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.
