A note on the minimal pairwise distance in optimal Lennard-Jones N-body clusters

Good a-priori bounds on the smallest pairwise distance rmin(LJNgmin) for a three-dimensional (3D) Lennard-Jones N-body cluster of globally minimal energy can significantly reduce the computational search space in the NP-hard problem to find this configuration. In this contribution the virial theorem is exploited for this purpose. We prove that if a configuration C(N) is a member of LJNequ (the stationary points), then rmin(C(N))≤rmin(LJ2gmin). It is also shown that if C(N)∈ LJNgmin⊂ LJNequ, equality holds if and only if N∈{2,3,4}. We conjecture that rmin(LJNgmin)>1 in units for which rmin(LJ2gmin)=216≈1.122462048. This conjectured lower bound, if correct, would improve the best lower bound currently known, rmin(LJNgmin)≥0.767764, by about 25%. In these units the smallest minimal pair distance found through numerical searches for LJNgmin with N≤1000 is rmin(LJ923gmin)≈1.01361, so the conjectured lower bound would presumably be close to optimal. From the virial theorem we obtain an identity for any C(N)∈LJNequ, which expresses rmin(C(N)) in terms of the distribution of relative distances in C(N). This result reveals interesting connections with the Erdős distance, and related problems.