Theory of metallic clusters: Asymptotic size dependence of electronic properties

For a spherical metallic cluster of large radius R, the total energy is E=\ensuremath{\alpha}4\ensuremath{\pi}${\mathit{R}}^{3}$/ 3+\ensuremath{\sigma}4\ensuremath{\pi}${\mathit{R}}^{2}$+\ensuremath{\gamma}2\ensuremath{\pi}R, the chemical potential is \ensuremath{\mu}=-W-c/R, and the first ionization energy I and electron affinity A are -\ensuremath{\mu}\ifmmode\pm\else\textpm\fi{}1/2(R+d). By solving the Euler equation within the Thomas-Fermi-Dirac-Gombas-Weizs\"acker-4 approximation for jellium spheres with up to ${10}^{6}$ electrons, we extract the surface energy \ensuremath{\sigma}, curvature energy \ensuremath{\gamma}, work function W, and constants c and d. The constant c is not zero, but neither is it -1/8, the prediction of the image-potential argument. We trace c to the second- and fourth-order density-gradient terms in the kinetic energy, which are present even in systems with no image potential. However, the constant d is found to be the distance from a planar surface to its image plane. In the absence of shell-structure oscillations, the asymptotic forms hold accurately even for very small clusters; this fact suggests a way to extract the curvature energy of a real metal from its surface and monovacancy-formation energies. We also discuss asymptotic ${\mathit{R}}^{\mathrm{\ensuremath{-}}1}$ corrections to the electron density profile and electrostatic potential of a planar surface.