Exact "exact exchange" potential of two- and one-dimensional electron gases beyond the asymptotic limit

TL;DR

This paper analytically derives the exact exchange potential for 1D and 2D electron gases beyond the known asymptotic limit, revealing deviations relevant for surface and nano-science applications.

Contribution

It provides the first analytical expressions for the exact exchange potential near low-dimensional electron gases beyond the asymptotic regime.

Findings

Potential follows -e^2/r_perp asymptotically at large distances

Deviations occur at distances comparable to the Fermi wavelength

Results serve as benchmarks for numerical DFT methods

Abstract

The exchange-correlation potential experienced by an electron in the free space adjacent to a solid surface or to a low-dimensional system defines the fundamental image states and is generally important in surface- and nano-science. Here we determine the potential near the two- and one-dimensional electron gases (EG), doing this analytically at the level of the exact exchange of the density-functional theory (DFT). We find that, at $r_\perp\gg k_F^{-1}$, where $r_\perp$ is the distance from the EG and $k_F$ is the Fermi radius, the potential obeys the already known asymptotic $-e^2/r_\perp$, while at $r_\perp \lesssim k_F^{-1}$, but {\em still in vacuum}, qualitative and quantitative deviations of the exchange potential from the asymptotic law occur. The playground of the excitations to the low-lying image states falls into the latter regime, causing significant departure from the Rydberg series. In general, our analytical exchange potentials establish benchmarks for numerical approaches in the low-dimensional science, where DFT is by far the most common tool.