Discrete Transformations in the Thomson Problem Tim LaFave Jr. Abstract A significantly lower upper limit to minimum energy solutions of the electrostatic Thomson Problem is reported. A point charge is introduced to the origin of each N-charge solution. This raises the total energy by N as an upper limit to each (N + 1)-charge solution. Minimization of energy to U(N + 1) is well fit with 0.5518(3/2)√N +1/2 for up to N=500. The energy distribution due to this displacement exhibits correspondences− with shell-filling behavior in atomic systems. This work may aid development of more efficient and innovative numerical search algorithms to obtain N-charge configurations having global energy minima and yield new insights to atomic structure. Keywords: “T. LaFave Jr. J. Electrostatics 72(1) 39-43 (2014)”. DOI: 10.1016/j.elstat.2013.11.007 1. Introduction within a continuous charged shell of unit radius hav- ing total charge, N. With this interpretation, the The Thomson Problem has drawn considerable half-integer terms may correspond to self-energies interest since the mid-1900s (1) having found use of N uniformly charged disks that are removed to in modeling fullerenes, (2,3) drug encapsulants, (4) yield the final minimized energy of discrete charges. spherical viruses, (5) and crystalline order on curved If the N 2/2 term is associated with the random surfaces. (6) In the past few decades several compu- distribution of point charges, the half-integer terms tational algorithms have yielded precise numerical may be related to correlation energies of surface solutions for many-N electron systems. (7–18) The Coulomb equilibrium states. (7) These ascriptions Thomson Problem has emerged as a benchmark of energy terms to physical entities have guided for global optimization algorithms (16,17) though its the development of fairly useful minimization al- general solution remains unknown. (19) gorithms. Numerical solutions of the Thomson Problem are Here, the discrete derivative of N 2/2 is shown those for which the total Coulomb repulsion energy, to correspond to the introduction of a single point charge, q0, at the origin of a given N-charge solu- N tion, and the discrete derivative of the remaining 1 U(N)= , (1) half-integer term(s) accounts for energy needed to ri rj displace q to the unit sphere. This yields each Xi<j | − | 0 subsequent (N + 1)-charge solution of the Thom- is a minimum for each N-charge system with ri and son Problem. In this manner, the upper limit of rj constrained to the surface of a unit sphere. each minimized (N + 1) energy solution is given by The initial condition of some minimization al- U(N)+ N. This represents a well-defined charge arXiv:1403.2592v1 [physics.class-ph] 22 Feb 2014 gorithms is the random distribution of N point configuration with a significantly lower energy than charges on the unit sphere. This sets a relatively random charge distributions and may be useful as (8) distant upper energy limit, Ur(N)= N(N 1)/2, initial conditions in relatively more efficient energy from which numerous iterations progress− toward a minimization algorithms. global minimum for each N-charge system. The Notably, the distribution of energy solutions of distribution of numerical solutions has been fit the Thomson problem is “systematic” (8) and not with empirical functions including U(N)= N 2/2+ random. Here, having ignored the linear term in aN 3/2 (7) and U(N) = N 2/2+ aN 3/2 + bN 1/2. (8) the discrete derivative, the remaining “systematic” The quadratic term is ascribed to energy stored distribution of energy associated with the displace- Preprint submitted to Elsevier June 15, 2021 ment of q0 to the surface of the unit sphere demon- 2 strably exhibits features uniquely correspondent N /2 with numerous shell-filling features throughout the N(N−1)/2 + periodic table of natural atomic systems. (20–22) U(N+1 ) + Hence, introduction of q0 to the Thomson Prob- U(N ) lem represents an intriguingly fresh perspective of + U(N+1) this extensively useful mathematical model and the Potential Energy U(N−1 ) classical underpinnings of atomic structure. U(N) Numerical solutions of the Thomson Problem for U(N−1) up to 500 charges used in this communication have N−1 N N+1 been obtained from a continually updated database Number of point charges maintained by Syracuse University. (23) Figure 1: The extreme upper energy limit of the Thomson 2. Discrete Derivatives Problem is given by N 2/2 for a continuous charge shell fol- lowed by N(N 1)/2, the energy associated with a random − + Two upper energy bounds of some interest are distribution of N point charges. Significantly lower, U(N ), those associated with a continuous charge shell (8) the energy of a given N-charge solution of the Thomson Problem with one charge at its origin is readily obtained consisting of infinite-many N charges such that by U(N) + N, where U(N) are solutions of the Thomson Problem. N 2 U∞(N)= (2) 2 imposing additional arguments. If these intermedi- ate values of f(N) should have physical usefulness, and the random distribution of N discrete point for instance if an effective fractional charge on the charges across the surface of the unit sphere unit sphere surface is admitted as a discrete charge approaches or leaves the surface, f(N) should have local minima at integer values of N. Though these N N 2 N Ur(N)= (N 1) = . (3) fit functions, Eqs. 4 and 5 have no local minima at 2 − 2 − 2 integer values of N, they are coarsely instructive. Plots of Eqs. 2 and 3 are shown in Fig. 1 together It is potentially more fruitful to design a fit with a few numerical solutions (open circles) of the function in accordance with the physical nature of Thomson Problem for illustration. electrostatic charge configurations. In particular, The minimized global potential energy solutions knowledge of f(N) at integer values of N, given for up to N=65 point charges were previously fit the discrete nature of point charges is paramount. using, (7) Consider the discrete derivative of f(N) at integer values of N, 2 N 3/2 f1(N)= + aN (4) ∆f (N) f(N + ∆N) f(N) 2 i = − (6) ∆N ∆N in which a = 0.5510, and for up to N 100 of Eq. 4, which yields, after binomial expansion of using (8) − ∼ the half-integer term, 2 N 3/2 1/2 f2(N)= + aN + bN . (5) 1 2 ∆f1(N) = N + 2 These smooth, continuous fit functions are gener- a 1 3 − 1 1 − 3 + 3N 2 + N 2 N 2 + (7) ally unrepresentative of the absolute minimum en- 2 4 − 8 ··· ergy configurations of discrete charges due to a vari- ety of issues. Among them, discrete charges cannot for ∆N = 1. For comparison, the derivative of be infinitesimally subdivided so all points, f(N), for Eq. 5, with binomial expansion of the half-integer non-integer N have no physical significance without terms, 2 [N+ ] [N+ ] 1 1 1 7 1 2 − 2 ∆f2(N) = N + + 3aN + (a + b)N 2 2 4 q 0 N 1 − 3 N + + (a b)N 2 + (8) U (N+1) 8 − ··· ∆ for ∆N = 1. The first set of parentheses in Eqs. 7 U’(N+1) ∆ [N+1] and 8 is the discrete derivative of N 2/2, the energy of a continuous charge shell, Eq. 2. Energy U(N+1) q ∆ 3. Discrete Transformation Energies 0 To understand the physical charge distribu- tion represented by the discrete energy differences, [N] Eqs. 7 and 8, consider an N-charge solution, such N N+1 as that of [N] shown in Fig. 2 for which the total Charge Number minimized electrostatic energy is U(N) as shown in Fig. 1. The discrete energy changes needed to Figure 2: A well-defined intermediate charge configuration, + obtain the solution U(N + 1) in Fig. 1 may be ob- [N ], with q0 at the origin of [N] linearly increases the + tained by introducing an (N + 1)th point charge, energy by N. Consequently, U(N ) is readily accessible. + Displacement of q0 to the unit sphere surface and global q0, to the origin denoted [N ] in Fig. 2. Its contri- minimization of energy yields the new [N + 1] solution with bution to the total energy, given equal interaction ∆U +(N + 1) √N. with all N charges on the unit sphere, is N. The ∝ total energy, U(N +), (Fig. 1) is U(N)+ N as illus- trated in Fig. 2. Thus, the first term of each dis- as shown in Fig. 2. Consequently, ∆U +(N + 1) crete derivative, ∆f (N), is due to the appearance i accounts for the half-integer terms in Eqs. 7 and of q at the origin. The remaining terms are due 0 8 and the +1/2 term associated with the discrete to the displacement of q to the unit sphere surface 0 derivative of N 2/2. and subsequent (or in situ) global minimization of To check the validity of the discrete transforma- energy, U(N + 1) in Fig. 1, to yield the (N + 1) tions from each N-charge solution of the Thomson configuration (Fig. 2). Problem to its adjacent (N + 1)-charge solution, In the strictest spirit of the Thomson Problem, ∆U +(N + 1) in Eq. 9 is fit with the remaining introduction of q to the origin of the unit sphere 0 terms of the discrete derivatives of Eqs. 7 and 8.