From Sticky-Hard-Sphere to Lennard-Jones-Type Clusters

A relation ${\mathcal{M}}_{\mathrm{S}\mathrm{H}\mathrm{S}\to \mathrm{L}\mathrm{J}}$ between the set of non-isomorphic sticky hard sphere clusters ${\mathcal{M}}_{\mathrm{S}\mathrm{H}\mathrm{S}}$ and the sets of local energy minima ${\mathcal{M}}_{LJ}$ of the $(m,n)$-Lennard-Jones potential $V^$\mathrm{L}\mathrm{J}$_{mn}® = \frac{\varepsilon}{n-m} [ m r^{-n} - n r^{-m} ]$ is established. The number of nonisomorphic stable clusters depends strongly and nontrivially on both $m$ and $n$, and increases exponentially with increasing cluster size $N$ for $N\gtrsim 10$. While the map from ${\mathcal{M}}_{\mathrm{S}\mathrm{H}\mathrm{S}}\to {\mathcal{M}}_{\mathrm{S}\mathrm{H}\mathrm{S}\to \mathrm{L}\mathrm{J}}$ is non-injective and non-surjective, the number of Lennard-Jones structures missing from the map is relatively small for cluster sizes up to $N=13$, and most of the missing structures correspond to energetically unfavourable minima even for fairly low $(m,n)$. Furthermore, even the softest Lennard-Jones potential predicts that the coordination of 13 spheres around a central sphere is problematic (the Gregory-Newton problem). A more realistic extended Lennard-Jones potential chosen from coupled-cluster calculations for a rare gas dimer leads to a substantial increase in the number of nonisomorphic clusters, even though the potential curve is very similar to a (6,12)-Lennard-Jones potential.