Correspondences between the classical electrostatic Thomson problem and atomic electronic structure

Abstract

Correspondences between the Thomson Problem and atomic electron shell-filling patterns are observed as systematic non-uniformities in the distribution of potential energy necessary to change configurations of N ≤ 100 electrons into discrete geometries of neighboring N − 1 systems. These non-uniformities yield electron energy pairs, intra-subshell pattern similarities with empirical ionization energy, and a salient pattern that coincides with size-normalized empirical ionization energies. Spatial symmetry limitations on discrete charges constrained to a spherical volume are conjectured as underlying physical mechanisms responsible for shell-filling patterns in atomic electronic structure and the Periodic Law.

Minimizing Energy of P oint Charges on a Sphere using Symbiotic Organisms Search Algorithm

“…Thomson's problem ranks 7 in the famous list of Steve Smale's [2], 18 problems for our century, which have proved to be useful in many scientific and technological domains ranging from Biology [3] to telecommunication. While Thomson's plum pudding model could not withstand the experimental tests, the irregularities found in the numerical computations of energy values have been found to correspond with electron-shell filling in atoms [4].…”

“…In section (3), an overview of the SOS algorithm used to solve Thomson's problem is presented. The implementation details of proposed algorithm and computational results are elaborated in sections (4) and (5) respectively. The conclusions drawn from the present study are elucidated in section (6).…”

Optimal $$N$$ N -Point Configurations on the Sphere: “Magic” Numbers and Smale’s 7th Problem

“…It would also be interesting to try to answer analogues of our questions formulated for the s-Fekete problem on other compact manifolds M. In particular, we expect that the analogue of our list of questions Q 1 -Q14 for M = S 1 33 By this we only mean a vague qualitative reminiscence. Of course, Thomson [Tho04] may have hoped to find the actual quantitative pattern of the periodic table of the chemists; for a most recent inquiry in this spirit, see [LaF13].…”

“…Originally Smale's 7th problem for the 21st century [Sma98] was formulated for the logarithmic energy, see below. Lastly, the value s = 1 yields the Coulomb pair-energy of "three-dimensional unit point charges" associated with the so-called Thomson problem (see [Tho04,Why52,ErHo97,Aetal97,Petal97,BCGMZ,LaF13]). …”

Distributing many points on spheres: Minimal energy and designs

“…Recently, T. LaFave Jr [138]. investigated correspondences between the Thomson problem and atomic electronic structure, and he applied discrete transformations to the Thomson problem in[139] to study the minimal Coulomb energy 6.…”