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

Good a-priori bounds on the smallest pairwise distance $r_{\mathrm{m}\mathrm{i}\mathrm{n}}(L{J}_N^{\mathrm{g}\mathrm{m}\mathrm{i}\mathrm{n}})$ 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 $L{J}_N^{\mathrm{e}\mathrm{q}\mathrm{u}}$ (the stationary points), then $r_{\mathrm{m}\mathrm{i}\mathrm{n}}(C^{(N)})\le r_{\mathrm{m}\mathrm{i}\mathrm{n}}(L{J}_2^{\mathrm{g}\mathrm{m}\mathrm{i}\mathrm{n}})$. It is also shown that if $C^{(N)}\in$ LJ${}_N^{\mathrm{g}\mathrm{m}\mathrm{i}\mathrm{n}}\subset$ LJ${}_N^{\mathrm{e}\mathrm{q}\mathrm{u}}$, equality holds if and only if $N\in \{2,3,4\}$. We conjecture that $r_{\mathrm{m}\mathrm{i}\mathrm{n}}(L{J}_N^{\mathrm{g}\mathrm{m}\mathrm{i}\mathrm{n}})>1$ in units for which $r_{\mathrm{m}\mathrm{i}\mathrm{n}}(L{J}_2^{\mathrm{g}\mathrm{m}\mathrm{i}\mathrm{n}})=2^{\frac{1}{6}}\approx 1.122462048$. This conjectured lower bound, if correct, would improve the best lower bound currently known, $r_{\mathrm{m}\mathrm{i}\mathrm{n}}(L{J}_N^{\mathrm{g}\mathrm{m}\mathrm{i}\mathrm{n}})\ge 0.767764$, by about 25$\mathrm{\%}$. In these units the smallest minimal pair distance found through numerical searches for LJ${}_N^{\mathrm{g}\mathrm{m}\mathrm{i}\mathrm{n}}$ with $N\le 1000$ is $r_{\mathrm{m}\mathrm{i}\mathrm{n}}(L{J}_{923}^{\mathrm{g}\mathrm{m}\mathrm{i}\mathrm{n}})\approx 1.01361$, so the conjectured lower bound would presumably be close to optimal. From the virial theorem we obtain an identity for any $C^{(N)}\in L{J}_N^{\mathrm{e}\mathrm{q}\mathrm{u}}$, which expresses $r_{\mathrm{m}\mathrm{i}\mathrm{n}}(C^{(N)})$ in terms of the distribution of relative distances in $C^{(N)}$. This result reveals interesting connections with the Erdős distance, and related problems.