- Title
- Continued fractions and the Thomson problem
- Creator
- Moscato, Pablo; Haque, Mohammad Nazmul; Moscato, Anna
- Relation
- ARC.DP200102364 http://purl.org/au-research/grants/arc/DP200102364
- Relation
- Science Reports Vol. 13, Issue 1, no. 7272
- Publisher Link
- http://dx.doi.org/10.1038/s41598-023-33744-5
- Publisher
- Nature Publishing Group
- Resource Type
- journal article
- Date
- 2023
- Description
- 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 x 10−8 for the model of the normalized energy (En(n)≡eg(n)≡2E(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 x 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.
- Subject
- energy configuration; sequences; integers; formula
- Identifier
- http://hdl.handle.net/1959.13/1481731
- Identifier
- uon:50785
- Identifier
- ISSN:2045-2322
- Rights
- © The Author(s) 2023. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http:// creativecommons. org/ licenses/ by/4.0/.
- Language
- eng
- Full Text
- Reviewed
- Hits: 1106
- Visitors: 1103
- Downloads: 8
| Thumbnail | File | Description | Size | Format | |||
|---|---|---|---|---|---|---|---|
| View Details Download | ATTACHMENT02 | Publisher version (open access) | 1 MB | Adobe Acrobat PDF | View Details Download |