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

The asymptotic structure of the Kohn-Sham exchange potential v(x) (r) in the classically forbidden region of a metal surface is investigated, together with that of the Slater exchange potential V-x(S) (r) and those of the approximate Krieger-Li-Iafrate V-x(KLI) (r) and Harbola-Sahni W-x(r) exchange potentials. Particularly, the former is shown to have the form of v(x)(z ->infinity) = -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, a controversy raised recently gets resolved. The structure of the exchange hole rho(x) (r,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)(r), V-x(S) (r), V-x(KLI) (r), and W-x(r) at a slab metal surface are also investigated. Particularly, v(x)(z ->infinity)=-1/z in the slab case. The distinction, in this respect, between the semi-infinite and the slab metal surfaces is elucidated.