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

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.