennard-Jones potential and