Variational calculations of low-index crystal face-dependent surface energies and work functions of simple metals

Surface energies and work functions are calculated for the (110), (100), and (111) faces of the bcc metals Li, Ba, Na, K, Rb, and Cs and the fcc metals Al, Pb, Ca, Sr, and for the (0001) faces of the hcp metals Zn and Mg. In the Kohn-Sham energy functional employed, the crystal lattice of ions is represented by the Ashcroft pseudopotential, and nonlocal exchange-correlation energy corrections included via the wave-vector analysis method. The surface energies are determined by application of the Rayleigh-Ritz variational principle. The work functions are obtained for these energy-minimized densities by the variationally accurate, "displaced-profile change-in-self-consistent-field" (DP▵SCF) expression, which is tested for real metals here for the first time. The variational electronic densities employed are those generated by the linear-potential model, which permits the calculations to be primarily analytical. It is observed that the surface energies for each metal with the exception of Rb and Cs increase with decreasing packing density of the exposed crystal face, and that the Smoluchowski rule of decreasing work functions with decreasing packing density is obeyed by each metal except Al and Pb. An analysis of these numerical results indicates them to be superior to those of perturbation theory, and to be equivalent or generally superior to the results of other variational calculations for metals with ${r}_{s}\ensuremath{\gtrsim}3$. The Mahan-Schaich derivation for the work function of jellium metal is extended to include local ionic pseudopotentials, and the result shown to be equivalent to the DP▵SCF expression. A general expression for polycrystalline work functions is also derived, and an empirical formula given for the work function of alkali metals. It is further argued that it is meaningful, at least for the alkali metals, to compare the polycrystalline work function to the minimum work function, and this equivalence is demonstrated by comparison with experiment.