Adiabatic-Connection-Fluctuation-Dissipation approach to the long-range behavior of the exchange-correlation energy at metal surfaces: A numerical study for jellium slabs

A still open issue in many-body theory is the asymptotic behavior of the exchange-correlation energy and potential in the vacuum region of a metal surface. Here we report a numerical study of the position-dependent exchange-correlation energy for jellium slabs, as obtained by combining the formally exact adiabatic-connection-fluctuation-dissipation theorem with either time-dependent density-functional theory or an inhomogeneous Singwi-Tosi-Land-Sj\"olander approach. We find that the inclusion of correlation allows to obtain well-converged semi-infinite-jellium results (independent of the slab thickness) that exhibit an image-like asymptotic behavior close to the classical image potential $V_{im}(z)=-e^2/4z$.