Density-functional theory is well known to rely on approximations to the unknown exchange-correlation ($xc$) energy functional, which can be constructed from the satisfaction of a number of known properties. One of those properties is the asymptotic behavior of exchange and correlation in localized or extended many-electron systems. However, the actual asymptotics of the $xc$ energy and potential in the vacuum region of a metal surface have remained elusive over the years. Here, we report what we consider to be a final word for the asymptotics of the $xc$ energy per particle ${ϵ}_{xc}[n](z)$ and the $xc$ Kohn-Sham potential $v_{xc}[n](z)$ in the vacuum side of both an extended semi-infinite metal and a localized metal slab, which are always universal (i.e., electron-density independent), negative, and inversely proportional to the distance from the surface. We find that, contrary to previous conclusions by other authors, while for extended systems the asymptotics are dominated by a type of correlation, for metal slabs they are dominated by exchange, as in the case of systems that are finite in all directions, like atoms, molecules, and metal clusters.

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• Approximation methods for many-body systems
• Density functional theory
• First-principles calculations