797 publications from this institution
We present a Thomas-Fermi-inspired density scaling under which electron densities of atomic, molecular, or condensed matter become both large and slowly varying, so that semiclassical approximations and second-order density gradient expansions are asymptotically exact for the kinetic and exchange energies. Thus, even for atoms and molecules, density-functional approximations should recover the universal second-order gradient expansions in this limit. We also explain why common generalized gradient approximations for exchange do not.
We report a three-dimensional wave-vector analysis of the jellium exchange-correlation (xc) surface energy in the random-phase approximation (RPA). The RPA accurately describes long-range xc effects which are challenging for semilocal approximations, since it includes the universal small-wave-vector behavior derived by Langreth and Perdew. We use these rigorous RPA calculations for jellium slabs to test RPA versions of nonempirical semilocal density-functional approximations for the xc energy. The local spin density approximation displays canceling errors in the small- and intermediate-wave-vector regions. The Perdew-Burke-Ernzerhof generalized gradient approximation improves the analysis for intermediate wave vectors, but remains too low for small wave vectors (implying too-low jellium xc surface energies). The nonempirical metageneralized gradient approximation of Tao, Perdew, Staroverov, and Scuseria gives a realistic wave-vector analysis, even for small wave vectors or long-range effects. We also study the effects of slab thickness and of short-range corrections to the RPA.
We show that the poor performance of approximate Kohn-Sham (KS) density functional theory for highly charged atomic ions is improved dramatically by ensuring that (i) the exchange functional recovers the correct leading term in the $Z$ expansion of the exchange energy ($Z$ is the nuclear charge) and (ii) the correlation functional is bounded under uniform scaling of the density to the high-density limit---i.e., ${\mathrm{lim}}_{\ensuremath{\lambda}\ensuremath{\rightarrow}\ensuremath{\infty}}{E}_{\mathrm{c}}^{\mathrm{KS}}[{n}_{\ensuremath{\lambda}}]>\ensuremath{-}\ensuremath{\infty}$, where ${n}_{\ensuremath{\lambda}}(\mathbf{r})={\ensuremath{\lambda}}^{3}n(\ensuremath{\lambda}\mathbf{r})$. The performance of several density functionals (BLYP, BP86, VS98, HCTH/407, PBE, PKZB, and TPSS) is compared for the 4-, 10-, and 18-electron atomic series spanning values of $Z$ from 4 to 28. Especially accurate results are obtained with the nonempirical metageneralized gradient approximation of Tao, Perdew, Staroverov, and Scuseria (TPSS). High-$Z$ limits of selected exchange and correlation functionals are evaluated and compared with the exact values.
Tao, Perdew, Staroverov, and Scuseria (TPSS) have constructed a nonempirical meta-generalized gradient approximation (meta-GGA) [Phys. Rev. Lett. 91, 146401 (2003)] for the exchange-correlation energy, imposing exact constraints relevant to the paradigm densities of condensed matter physics and quantum chemistry. Results of their extensive tests on molecules, solids, and solid surfaces are encouraging, suggesting that this density functional achieves uniform accuracy for diverse properties and systems. In the present work, this functional is explained and details of its construction are presented. In particular, the functional is constructed to yield accurate energies under uniform coordinate scaling to the low-density or strong-interaction limit. Its nonlocality is displayed by plotting the factor F(xc) that gives the enhancement relative to the local density approximation for exchange. We also discuss an apparently harmless order-of-limits problem in the meta-GGA. The performance of this functional is investigated for exchange and correlation energies and shell-removal energies of atoms and ions. Non-self-consistent molecular atomization energies and bond lengths of the TPSS meta-GGA, calculated with GGA orbitals and densities, agree well with those calculated self-consistently. We suggest that satisfaction of additional exact constraints on higher rungs of a ladder of density functional approximations can lead to further progress.
To boost the accuracy of electronic structure calculations, the exchange-correlation energy may be constructed from the Kohn-Sham orbitals. A formally exact construction is the density-functional perturbation series, which appears to diverge for many real systems. We predict the radius of convergence and resum this series, using only exact exchange and second-order correlation plus explicit density functionals for the strong-interaction limit. Our new correlation functional, along with exact exchange, predicts atomization energies with competitive accuracy and without the usual error cancellation.
Semilocal 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 semilocal functionals, it is often worse for equilibrium properties of sp-bonded molecules and solids. Nonempirical semilocal 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 local spin density approximation (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 LSDA-SIC and about -3.5% for nonempirical generalized gradient [Perdew-Burke-Ernzerhof (PBE)-SIC] and meta-generalized gradient strongly constrained and appropriately normed (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.
The strong-interaction limit of density-functional (DF) theory is simple and provides information required for an accurate resummation of DF perturbation theory. Here we derive the point-charge-plus-continuum (PC) model for that limit, and its gradient expansion. The exchange-correlation (xc) energy ${E}_{\mathrm{xc}}[\ensuremath{\rho}]\ensuremath{\equiv}{\ensuremath{\int}}_{0}^{1}d\ensuremath{\alpha}{W}_{\ensuremath{\alpha}}[\ensuremath{\rho}]$ follows from the xc potential energies ${W}_{\ensuremath{\alpha}}$ at different interaction strengths $\ensuremath{\alpha}>~0$ [but at fixed density $\ensuremath{\rho}(\mathbf{r})].$ For small $\ensuremath{\alpha}\ensuremath{\approx}0,$ the integrand ${W}_{\ensuremath{\alpha}}$ is obtained accurately from perturbation theory, but the perturbation expansion requires resummation for moderate and large $\ensuremath{\alpha}.$ For that purpose, we present density functionals for the coefficients in the asymptotic expansion ${W}_{\ensuremath{\alpha}}\ensuremath{\rightarrow}{W}_{\ensuremath{\infty}}{+W}_{\ensuremath{\infty}}^{\ensuremath{'}}{\ensuremath{\alpha}}^{\ensuremath{-}1/2}$ for $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\alpha}}\ensuremath{\infty}$ in the PC model. ${W}_{\ensuremath{\infty}}^{\mathrm{PC}}$ arises from strict correlation, and ${W}_{\ensuremath{\infty}}^{\ensuremath{'}\mathrm{PC}}$ from zero-point vibration of the electrons around their strictly correlated distributions. The PC values for ${W}_{\ensuremath{\infty}}$ and ${W}_{\ensuremath{\infty}}^{\ensuremath{'}}$ agree with those from a self-correlation-free meta-generalized gradient approximation, both for atoms and for atomization energies of molecules. We also (i) explain the difference between the PC cell and the exchange-correlation hole, (ii) present a density-functional measure of correlation strength, (iii) describe the electron localization and spin polarization energy in a highly stretched ${\mathrm{H}}_{2}$ molecule, and (iv) discuss the soft-plasmon instability of the low-density uniform electron gas.
For exchange-correlation functionals that depend explicitly on the Kohn-Sham orbitals, the potential V(xcsigma)(r) must be obtained as the solution of the optimized effective potential (OEP) integral equation. This is very demanding and has limited the use of orbital functionals. We demonstrate that instead the OEP can be obtained iteratively by solving the partial differential equations for the orbital shifts that exactify the Krieger-Li-Iafrate approximation. Unoccupied orbitals do not need to be calculated. Accuracy and efficiency of the method are shown for atoms and clusters using the exact-exchange energy. Counterintuitive asymptotic limits of the exact OEP are presented.
Generalized gradient approximations (GGA's) seek to improve upon the accuracy of the local-spin-density (LSD) approximation in electronic-structure calculations. Perdew and Wang have developed a GGA based on real-space cutoff of the spurious long-range components of the second-order gradient expansion for the exchange-correlation hole. We have found that this density functional performs well in numerical tests for a variety of systems: (1) Total energies of 30 atoms are highly accurate. (2) Ionization energies and electron affinities are improved in a statistical sense, although significant interconfigurational and interterm errors remain. (3) Accurate atomization energies are found for seven hydrocarbon molecules, with a rms error per bond of 0.1 eV, compared with 0.7 eV for the LSD approximation and 2.4 eV for the Hartree-Fock approximation. (4) For atoms and molecules, there is a cancellation of error between density functionals for exchange and correlation, which is most striking whenever the Hartree-Fock result is furthest from experiment. (5) The surprising LSD underestimation of the lattice constants of Li and Na by 3--4 % is corrected, and the magnetic ground state of solid Fe is restored. (6) The work function, surface energy (neglecting the long-range contribution), and curvature energy of a metallic surface are all slightly reduced in comparison with LSD. Taking account of the positive long-range contribution, we find surface and curvature energies in good agreement with experimental or exact values. Finally, a way is found to visualize and understand the nonlocality of exchange and correlation, its origins, and its physical effects.
The Perdew-Zunger self-interaction correction cures many common problems associated with semilocal density functionals, but suffers from a size-extensivity problem when Kohn-Sham orbitals are used in the correction. Fermi-L\"owdin-orbital self-interaction correction (FLOSIC) solves the size-extensivity problem, allowing its use in periodic systems and resulting in better accuracy in finite systems. Although the previously published FLOSIC algorithm Pederson et al., J. Chem. Phys. 140, 121103 (2014). appears to work well in many cases, it is not fully self-consistent. This would be particularly problematic for systems where the occupied manifold is strongly changed by the correction. In this paper, we demonstrate a different algorithm for FLOSIC to achieve full self-consistency with only marginal increase of computational cost. The resulting total energies are found to be lower than previously reported non-self-consistent results.
