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

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