Evaluation of the exchange-correlation potential at a metal surface from many-body perturbation theory

We report a first-principles evaluation of the exchange-correlation potential ($V_{\mathrm{XC}}$) at a metal surface. An integral equation relating $V_{\mathrm{XC}}$ and the nonlocal electron self-energy (${\Sigma }_{\mathrm{XC}}$) is solved numerically for a free-electron metal surface, with use of a static approximation for ${\Sigma }_{\mathrm{XC}}$. The strongly inhomogeneous nature of the electron density profile at the surface is treated exactly, i.e., without invoking the usual local-density approximation. Our result for $V_{\mathrm{XC}}$ has the correct imagelike asymptotic behavior; it derives implicitly from a nonlocal exchange-correlation energy functional. We study the effect of nonlocality on the position of the effective image plane ($z_0$) from an analysis of the image tail of $V_{\mathrm{XC}}$ and also from linear-response theory. The difference in the values of $z_0$ obtained by both methods for low metallic densities is attributed to electron-overlap effects.