A sensible estimate for the stability constant of the Lennard-Jones potential

TL;DR

This paper provides a tighter upper bound for the stability constant of the Lennard-Jones potential in three dimensions, improving previous bounds and offering potential applicability to similar potentials.

Contribution

The authors establish a significantly improved upper bound for the Lennard-Jones stability constant and refine the lower bound for minimal interparticle distance in optimal configurations.

Findings

Stability constant of Lennard-Jones potential is less than 14.316.

Lower bound for minimal interparticle distance is 0.684.

Method may apply to Morse and similar potentials.

Abstract

We show that the stability constant of the Lennard-Jones potential in $\mathbb R^3$, $\Phi(x)=\lVert x \rVert^{-12}-2 \lVert x \rVert^{-6}$, is smaller than $14.316$. This is remarkably smaller than the best previously known bound. Our method is probably applicable to other similar potentials such as the Morse potentials. We also improve slightly, in the Lennard-Jones case, the lower bound for the minimum interparticle distance of an optimal $n$-particle configuration to $0.684$.