Exchange potential from functional differentiation in self-consistent field theory of a jellium-type metal surface

A form of the exchange potential for the jellium metal surface is derived through functional differentiation of the exchange-energy functional by considering variations restricted to a particular class of densities. The potential satisfies the virial-theorem sum rule, possesses the correct scaling property, and satisfies the second-derivative condition. It is the sum of two parts: the first being explicitly written in terms of the Slater potential created by the Fermi hole, the electronic density, and their gradients. This contribution is shown to contain already the exact asymptotic properties of the full exchange potential, viz., the image potential structure in the vacuum and the Gasper-Kohn-Sham value in the metal bulk. As a consequence of the restricted variations, the second contribution is not explicitly known. Nevertheless, it is short ranged, satisfies itself an important sum rule, and has the required scaling property. For slowly varying densities, it depends upon terms of O(∇ρ,${\mathrm{\nabla }}^2$ρ). The known component itself also satisfies the virial theorem and the scaling condition. Furthermore, within an approximation for the functional derivative of the Slater potential, this component satisfies the second functional derivative condition as well as the sum rule relating the exchange potential to its functional derivative.