Spherical voids in the stabilized jellium model: Rigorous theorems and Padé representation of the void-formation energy

We consider the energy needed to form a spherical hole or void in a simple metal, modeled as ordinary jellium or stabilized jellium. (Only the latter model correctly predicts positive formation energies for voids in high-density metals.) First we present two Hellmann-Feynman theorems for the void-formation energy 4\ensuremath{\pi}${\mathit{R}}^{2}$${\mathrm{\ensuremath{\sigma}}}_{\mathit{R}}^{\mathit{v}}$(n\ifmmode\bar\else\textasciimacron\fi{}) as a function of the void radius R and the positive-background density n\ifmmode\bar\else\textasciimacron\fi{}, which may be used to check the self-consistency of numerical calculations. They are special cases of more-general relationships for partially emptied or partially stabilized voids. The difference between these two theorems has an analog for spherical clusters. Next we link the small-R expansion of the void surface energy (from perturbation theory) with the large-R expansion (from the liquid drop model) by means of a Pad\'e approximant without adjustable parameters. For a range of sizes (including the monovacancy and its ``antiparticle,'' the atom), we compare void formation energies and cohesive energies calculated by the liquid drop expansion (sum of volume, surface, and curvature energy terms), by the Pad\'e form, and by self-consistent Kohn-Sham calculations within the local-density approximation, against experimental values. Thus we confirm that the domain of validity of the liquid drop model extends down almost to the atomic scale of sizes. From the Pad\'e formula, we estimate the next term of the liquid drop expansion beyond the curvature energy term. The Pad\'e form suggests a ``generalized liquid drop model,'' which we use to estimate the edge and step-formation energies on an Al (111) surface.