From sticky-hard-sphere to Lennard-Jones-type clusters

A relation M-SHS -> LJ between the set of nonisomorphic sticky-hard-sphere clusters M-SHS and the sets of local energy minima M-LJ of the (m, n)-Lennard-Jones potential V-mn(LJ)(r) = epsilon/n-m [mr(-n) - nr(-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 greater than or similar to 10. While the map from M-SHS -> M-SHS -> LJ is noninjective and nonsurjective, 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 unfavorable 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.