Author:
Moscato Pablo,Haque Mohammad Nazmul,Moscato Anna
Abstract
AbstractWe 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.
Funder
Australian Research Council
Maitland Cancer Appeal
Publisher
Springer Science and Business Media LLC
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