Surfaces of real metals by the variational self-consistent method

We present a self-consistent calculation of the ground state of the metallic planar surface in which the discrete-lattice perturbation $\delta v\left(\overrightarrow{\mathrm{r}}\right)$ is treated variationally, with use of a single variational parameter, rather than perturbatively. Our calculation, which reduces to the perturbation theory of Lang and Kohn in the limit of weak $\delta v\left(\overrightarrow{\mathrm{r}}\right)$ and retains most of the simplicity and broad utility of their approach, shows that for many metal surfaces $\delta v\left(\overrightarrow{\mathrm{r}}\right)$ is not a weak perturbation: The electron density profiles at real metal surfaces are often unlike those of the jellium model and in fact show a strong dependence on the choice of exposed crystallographic face. This face dependence, which is rather simply related to the average value ${〈\delta v〉}_{\mathrm{av}}$ of the discrete-lattice perturbation over the volume of the semi-infinite crystal may have important consequences in the calculation of many surface-related properties, including chemisorption, to which our method is easily applicable. We calculate the face-dependent surface energies, density profiles, and work functions of nine simple metals. Calculated surface energies, including the correction to the local-density approximation for exchange and correlation given by the method of wave-vector analysis, are in good agreement with measured surface tensions for those seven of the nine metals in which the ionic pseudopotential gives a good account of the bulk binding energy. Problems still to be considered include improvements in the pseudopotential, lattice relaxation at the surface, and variation of the electron density over planes parallel to the surface.