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

Abstract

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 ͑DMC͒ energies of Sottile and Ballone ͑2001͒ at the correlated level. The density functionals tested are the local spin density ͑LSD͒, generalized gradient ͑GGA͒, and meta-generalized gradient ͑meta-GGA͒ 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 accurate͒ than those extrapolated by Acioli and Ceperley ͑1996͒ 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-GGA͒, not in the surface energy ͑as it is in GGA͒. Additionally, a formula is given for the interpolation and extrapolation of the surface energy xc as a function of the bulk density parameter r s .

Metal surface energy: Persistent cancellation of short-range correlation effects beyond the random phase approximation

“…Usually GGA works better than LDA, but not for the jellium surface energy where long-range effects are especially important. In the present work, as in four others, 10,31,34,35 we have found that the jellium xc surface energy is only a few percent higher than it is in LDA. These closely-agreeing methods include two different short-range corrections to RPA (present work using a non-local xc kernel and Ref.…”

“…10), extraction of a surface energy from DMC energies for jellium spheres (Ref. 34), and a meta-GGA density functional (Ref. 7).…”

Extraction of aluminium surface energies from slab calculations: perturbative and non-perturbative approaches

“…On the other hand, GGA-PBE surface energies are generally lower [14]. This paradox has been discussed by Almeida et al [15]. Meta-GGA-PKZB has been employed to improve the GGA surface energy of the jellium model [14].…”

Simple impurity embedded in a spherical jellium: Approximations of density functional theory compared to quantum Monte Carlo benchmarks

“…These results have been later compared against several DFT approximations 19,20 . Surface correlation energies 18 and surface exchange-correlation energies 19,20 have been obtained using extrapolated values to infinite size.…”

“…These results have been later compared against several DFT approximations 19,20 . Surface correlation energies 18 and surface exchange-correlation energies 19,20 have been obtained using extrapolated values to infinite size. In general, the values obtained for correlation surface energies obtained with spheres are consistent with the latest DMC results for jellium slabs 21 and other methods, but differ to some extent from the original DMC results from slab geometries 22 .…”