Theory of metallic clusters: Asymptotic size dependence of electronic properties

For a spherical metallic cluster of large radius R, the total energy is E={α}4{π}R³/ 3+{σ}4{π}R²+{γ}2{π}R, the chemical potential is {μ}=-W-c/R, and the first ionization energy I and electron affinity A are -{μ}±{}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 10⁶ electrons, we extract the surface energy {σ}, curvature energy {γ}, 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 R^-1 corrections to the electron density profile and electrostatic potential of a planar surface.