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

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 $0.684$.