Correlation energy of two electrons in a ball

TL;DR

This paper investigates the ground-state correlation energy of two electrons confined in a D-dimensional ball, providing accurate calculations and testing a conjecture about its behavior in the large-D limit.

Contribution

It offers precise calculations of correlation energies for two electrons in a spherical confinement and tests a recent conjecture on their asymptotic behavior as dimension increases.

Findings

Accurate correlation energy results for various dimensions.

Confirmation of the conjecture that $E_c^{(0)} o -rac{ ext{delta}^2}{8}$ as D increases.

Insights into the behavior of confined two-electron systems in high dimensions.

Abstract

We study the ground-state correlation energy $E_{\rm c}$ of two electrons of opposite spin confined within a $D$-dimensional ball ($D \ge 2$) of radius $R$. In the high-density regime, we report accurate results for the exact and restricted Hartree-Fock energy, using a Hylleraas-type expansion for the former and a simple polynomial basis set for the latter. By investigating the exact limiting correlation energy $E_{\rm c}^{(0)} = \lim_{R \to 0} \Ec$ for various values of $D$, we test our recent conjecture [J. Chem. Phys. {\bf 131} (2009) 241101] that, in the large-$D$ limit, $E_{\rm c}^{(0)} \sim -\delta^2/8$ for any spherically-symmetric confining external potential, where $\delta=1/(D-1)$.