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We report a three-dimensional wave-vector analysis of the jellium exchange-correlation (xc) surface energy in the random-phase approximation (RPA). The RPA accurately describes long-range xc effects which are challenging for semilocal approximations, since it includes the universal small-wave-vector behavior derived by Langreth and Perdew. We use these rigorous RPA calculations for jellium slabs to test RPA versions of nonempirical semilocal density-functional approximations for the xc energy. The local spin density approximation displays canceling errors in the small- and intermediate-wave-vector regions. The Perdew-Burke-Ernzerhof generalized gradient approximation improves the analysis for intermediate wave vectors, but remains too low for small wave vectors (implying too-low jellium xc surface energies). The nonempirical metageneralized gradient approximation of Tao, Perdew, Staroverov, and Scuseria gives a realistic wave-vector analysis, even for small wave vectors or long-range effects. We also study the effects of slab thickness and of short-range corrections to the RPA.
Pitarke et al. (2006) studied this question.