A Tale of Two Electrons: Correlation at High Density

TL;DR

This paper reviews recent advances in calculating the high-density correlation energy in various two-electron systems, revealing a universal behavior in high dimensions and comparing different models like helium, hookium, spherium, and ballium.

Contribution

It provides new insights into the high-density correlation energy across multiple two-electron systems and derives a universal formula in the large-dimension limit.

Findings

High-density correlation energy $ ext{E}_c$ is analyzed for various two-electron systems.

A universal asymptotic form $ ext{E}_c o -1/(8D^2)$ is derived for large dimensions.

Results explain the similarity of $ ext{E}_c$ in different systems at D=3.

Abstract

We review our recent progress in the determination of the high-density correlation energy $\Ec$ in two-electron systems. Several two-electron systems are considered, such as the well known helium-like ions (helium), and the Hooke's law atom (hookium). We also present results regarding two electrons on the surface of a sphere (spherium), and two electrons trapped in a spherical box (ballium). We also show that, in the large-dimension limit, the high-density correlation energy of two opposite-spin electrons interacting {\em via} a Coulomb potential is given by $\Ec \sim -1/(8D^2)$ for any radial external potential $V(r)$, where $D$ is the dimensionality of the space. This result explains the similarity of $\Ec$ in the previous two-electron systems for D=3.