New alternatives to the Lennard-Jones potential

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 k16-r+k248-r+k372-r (with k1,k2,k3 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.