  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.
