# A Sensible Estimate for the Stability Constant of the Lennard-Jones Potential

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## Abstract

We show that the stability constant of the Lennard-Jones potential in \(\mathbb {R}^{3}\), \(\Phi (x)=\Vert x\Vert _2^{-12}-2\Vert x\Vert _2^{-6}\), is smaller than 14.316. This is remarkably smaller than the best previously known bound. Our method is very elementary, and 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.

## Keywords

Stability constant Lennard-Jones potential Statistical mechanics## Notes

### Acknowledgments

I am totally grateful to Aldo Procacci. He taught me about this problem and much more. This work has been supported by the argentinian state organism Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET).

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