Surface and curvature energies from jellium spheres: Density functional hierarchy and quantum Monte Carlo

We consider spherical jellium clusters with up to 200 electrons as a testing ground for density functional approximations to the exchange-correlation energy of a many-electron ground state. As nearly-exact standards, we employ Hartree--Fock energies at the exchange-only level and the diffusion Monte Carlo (DMC) energies of Sottile and Ballone (2001) at the correlated level. The density functionals tested are the local spin density (LSD), generalized gradient (GGA), and meta-generalized gradient (meta-GGA) approximations; the latter gives the most accurate results. By fitting the deviation from the LSD energy of closed-shell clusters to the predictions of the liquid drop model, we extract the exchange-correlation surface energies and curvature energies of a semi-infinite jellium from the energies of finite clusters. For the density functionals, the surface energies so extracted agree closely with those calculated directly for a single planar surface. But for the diffusion Monte Carlo method, the surface energies so extracted are considerably lower (and we suspect more accurate) than those extrapolated by Acioli and Ceperley (1996) from their DMC supercell calculations. The errors of the LSD, GGA, and meta-GGA surface and curvature energies are estimated, and are found to be consistently small for both properties only at the meta-GGA level. These errors are qualitatively related to relative performances of the various density functionals for the calculation of atomization energies: the proper self-interaction correction to the LSD for a one-electron atom is in the curvature energy (as it is in meta-GGA), not in the surface energy (as it is in GGA). Additionally, a formula is given for the interpolation and extrapolation of the surface energy ${\ensuremath{\sigma}}_{\mathrm{xc}}$ as a function of the bulk density parameter ${r}_{s}.$