Continued fractions and the Thomson problem

Abstract

We introduce new analytical approximations of the minimum electrostatic energy configuration of n electrons, E(n), when they are constrained to be on the surface of a unit sphere. Using 453 putative optimal configurations, we searched for approximations of the form E(n)=(n2/2)eg(n) where g(n) was obtained via a memetic algorithm that searched for truncated analytic continued fractions finally obtaining one with Mean Squared Error equal to 5.5549 × 10 - 8 for the model of the normalized energy (En(n) ≡ eg(n)≡ 2 E(n) / n2). Using the Online Encyclopedia of Integer Sequences, we searched over 350,000 sequences and, for small values of n, we identified a strong correlation of the highest residual of our best approximations with the sequence of integers n defined by the condition that n2+ 12 is a prime. We also observed an interesting correlation with the behavior of the smallest angle α(n) , measured in radians, subtended by the vectors associated with the nearest pair of electrons in the optimal configuration. When using both n and α(n) as variables a very simple approximation formula for En(n) was obtained with MSE= 7.9963 × 10 - 8 and MSE= 73.2349 for E(n). When expanded as a power series in infinity, we observe that an unknown constant of an expansion as a function of n- 1 / 2 of E(n) first proposed by Glasser and Every in 1992 as - 1.1039 , and later refined by Morris, Deaven and Ho as - 1.104616 in 1996, may actually be very close to −1.10462553440167 when the assumed optima for n≤ 200 are used.