Lower Bounds on the Exchange-Correlation Energy in Reduced Dimensions

TL;DR

This paper derives and tests bounds on the exchange-correlation energy for many-electron systems across different dimensions, providing new estimates and insights into low-dimensional quantum systems.

Contribution

It introduces universal bounds on exchange-correlation energy in various dimensions and provides the tightest numerical prefactor estimate based on electron gas properties.

Findings

Bounds are derived for 3D, 2D, 1D, and quasi-1D systems.

Numerical tests support the validity of the bounds.

Evidence of a dimensional crossover between two and one dimensions.

Abstract

Bounds on the exchange-correlation energy of many-electron systems are derived and tested. By using universal scaling properties of the electron-electron interaction, we obtain the exponent of the bounds in three, two, one, and quasi-one dimensions. From the properties of the electron gas in the dilute regime, the tightest estimate to date is given for the numerical prefactor of the bound, which is crucial in practical applications. Numerical tests on various low-dimensional systems are in line with the bounds obtained, and give evidence of an interesting dimensional crossover between two and one dimensions.