Hooke's law correlation in two-electron systems

TL;DR

This paper investigates the properties of Hooke's law correlation energy in two-electron systems, comparing harmonic and Coulomb interactions in different models and dimensions, revealing exact solutions and differences in correlation behaviors.

Contribution

It provides a detailed analysis of Hooke's law correlation energy in two-electron models, including exact solutions for specific dimensions and comparisons with Coulombic systems.

Findings

Exact solutions for spherium model at D=1 and 3 using Mathieu functions

Differences in correlation energy behavior between harmonic and Coulomb interactions

Insights into weakly and strongly correlated electron regimes

Abstract

We study the properties of the Hooke's law correlation energy ($\Ec$), defined as the correlation energy when two electrons interact {\em via} a harmonic potential in a $D$-dimensional space. More precisely, we investigate the $^1S$ ground state properties of two model systems: the Moshinsky atom (in which the electrons move in a quadratic potential) and the spherium model (in which they move on the surface of a sphere). A comparison with their Coulombic counterparts is made, which highlights the main differences of the $\Ec$ in both the weakly and strongly correlated limits. Moreover, we show that the Schr\"odinger equation of the spherium model is exactly solvable for two values of the dimension ($D = 1 \text{and} 3$), and that the exact wave function is based on Mathieu functions.