Nonlocal relation between kinetic and exchange energy densities in the Bardeen model of a planar metal surface

The infinite barrier model of a planar metal surface, going back to Bardeen, is shown to permit the kinetic energy density to be written explicitly in terms of the ground-state electron density and its first derivative. The exchange energy density is then written in what proves to be an explicitly nonlocal form in terms of the electron density. Finally, by integrating the first-order equation for the electron density in terms of the kinetic energy density, a precise relation is established in this metal surface model between exchange and kinetic energy densities.