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

A three-dimensional harmonic oscillator consisting of $N\ge 2$ 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: $E_{gs}=\omega N^{4/3}{f}_{gs}\left(\beta N^{1/2}\right)$, where $\omega$ 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.

Figure 1

Density-functional (PBE) and PIMC results for the scaled ground-state energies of HAs as a function of $z={(N/\omega )}^{1/2}$ (symbols). The solid line represents the function $f_{gs}\left(z\right)$ of Eq. (7). The fitting region contains systems with $\omega =1.0$ and $\omega =0.5$. The control region contains systems with $\omega =0.1$.

Figure 2

LDA results (symbols) for the scaled ground-state energies of Hooke atoms with varying confinement strengths up to $N\approx 1500$ in comparison with the scaling function $f_{gs}\left(z\right)$ (solid line) of Eq. (7). Inset: Additional numerical results from Refs. [32, 33, 34].

Figure 3

Upper panel: Relative error of numerical PBE calculations with respect to ground-state energies resulting from Eq. (7). Lower panel: Same for LDA calculations. Hooke atoms with varying confinement strengths and particle numbers up to $N=1500$ were considered (see main text for details).

Figure 4

Scaled chemical potentials (PBE results) of Hooke atoms as a function of $z$. The solid line represents the function $f_{\mu }$ in Eq. (10). Inset: Results from additional coupled-cluster (singles and doubles) calculations of systems with $\{N=18,58\}$ and confinement strengths ranging from 0.1 to 25.0 a.u. [32].

Figure 5

Addition energies of few-particle Hooke atoms as a function of $zN$ (symbols). The results correspond to the local-density approximation (LDA) within density-functional theory, variational quantum Monte Carlo (VMC), and diffusion Monte Carlo (DMC). The solid line represents the function $\mathrm{\Delta }\mu (N,\omega =0.5)$ in Eq. (14).