Asymptotic behavior of the Kohn-Sham exchange potential at a metal surface

TL;DR

This paper investigates the asymptotic behavior of various exchange potentials at metal surfaces, confirming the -α/x decay form and resolving recent controversies, with implications for surface electronic structure modeling.

Contribution

It provides a detailed analysis of the asymptotic forms of exchange potentials at metal surfaces, confirming the -α/z decay and clarifying differences between semi-infinite and slab models.

Findings

Exchange potential v_x(z) behaves as -α_x/z at large distances.

The asymptotic form v_x(z) = -1/z is confirmed for slab metals.

Delocalization of the exchange hole in the bulk is demonstrated.

Abstract

The asymptotic structure of the Kohn-Sham exchange potential $v_x ({\bf r})$ in the classically forbidden region of a metal surface is investigated, together with that of the Slater exchange potential $V_x^S ({\bf r})$ and those of the approximate Krieger-Li-Iafrate $V_x^{KLI} ({\bf r})$ and Harbola-Sahni $W_x ({\bf r})$ exchange potentials. Particularly, the former is shown to have the form of $v_{x} (z \to \infty) = - \alpha_x / z$ with $\alpha_x$ a constant dependent only of bulk electron density. The same result in previous work is thus confirmed; in the meanwhile controversy raised recently gets resolved. The structure of the exchange hole $\rho_x ({\bf r}, {\bf r}')$ is examined, and the delocalization of it in the metal bulk when the electron is at large distance from the metal surface is demonstrated with analytical expressions. The asymptotic structures of $v_x ({\bf r})$, $V_x^S ({\bf r})$, $V_x^{KLI} ({\bf r})$, and $W_x ({\bf r})$ at a slab metal surface are also investigated. Particularly, $v_{x} (z \to \infty) = - 1/ z$ in the slab case. The distinction in this respect between the semi-infinite and the slab metal surfaces is elucidated.