New alternatives to the Lennard-Jones potential

Abstract

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 $$r^{-6}$$ r - 6 and to a very simple $$depth=1$$ d e p t h = 1 truncated continued fraction with integer coefficients and depending on $$n^{-r}$$ n - r only, where n is a natural number (with $$n=13$$ n = 13 for Xe, $$n=16$$ n = 16 for Kr, $$n=17$$ n = 17 for Ar, and $$n=27$$ n = 27 for Neon). For Helium (He), the data is well approximated with a function having only one variable $$n^{-r}$$ n - r with $$n=31$$ n = 31 and a truncated continued fraction with $$depth=2$$ d e p t h = 2 (i.e., the third convergent of the expansion). Also, for He, we have found an interesting $$depth=0$$ d e p t h = 0 result, a Dirichlet polynomial of the form $$k_1 \, 6^{-r} + k_2 \, 48^{-r} + k_3 \, 72^{-r}$$ k 1 6 - r + k 2 48 - r + k 3 72 - r (with $$k_1, k_2, k_3$$ 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

• Lennard-Jones potential
• Dirichlet polynomial
• Symbolic regression
• Analytic continued fraction
• Memetic algorithm