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

TL;DR

The work establishes an $N$-independent upper bound on the minimal interparticle distance for Lennard-Jones equilibrium configurations, showing $r_{min}({\cal C}^{(N)}) \le r_{min}({\cal C}^{(2)}_{gmin}) = 2^{1/6}$ with equality only for $N\in\{2,3,4\}$. It then motivates a conjecture that globally minimal LJ clusters satisfy $r_{min}({\cal C}^{(N)}) \ge 1$ (in the same units), supported by numerical data up to $N=1000$ (e.g., $N=923$ yields $r_{min}\approx 1.01361$), suggesting a substantial improvement over the best known bound. A key methodological advance is the virial identity, recast as $r_{min}({\cal C}^{(N)}) = r_{min}({\cal C}^{(2)}_{gmin}) (R({\cal C}^{(N)})/A({\cal C}^{(N)}))^{1/6}$, which ties the minimal distance to the distance spectrum via $R/A$, and invites connections to Erdős-type distance problems. The paper also analyzes distance spectra from selected LJ minima, revealing structural motifs (icosahedral, cuboctahedral, decahedral) and offering a spectral lens on geometry-energetics, with potential to prune NP-hard searches for LJ global minima.

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 $N\in\{2,3,4\}$. 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 Erdős distance, and related problems.

Figures (7)

• Figure 1: Minimum distance in Lennard-Jones clusters with globally minimal energy in the size range from 2 to 1000 atoms. The points are joined to guide the eye. Structures based on the truncated cuboctahedron (O), the Leary tetrahedron (L) Leary, and Marks decahedra (D) marks84 are highlighted. In each case the minimum distance is significantly larger than for the neighbouring clusters based on Mackay icosahedra mackay62.
• Figure 2: Distance spectra of the LJ$_N$ global minima for $N\in\{12,13,14\}$ (top-down). Visibly no distance is smaller than 1.
• Figure 3: Distance spectra of the LJ$_N$ global minima for $N\in\{54,55,56\}$ (top-down). Again, no distance is smaller than 1.
• Figure 4: Distance spectra of putative LJ$_{N}$ global minima for $N\in\{146,147,148\}$.
• Figure 5: Distance spectra of putative LJ$_{N}$ global minima for $N\in\{309,561,923\}$.