Localized versus extended systems in density functional theory: Some lessons from the Kohn-Sham exact exchange potential

A long-standing puzzle in density functional theory is the issue of the long-range behavior of the Kohn-Sham exchange-correlation potential at metal surfaces. As an important step toward its solution, it is proven here, through a rigorous asymptotic analysis and an accurate numerical solution of the optimized-effective-potential integral equation, that the Kohn-Sham exact exchange potential decays as $\text{ln}(z)/z$ far into the vacuum side of an extended semi-infinite jellium. In contrast with the situation in localized systems, such as atoms, molecules, and slabs, this dominant contribution does not arise from the so-called Slater potential. This exact exchange result provides a strong constraint on the suitability of approximate correlation-energy functionals.