Publications

With more explanation than usual and without appeal to Janak's theorem, we review the statement and proof of the ionization potential theorems for the exact Kohn-Sham density-functional theory of a many-electron system: (1) For any average electron number $N$ between the integers $Z\ensuremath{-}1$ and $Z,$ and thus for $N\ensuremath{\rightarrow}Z$ from below, the highest occupied or partly occupied Kohn-Sham orbital energy is minus the ionization energy of the $Z$-electron system. (2) For $Z\ensuremath{-}1<N<Z,$ the exact Kohn-Sham effective potential ${v}_{s}(\mathbf{r})$ tends to zero as $|\mathbf{r}|\ensuremath{\rightarrow}\ensuremath{\infty}.$ We then argue that an objection to these theorems. [L. Kleinman, Phys. Rev. B 56, 12 042 (1997)] overlooks a crucial step in the proof of theorem (2): The asymptotic exponential decay of the exact electron density of the $Z$-electron system is controlled by the exact ionization energy, but the decay of an approximate density is not controlled by the approximate ionization energy. We review relevant evidence from the numerical construction of the exact Kohn-Sham potential. In particular, we point out a model two-electron problem for which the ionization potential theorems are exactly confirmed. Finally, we comment on related issues: the self-interaction correction, the discontinuity of the exact Kohn-Sham potential as $N$ passes through the integer $Z,$ and the generalized sum rule on the exchange-correlation hole.

Abstract Review: 350 refs.

Although every stationary-state density ${n}_{i}$(r\ensuremath{\rightarrow}) of a many-particle system is not an extremum of the ground-state density functional ${E}_{v}$[n], every extremum of ${E}_{v}$[n] [i.e., every solution of the Euler equation \ensuremath{\delta}${E}_{v}$/\ensuremath{\delta}n(r\ensuremath{\rightarrow})=\ensuremath{\lambda}] is a stationary-state density ${n}_{i}$(r\ensuremath{\rightarrow}). Always, ${E}_{v}$[${n}_{i}$]\ensuremath{\le}${E}_{i}$, where ${E}_{i}$ is the lowest stationary-state energy for density ${n}_{i}$(r\ensuremath{\rightarrow}); the equality holds if and only if ${n}_{i}$(r\ensuremath{\rightarrow}) is an extremum of ${E}_{v}$[n]. The extrema lying above the absolute minimum are excited-state densities which fail to be pure-state v-representable. Surprisingly, infinitesimal number-conserving density variations \ensuremath{\delta}n(r\ensuremath{\rightarrow}) about an extremum n(r\ensuremath{\rightarrow}) do not lead to energy variations \ensuremath{\delta}${E}_{v}$ of order (\ensuremath{\delta}n${)}^{2}$ when \ensuremath{\delta}n(r\ensuremath{\rightarrow})/${n}^{1/2}$(r\ensuremath{\rightarrow}) fails to be square-integrable; in fact, variations \ensuremath{\delta}${E}_{v}$ of order \ensuremath{\Vert}\ensuremath{\delta}n\ensuremath{\Vert} about the ground state are exemplified by the recently discovered ``derivative discontinuities of the energy.'' This unconventional behavior of ${E}_{v}$[n] may be traced in part to an asymptotic divergence of ${\ensuremath{\delta}}^{2}$${E}_{v}$/\ensuremath{\delta}n(r\ensuremath{\rightarrow})\ensuremath{\delta}n(r\ensuremath{\rightarrow}'). Conditions are presented under which a self-consistent solution of the Kohn-Sham single-particle problem represents an extremum of ${E}_{v}$[n]. The multiplets of the ground-state orbital configuration of the carbon atom are examined. The local-density and Langreth-Mehl approximations are found to yield a remarkably accurate account of the degeneracy of the various ground-state densities for this system, but no estimate of the multiplet splitting is obtained. Finally, aspects of v-representability are discussed, with emphasis on the iron atom.

Semi-local density functionals for the exchange-correlation energy of a many-electron system cannot be exact for all one-electron densities. In 1981, Perdew and Zunger (PZ) subtracted the fully-nonlocal self-interaction error orbital-by-orbital, making the corrected functional exact for all collections of separated one-electron densities, and making no correction to the exact functional. Although the PZ self-interaction correction (SIC) eliminates many errors of semi-local functionals, it is often worse for equilibrium properties of sp-bonded molecules and solids. Non-empirical semi-local functionals are usually designed to be exact for electron gases of uniform density, and thus also make 0% error for neutral atoms in the limit of large atomic number Z, but PZ SIC is not so designed. For localized SIC orbitals, we show analytically that the LSDA-SIC correlation energy per electron of the uniform gas in the high-density limit makes an error of -50% in the spin-unpolarized case, and -100% in the fully-spin-polarized case. Then we extrapolate from the Ne, Ar, Kr, and Xe atoms to estimate the relative errors of the PZ SIC exchange-correlation energies (with localized SIC orbitals) in the limit of large atomic number: about +5.5% for the local spin density approximation (LSDA-SIC), and about -3.5% for nonempirical generalized gradient (PBE-SIC) and meta-generalized gradient (SCAN-SIC) approximations. The SIC errors are considerably larger than those that have been estimated for LSDA-SIC by approximating the localized SIC orbitals for the uniform gas, and may explain the errors of PZ SIC for equilibrium properties, opening the door to a generalized SIC that is more widely accurate.

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 ${r}_{s}.$

A simple and accurate analytic model is derived for the correlation energy of any atom or ion within three densityfunctional approximations based on the uniform electron gas: the local-spin-density approximation (LSD), the selfinteraction corrected (SIC) version of LSD, and the antiparallel-spin LSD of Stoll et al. The last two approximations give good results for the correlation energies of neutral atoms, in contrast to LSD which overestimates these energies by a factor of 2. However, all three approximations show an incorrect $\mathrm{ln}Z$ leading behavior when the nuclear charge $Z$ tends to infinity at fixed electron number $N$. It is hard to see how any a priori electron-gas approximation can reproduce the exact leading behavior, which is constant or linear in $Z$ according to the value of $N$.

The local-density approximation for the exchange-correlation potential understimates the fundamental band gaps of semiconductors and insulators by about 40%. It is argued here that underestimation of the gap width is also to be expected from the unknown exact potential of Kohn-Sham density-functional theory, because of derivative discontinuities of the exchange-correlation energy. The need for an energy-dependent potential in band theory is emphasized. The center of the gap, however, is predicted exactly by the Kohn-Sham band structure.

In Table II, ⌬ c,srLSD for CH 4 should be Ϫ0.8 kcal/mole, not 4.3.This is a typographical error and does not affect our results or conclusions.

The ground-state energy and density of a many-electron system are often calculated by Kohn-Sham density functional theory. We describe a ladder of approximations for the exchange-correlation energy as a functional of the electron density. At the lowest rung of this ladder, the contribution to the energy from a volume element of 3-dimensional space is determined by the local density there. Higher rungs or levels incorporate increasingly complex ingredients constructed from the density or the Kohn-Sham orbitals in or around this volume element. We identify which additional exact conditions can be satisfied at each level, and discuss the extent to which the functionals at each level may be constructed without empirical input. We also discuss the research that remains to be done at the exact-exchange level, and present our “dreams of a final theory.” “Jacob left Beer-sheba and went toward Haran. He came to a certain place and stayed there for the night, because the sun had set. Taking one of the stones of the place, he put it under his head and lay down in that place. And he dreamed that there was a ladder set up on the earth, the top of it reaching to heaven; and the angels of God were ascending and descending on it.”

The accuracy and efficiency of a density functional is dependent on the basic ingredients it uses and how the ingredients are built into the functional as a whole. An iso-orbital indicator based on the electron density, its gradients, and the kinetic energy density, has proven an essential dimensionless variable that allows density functionals to recognize and correctly treat various types of chemical bonding, both strong and weak. Density functionals constructed around the iso-orbital indicator usually require dense real-space grids for numerical implementation that deteriorate computational efficiency, with poor grid convergence compromising the improved accuracy. Here, an improved iso-orbital indicator is proposed based on the same ingredients that retains the capability to identify the same chemical bonds while significantly relieving the requirement of dense grids. Furthermore, the improved iso-orbital indicator gives an improved recognition for tail regions of electron densities and is divergence-free for the exchange-correlation potential. The improved iso-orbital indicator is therefore expected to be the prime choice for further density functional development.