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

Good a-priori bounds on the smallest pairwise distance $r_{\rm{{min}}}(\mbox{LJ}_N^{\rm{gmin}})$ 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 $\mbox{LJ}_N^{\rm{equ}}$ (the stationary points), then $r_{\rm{{min}}}({C}^{(N)}) \leq r_{\rm{{min}}}(\mbox{LJ}_2^{\rm{gmin}})$. It is also shown that if ${C}^{(N)}\in$ LJ$_N^{\rm{gmin}}\subset$ LJ$_N^{\rm{equ}}$, equality holds if and only if $N\in\{2,3,4\}$. We conjecture that $r_{\rm{{min}}}(\mbox{LJ}_N^{\rm{gmin}}) >1$ in units for which $r_{\rm{{min}}}(\mbox{LJ}_2^{\rm{gmin}})= 2^\frac16 \approx 1.122462048$. This conjectured lower bound, if correct, would improve the best lower bound currently known, $r_{\rm{{min}}}(\mbox{LJ}_N^{\rm{gmin}})\geq 0.767764$, by about 25$\%$. In these units the smallest minimal pair distance found through numerical searches for LJ$_N^{\rm{gmin}}$ with $N\leq 1000$ is $r_{\rm{{min}}}(\mbox{LJ}_{923}^{\rm{gmin}}) \approx 1.01361$, so the conjectured lower bound would presumably be close to optimal. From the virial theorem we obtain an identity for any ${C}^{(N)}\in \mbox{LJ}_N^{\rm{equ}}$, which expresses $r_{\rm{{min}}}({C}^{(N)})$ in terms of the distribution of relative distances in ${C}^{(N)}$. This result reveals interesting connections with the Erd\H{o}s distance, and related problems.