Universal scaling relations for the energies of many-electron Hooke atoms

A three-dimensional harmonic oscillator consisting of N >= 2 Coulomb-interacting charged particles, often called a (many-electron) Hooke atom, is a popular model in computational physics for, e.g., semiconductor quantum dots and ultracold ions. Starting from Thomas-Fermi theory, we show that the ground-state energy of such a system satisfies a nontrivial relation: E-gs = omega N-4/3 f(gs) ( beta N-1/2), where. is the oscillator strength, beta is the ratio between Coulomb and oscillator characteristic energies, and f(gs) is a universal function. We perform extensive numerical calculations to verify the applicability of the relation. In addition, we show that the chemical potentials and addition energies also satisfy approximate scaling relations. In all cases, analytic expressions for the universal functions are provided. The results have predictive power in estimating the key ground-state properties of the system in the large-N limit, and can be used in the development of approximative methods in electronic structure theory.