Energy surface, chemical potentials, Kohn-Sham energies in spin-polarized density functional theory

On the basis of the zero-temperature grand canonical ensemble generalization of the energy E[N,N_s,v,B] for fractional particle N and spin N_s numbers, the energy surface over the (N,N_s) plane is displayed and analyzed in the case of homogeneous external magnetic fields B(r). The (negative of the) left/right-side derivatives of the energy with respect to N, N_up, and N_down give the fixed-N_s, spin-up, and spin-down ionization potentials/electron affinities, respectively, while the derivative of E[N,N_s,v,B] with respect to N_s gives the (signed) half excitation energy to a state with N_s increased (or decreased) by 2. The highest occupied and lowest unoccupied Kohn-Sham spin-orbital energies are identified as the corresponding spin-up and spin-down ionization potentials and electron affinities. The excitation energies to the states with N_s+2, N_s-2, can be obtained as the differences between the lowest unoccupied and the opposite-spin highest occupied spin-orbital energies, if the (N,N_s) representation of the Kohn-Sham spin-potentials is used. The cases where the convexity condition on the energy does not hold are also discussed. Finally, the discontinuities of the energy derivatives and the Kohn-Sham potential are analyzed and related.Comment: 35 pages, to appear in JCP; text made more precise, Aufbau discussed, T_s derivative discontinuities given too, two Appendices adde

Empirical analysis of the Lieb-Oxford bound in ions and molecules

Universal properties of the Coulomb interaction energy apply to all many-electron systems. Bounds on the exchange-correlation energy, inparticular, are important for the construction of improved density functionals. Here we investigate one such universal property -- the Lieb-Oxford lower bound -- for ionic and molecular systems. In recent work [J. Chem. Phys. 127, 054106 (2007)], we observed that for atoms and electron liquids this bound may be substantially tightened. Calculations for a few ions and molecules suggested the same tendency, but were not conclusive due to the small number of systems considered. Here we extend that analysis to many different families of ions and molecules, and find that for these, too, the bound can be empirically tightened by a similar margin as for atoms and electron liquids. Tightening the Lieb-Oxford bound will have consequences for the performance of various approximate exchange-correlation functionals.Comment: 8 pages, 3 color figure

Correlation energy, pair-distribution functions and static structure factors of jellium

We discuss and clarify a simple and accurate interpolation scheme for the spin-resolved electron static structure factor (and corresponding pair correlation function) of the 3D unpolarized homogeneous electron gas which, along with some analytic properties of the spin-resolved pair-correlation functions, we have just published. We compare our results with the very recent spin-resolved scheme by Schmidt et al., and focus our attention on the spin-resolved correlation energies and the high-density limit of the correlation functions.Comment: 8 pages, 3 figures. Proceedings of the conference on Statistical Mechanics and Strongly Correlated Systems (Bachelet, Parisi & Vulpiani Eds.) to appear as a special issue of Physica A (Elsevier, Amsterdam 2000

Two Avenues to Self-Interaction Correction within Kohn-Sham Theory: Unitary Invariance is the Shortcut

The most widely-used density functionals for the exchange-correlation energy are inexact for one-electron systems. Their self-interaction errors can be severe in some applications. The problem is not only to correct the self-interaction error, but to do so in a way that will not violate size-consistency and will not go outside the standard Kohn-Sham density functional theory. The solution via the optimized effective potential (OEP) method will be discussed, first for the Perdew-Zunger self-interaction correction (whose performance for molecules is briefly summarized) and then for the more modern self-interaction corrections based upon unitarily-invariant indicators of iso-orbital regions. For the latter approaches, the OEP construction is greatly simplified. The kinetic-energy-based iso-orbital indicator \tau^W_\sigma(\re)/\tau_\sigma(\re) will be discussed and plotted, along with an alternative exchange-based indicator

Nonempirical Density Functionals Investigated for Jellium: Spin-Polarized Surfaces, Spherical Clusters, and Bulk Linear Response

Earlier tests show that the Tao-Perdew-Staroverov-Scuseria (TPSS) nonempirical meta-generalized gradient approximation (meta-GGA) for the exchange-correlation energy yields more accurate surface energies than the local spin density (LSD) approximation for spin-unpolarized jellium. In this study, work functions and surface energies of a jellium metal in the presence of ``internal'' and external magnetic fields are calculated with LSD, Perdew-Burke-Ernzerhof (PBE) GGA, and TPSS meta-GGA and its predecessor, the nearly nonempirical Perdew-Kurth-Zupan-Blaha (PKZB) meta-GGA, using self-consistent LSD orbitals and densities. The results show that: (i) For normal bulk densities, the surface correlation energy is the same in TPSS as in PBE, as it should be since TPSS strives to represent a self-correlation correction to PBE; (ii) Normal surface density profiles can be scaled uniformly to the low-density or strong-interaction limit, and TPSS provides an estimate for that limit that is consistent with (but probably more accurate than) other estimates; (iii) For both normal and low densities, TPSS provides the same description of surface magnetism as PBE, suggesting that these approximations may be generally equivalent for magnetism. The energies of jellium spheres with up to 106 electrons are calculated using density functionals and compared to those obtained with Diffusion Quantum Monte Carlo data, including our estimate for the fixed-node correction. Finally we calculate the linear response of bulk jellium using these density functionals, and find that not only LSD but also PBE GGA and TPSS meta-GGA yield a linear-response in good agreement with that of the Quantum Monte Carlo method, for wavevectors of the perturbing external potential up to twice the Fermi wavevector.Comment: 14 pages, 9 figure

Localization and delocalization errors in density functional theory and implications for band-gap prediction

The band-gap problem and other systematic failures of approximate functionals are explained from an analysis of total energy for fractional charges. The deviation from the correct intrinsic linear behavior in finite systems leads to delocalization and localization errors in large or bulk systems. Functionals whose energy is convex for fractional charges such as LDA display an incorrect apparent linearity in the bulk limit, due to the delocalization error. Concave functionals also have an incorrect apparent linearity in the bulk calculation, due to the localization error and imposed symmetry. This resolves an important paradox and opens the possibility to obtain accurate band-gaps from DFT.Comment: 4 pages 4 figure

Ab initio pseudopotentials for electronic structure calculations of poly-atomic systems using density-functional theory

The package fhi98PP allows one to generate norm-conserving pseudopotentials adapted to density-functional theory total-energy calculations for a multitude of elements throughout the periodic table, including first-row and transition metal elements. The package also facilitates a first assessment of the pseudopotentials' transferability, either in semilocal or fully separable form, by means of simple tests carried out for the free atom. Various parameterizations of the local-density approximation and the generalized gradient approximation for exchange and correlation are implemented.Comment: 44 pages, 5 Postscript figures, epsfig, elsart, psfrag, submitted to Comput. Phys. Commun. Other related publications can be found at http://www.rz-berlin.mpg.de/th/paper.htm

Density-Functional-Based Determination of the CH3-CH4 Hydrogen Exchange Reaction Barrier

Due to the overbinding that is inherent in existing {\em local} approximations to the density-functional formalism, certain reaction energies have not been accessible. Since the generalized gradient approximation significantly decreases the overbinding, prospects for density-functional-based reaction dynamics are promising. Results on the generalized-gradient based determination of the CH3-CH4 hydrogen exchange reaction are presented. Including all Born-Oppenheimer effects an energy barrier of 9.5 kcal/Mole is found which is a very significant improvement over the local-density approximation.Comment: 5 twocolumn pages (needs twocolumn.sty), revtex, 3 figures, To appear in Chem.Phys.Let

Exchange and Correlation in Open Systems of Fluctuating Electron Number

While the exact total energy of a separated open system varies linearly as a function of average electron number between adjacent integers, the energy predicted by semi-local density functional approximations curves upward and the exact-exchange-only or Hartree-Fock energy downward. As a result, semi-local density functionals fail for separated open systems of fluctuating electron number, as in stretched molecular ions A$_2^{+}$ and in solid transition metal oxides. We develop an exact-exchange theory and an exchange-hole sum rule that explain these failures and we propose a way to correct them via a local hybrid functional.Comment: 4 pages, 2 figure