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

arXiv:2511.15008v1 Announce Type: new
Abstract: 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 $Nin{2,3,4}$. 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 $Nleq 1000$ 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 ErdH{o}s distance, and related problems.