Continued fractions and the Thomson problem

Abstract

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 $$E(n) = (n^2/2) \, e^{g(n)}$$ E ( n ) = ( n 2 / 2 ) e g ( 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 $${5.5549 \times 10^{-8}}$$ 5.5549 × 10 - 8 for the model of the normalized energy ( $$E_n(n) \equiv e^{g(n)} \equiv 2E(n)/n^2$$ E n ( n ) ≡ e g ( n ) ≡ 2 E ( n ) / n 2 ). 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 $$n^2+12$$ n 2 + 12 is a prime. We also observed an interesting correlation with the behavior of the smallest angle $$\alpha (n)$$ α ( n ) , measured in radians, subtended by the vectors associated with the nearest pair of electrons in the optimal configuration. When using both $$\sqrt{n}$$ n and $$\alpha (n)$$ α ( n ) as variables a very simple approximation formula for $$E_n(n)$$ E n ( n ) was obtained with MSE= $$7.9963 \times 10^{-8}$$ 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 $$n^{-1/2}$$ n - 1 / 2 of E(n) first proposed by Glasser and Every in 1992 as $$-1.1039$$ - 1.1039 , and later refined by Morris, Deaven and Ho as $$-1.104616$$ - 1.104616 in 1996, may actually be very close to −1.10462553440167 when the assumed optima for $$n\le 200$$ n ≤ 200 are used.