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.
Categories: physics.atm-clus math-ph math.MP