Semilocal approximations to the Kohn-Sham exchange potential as applied to a metal surface

Several semilocal exchange potentials usually employed in the framework of density-functional theory (DFT) are tested and compared with their exact counterpart, Optimized Effective Potential (OEP), as applied to jellium-slab model a metal-vacuum interface. Driven by explicit dependence on ground-state density, its gradient, kinetic-energy three analyzed approach respective asymptotic limits faster than case OEP, all them having an scaling form $-\alpha\,e^2/z + V_{\infty}$, $\alpha < 1$. Here we provide leading analytic asymptotics under study, find that none exhibits OEP slab $-\;e^2/z$. While so-called Becke-Roussel potential's asymptote is close other two study material-dependent positive constant value far into vacuum, resulting considerably overestimated ionization potentials.