Correspondences Between the Classical Electrostatic Thomson

Correspondences Between the Classical Electrostatic Thomson Problem and Atomic Electronic Structure Tim LaFave Jr.1,∗ 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. 1. Introduction present paper builds on this previous work by iden- tifying numerous correspondences between the elec- Quantum mechanical treatments of electrons in trostatic Thomson Problem of distributing equal spherical quantum dots, or “artificial atoms”, 1 rou- point charges on a unit sphere and atomic electronic tinely exhibit correspondences to atomic-like shell- structure. filling patterns by the appearance of abrupt jumps Despite the diminished stature of J.J. Thomson’s or dips in calculated energy or capacitance distribu- classical “plum-pudding” model 19 among more ac- tions as electrons are added to or removed from the curate atomic models, the Thomson Problem has 2–10 system. Additionally, shell-filling is observed in attracted considerable attention since the mid- ion trap models in which ions are subject to a spher- twentieth century. 20 The Thomson Problem has 11–14 ical harmonic potential. An understanding of found use in practical applications including models physical mechanisms responsible for shell-filling is of spherical viruses, 21 fullerenes, 22,23 drug encapsu- useful to the engineering of tailorable electronic lant design, 24 and crystalline order on curved sur- properties of quantum dots and ion traps as well faces. 25 Numerical solutions for many-N electron as a better understanding of atomic electronic phe- systems have emerged in the last few decades us- nomena. ing a variety of computational algorithms. 26–36 The Electron shell-filling behavior has been observed Thomson Problem is now a benchmark for global in two-dimensional classical electrostatic models optimization algorithms, 34,35 yet its general solu- 15 using a parabolic potential. However, electro- tion remains an important unsolved mathematics static treatments of three-dimensional artificial problem. 37 atoms have fallen short of yielding any observ- 2 A symmetry-dependent electrostatic potential able shell-filling patterns. Recently, similarities be- energy distribution is obtained using numerical so- tween classical electrostatic properties of spheri- arXiv:1403.2591v1 [physics.class-ph] 22 Feb 2014 lutions of the Thomson Problem for N ≤ 100 elec- cal quantum dots and the distribution of empir- trons residing strictly on a unit sphere. This dis- ical ionization energies of neutral atoms were re- tribution exhibits many disparities (“jumps” and ported for N ≤ 32 electrons 16,17 when evaluated 18 “dips”) that appear to be randomly distributed. using the discrete charge dielectric model. The However, upon closer inspection these disparities appear in a “systematic” 38 pattern shown here to ∗ be consistent with the pattern of atomic electron [email protected] 1University of Texas at Dallas, Department of Electrical shell-filling as found in the form of the modern Engineering, Richardson, TX 75080 Periodic Table. A derivation of the symmetry- Preprint submitted to Elsevier November 6, 2012 dependent potential energy distribution is given. A detailed description of its many correspondences with atomic electronic structure is provided in sup- port of the conjecture that spatial symmetry limita- q tions on discrete charges constrained to a spherical 0 q volume of space, as within a spherical dielectric or 0 the central field of a nucleus, are underlying physical mechanisms responsible for electron shell-filling [5] [4+ ] in quantum dots, ion traps, and atomic electronic (a) (b) structure. Additionally, a pattern of the largest energy disparities is shown to coincide with size- normalized empirical ionization energy data with Figure 1: Discrete spatial symmetry changes. The 5-electron solution of the Thomson problem on (a) a unit sphere trans- discussion concerning relevant topological features forms into (b) the centered [4+] configuration having one of N-charge solutions and correspondence to shell- charge, q0, at the origin surrounded by the Thomson solu- filling in atoms and ion traps. The systematic pat- tion for 4-electrons on the unit sphere. tern of classical electrostatic symmetry-dependent energies consistent with atomic electron shell-filling such that the total number of electrons remains un- is anticipated given the variety of neighboring geo- changed, a single electron is moved from the unit metric electron orbital shapes obtained from quan- sphere to its origin. In general, the resulting elec- tum mechanics (s, p, d, and f orbitals). tron distribution is one electron, q , at the origin, For ease of verification, the reported results are 0 and the remaining N−1 electrons distributed on the based on data collected in an interactive database unit sphere having the [N−1] solution of the Thom- of numerical solutions of the Thomson Problem son Problem. This centered configuration may be hosted by Syracuse University 39 which may be com- denoted by [N − 1+], in which “+” indicates the pared with numerous other published sources. 26–36 presence of q0. The total energy of any transformed system may be expressed as a function of U([2]), the 2. Discrete Symmetry Changes energy of the two-electron solution of the Thomson Problem, In the absence of a positively-charged spherical volume, electrons in the “plum pudding” model re- pel each other in such a manner that they nat- U([N −1+]) = U([N −1]) + 2(N −1)U([2]) (2) urally form solutions of the Thomson Problem. 40 These solutions are obtained by minimizing the to- where the last term accounts for the interaction of tal Coulomb repulsion energy q0 with all N−1 electrons residing on the “Thomson sphere”. The [4+] solution shown schematically in Fig. 1b, N 1 consists of four electrons at vertices of a regular U(N)= X (1) |r − r | tetrahedron about q0 at the origin. Symmetrically, i<j i j the [4+] point group configuration is identical to the [4] point group configuration 12 of the Thomson of each N-electron system with ri and rj constrained to the surface of a unit sphere. An example problem as q0 interacts identically with all N − 1 of the 5-electron solution is shown in Fig. 1a. The charges on the Thomson sphere. Using Eq. 2, the minimum energy is obtained with an electron at energy difference as shown in Fig. 2 each “pole” of the unit sphere, and the remaining three electrons are located at vertices of an equi- ′ + lateral triangle about the “equator”. Herein, the ∆U (N) = U([N −1 ]) − U([N −1]) geometric configuration of each N-electron system = (N −1) is denoted in square brackets, [N]. To change the electrostatic electron configuration is isosymmetric. Here, U([2]) = 1/2 for a unit of a given [N] solution of the Thomson Problem to sphere. In general, U([N −1+]) > U([N]). There- the configuration of its neighboring [N−1] solution fore, the energy difference ∆U(N) associated with 2 exclusively with discrete geometric changes between + [N−1 ] [N] configurations and their neighboring [N −1+] centered configurations. This q transformation ∆ + 0 U (N) corresponds both with the single positive net-charge of a singly-ionized atom (charge sign symmetry) ∆U’(N) and the removal of the ionized electron through ′ [N] its mathematically-equivalent image charge, q0, as Energy may be realized within the context of a dielectric ∆ isosymmetry U(N) sphere. The energy distribution associated with this transformation, Fig. 3, yields a systematic pattern of non-uniformities consistent with electron shell-filling patterns in atoms, exhibit distinct [N−1] energy-pairs, feature intra-subshell energy patterns consistent with empirical intra-subshell ionization N−1 N Electron Number energies, and consists of a pattern of salient features consistent with size-normalized empirical ionization energies. Taken together, these many correspon- Figure 2: Discrete spatial symmetry changes in the Thomson dences characterize Fig. 3 as an electrostatic “fin- problem. Changing the symmetry of a given [N] configuration to the symmetry of its neighboring [N−1] configuration gerprint” of the periodic table of elements resulting while maintaining all N electrons involves an intermediate from a single “shell” of electrons in the Thomson [N −1+] centered configuration. This transition represents Problem. the symmetry-dependent component, ∆U +(N), of the energy needed to remove a single electron. The remaining transition from [N−1+] to[N−1] is the isosymmetric component ′ whose energy, ∆U (N), is linearly dependent on N. 3.1. Image Charges and Ionized Electrons the removal of an electron in the Thomson problem Symmetry-dependent potential energy differ- may be expressed, ences, ∆U +(N,ε), of Thomson Problem solutions treated within a sphere of dielectric constant ε = ′ 20ε and radius a = 100nm, for illustrative pur- ∆U(N) = ∆U (N) − ∆U +(N) 0 poses, are plotted in Fig. 3 (solid circles). This + = (N − 1) − ∆U (N) (3) yields a similar energy distribution as before (open circles) but with much more pronounced non- as shown in Fig. 2. Removal of an electron is de- uniformities owing partly to the minimization of pendent on a linear isosymmetric term in N and the total stored energy with respect to the varying a term resulting from a symmetry change between Thomson radius. Each charge, q , inside the dielec- neighboring point groups. Energy differences due i tric sphere may be replaced by a mathematically to this change in each N-electron system, ′ equivalent image charge, qi, outside the sphere.

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