Uniform electron gases: III. Low-density gases on three-dimensional spheres

TL;DR

This study combines variational Monte Carlo and Hartree-Fock methods to accurately analyze low-density, same-spin electron gases on a 3-sphere, revealing localized Wigner crystal formations and symmetric configurations related to Platonic solids.

Contribution

It introduces a combined VMC and HF approach for low-density electron gases on a 3-sphere, identifying special symmetric configurations and calculating vibrational corrections.

Findings

Identified 11 symmetric electron arrangements on the 3-sphere.

Discovered a new highly symmetric structure at n=13.

Calculated vibrational corrections to Thomson energies.

Abstract

By combining variational Monte Carlo (VMC) and complete-basis-set limit Hartree-Fock (HF) calculations, we have obtained near-exact correlation energies for low-density same-spin electrons on a three-dimensional sphere (3-sphere), i.e.~the surface of a four-dimensional ball. In the VMC calculations, we compare the efficacies of two types of one-electron basis functions for these strongly correlated systems, and analyze the energy convergence with respect to the quality of the Jastrow factor. The HF calculations employ spherical Gaussian functions (SGFs) which are the curved-space analogs of cartesian Gaussian functions. At low densities, the electrons become relatively localized into Wigner crystals, and the natural SGF centers are found by solving the Thomson problem (i.e. the minimum-energy arrangement of $n$ point charges) on the 3-sphere for various values of $n$. We have found 11 special values of $n$ whose Thomson sites are equivalent. Three of these are the vertices of four-dimensional Platonic solids --- the hyper-tetrahedron ($n=5$), the hyper-octahedron ($n=8$) and the 24-cell ($n=24$) --- and a fourth is a highly symmetric structure ($n=13$) which has not previously been reported. By calculating the harmonic frequencies of the electrons around their equilibrium positions, we also find the first-order vibrational corrections to the Thomson energy.