New alternatives to the Lennard-Jones potential

Abstract

We present a new method for approximating two-body interatomic potentials from existing ab initio data based on representing the unknown function as an analytic continued fraction. In this study, our method was first inspired by a representation of the unknown potential as a Dirichlet polynomial, i.e., the partial sum of some terms of a Dirichlet series. Our method allows for a close and computationally efficient approximation of the ab initio data for the noble gases Xenon (Xe), Krypton (Kr), Argon (Ar), and Neon (Ne), which are proportional to $r^{-6}$ and to a very simple $depth=1$ truncated continued fraction with integer coefficients and depending on $n^{-r}$ only, where n is a natural number (with $n=13$ for Xe, $n=16$ for Kr, $n=17$ for Ar, and $n=27$ for Neon). For Helium (He), the data is well approximated with a function having only one variable $n^{-r}$ with $n=31$ and a truncated continued fraction with $depth=2$ (i.e., the third convergent of the expansion). Also, for He, we have found an interesting $depth=0$ result, a Dirichlet polynomial of the form $k_1{6}^{-r}+k_2{48}^{-r}+k_3{72}^{-r}$ (with $k_1,k_2,k_3$ all integers), which provides a surprisingly good fit, not only in the attractive but also in the repulsive region. We also discuss lessons learned while facing the surprisingly challenging non-linear optimisation tasks in fitting these approximations and opportunities for parallelisation.

Keywords: Analytic continued fraction; Dirichlet polynomial; Lennard-Jones potential; Memetic algorithm; Symbolic regression.