Standard spin-density functionals for the exchange-correlation energy of a many-electron ground state make serious self-interaction errors which can be corrected by the Perdew-Zunger self-interaction correction (SIC). We propose a size-extensive construction of SIC orbitals which, unlike earlier constructions, makes SIC computationally efficient, and a true spin-density functional. The SIC orbitals are constructed from a unitary transformation that is explicitly dependent on the non-interacting one-particle density matrix. When this SIC is applied to the local spin-density approximation, improvements are found for the atomization energies of molecules.
Density functional theory (DFT) has been extensively used to model the properties of water. Albeit maintaining a good balance between accuracy and efficiency, no density functional has so far achieved the degree of accuracy necessary to correctly predict the properties of water across the entire phase diagram. Here, we present density-corrected SCAN (DC-SCAN) calculations for water which, minimizing density-driven errors, elevate the accuracy of the SCAN functional to that of “gold standard” coupled-cluster theory. Building upon the accuracy of DC-SCAN within a many-body formalism, we introduce a data-driven many-body potential energy function, MB-SCAN(DC), that quantitatively reproduces coupled cluster reference values for interaction, binding, and individual many-body energies of water clusters. Importantly, molecular dynamics simulations carried out with MB-SCAN(DC) also reproduce the properties of liquid water, which thus demonstrates that MB-SCAN(DC) is effectively the first DFT-based model that correctly describes water from the gas to the liquid phase.
For the surface energy of jellium at alkali-metal densities, the local-density approximation (LDA) and more advanced density-functional methods disagree strongly with the wave-function-based Fermi hypernetted-chain and diffusion Monte Carlo methods. We present a wave-vector interpolation correction to the generalized gradient approximation which gives jellium surface energies consistent with two other estimates based on advanced density functionals. LDA makes compensating errors at intermediate and small wave vectors. Studies of small jellium clusters also support the density-functional estimate for the jellium surface energy.
We present an analysis of local or semilocal density functionals for the exchange-correlation energy by decomposing them into their gradients rs (local Seitz radius), ζ (relative spin polarization), and s (reduced density gradient). We explain the numerical method pertaining to this kind of analysis and present results for a few atoms and ions. The atomic shell structure is prominent, and only the ranges 0 < rs < 10 and 0 < s < 3 are important. The low-density and large-gradient domains, where the approximations for the exchange-correlation energy are least trustworthy, have very little weight. © 1997 John Wiley & Sons, Inc.
We present the case for the nonempirical construction of density functional approximations for the exchange-correlation energy by the traditional method of "constraint satisfaction" without fitting to data sets, and present evidence that this approach has been successful on the first three rungs of "Jacob's ladder" of density functional approximations [local spin-density approximation (LSD), generalized gradient approximation (GGA), and meta-GGA]. We expect that this approach will also prove successful on the fourth and fifth rungs (hyper-GGA or hybrid and generalized random-phase approximation). In particular, we argue for the theoretical and practical importance of recovering the correct uniform density limit, which many semiempirical functionals fail to do. Among the beyond-LSD functionals now available to users, we recommend the nonempirical Perdew-Burke-Ernzerhof (PBE) GGA and the nonempirical Tao-Perdew-Staroverov-Scuseria (TPSS) meta-GGA, and their one-parameter hybrids with exact exchange. TPSS improvement over PBE is dramatic for atomization energies of molecules and surface energies of solids, and small or moderate for other properties. TPSS is now or soon will be available in standard codes such as GAUSSIAN, TURBOMOLE, NWCHEM, ADF, WIEN, VASP, etc. We also discuss old and new ideas to eliminate the self-interaction error that plagues the functionals on the first three rungs of the ladder, bring up other related issues, and close with a list of "do's and don't's" for software developers and users.
Ground-state Kohn-Sham density functional theory provides, in principle, the exact ground-state energy and electronic spin densities of real interacting electrons in a static external potential. In practice, the exact density functional for the exchange-correlation (xc) energy must be approximated in a computationally efficient way. About 20 mathematical properties of the exact xc functional are known. In this work, we review and discuss these known constraints on the xc energy and hole. By analyzing a sequence of increasingly sophisticated density functional approximations (DFAs), we argue that (<i>a</i>) the satisfaction of more exact constraints and appropriate norms makes a functional more predictive over the immense space of many-electron systems and (<i>b</i>) fitting to bonded systems yields an interpolative DFA that may not extrapolate well to systems unlike those in the fitting set. We discuss both how the class of well-described systems has grown along with constraint satisfaction and the possibilities for future functional development.
High-quality colloidal CdTe quantum wires having purposefully controlled diameters in the range 5-11 nm are grown by the solution-liquid-solid (SLS) method, using Bi nanoparticle catalysts, cadmium octadecylphosphonate and trioctylphosphine telluride as precursors, and a TOPO solvent. The wires adopt the wurtzite structure and grow along the [002] direction (parallel to the c axis). The size dependence of the effective band gaps in the wires is determined from the absorption spectra and compared to the experimental results for high-quality CdTe quantum dots. In contrast to the predictions of an effective-mass approximation, particle-in-a-box model, and previous experimental results from CdSe and InP dot-wire comparisons, the effective band gaps of CdTe dots and wires of like diameter are found to be experimentally indistinguishable. The present results are analyzed using density functional theory under the local-density approximation by implementing a charge-patching method. The higher-level theoretical analysis finds the general existence of a threshold diameter, above which dot and wire effective band gaps converge. The origin and magnitude of this threshold diameter are discussed.
We study the asymptotic expansion of the neutral-atom energy as the atomic number Z-->infinity, presenting a new method to extract the coefficients from oscillating numerical data. Recovery of the correct expansion yields a condition on the Kohn-Sham kinetic energy that is important for the accuracy of approximate kinetic energy functionals for atoms, molecules, and solids. For example, this determines the small gradient limit of any generalized gradient approximation and conflicts somewhat with the standard gradient expansion. Tests are performed on atoms, molecules, and jellium clusters using densities constructed from Kohn-Sham orbitals. We also give a modern, highly accurate parametrization of the Thomas-Fermi density of neutral atoms.
The ground-state density $n$ of a many-electron system obeys a Schr\"odinger-like differential equation for ${n}^{\frac{1}{2}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})$, which may be solved by standard Kohn-Sham programs. The exact local effective (nonexternal) potential, ${v}_{\mathrm{eff}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})$, is displayed explicitly in terms of wave-function expectation values, from which ${v}_{\mathrm{eff}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})>~0$ for all $\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}$. A derivation for $n$ as $|\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}|\ensuremath{\rightarrow}\ensuremath{\infty}$ implies that this new effective potential tends asymptotically to zero, as does the exact Kohn-Sham potential, with the highest occupied eigenvalue as the exact ionization energy. A new exact expression is also presented for the exchange-correlation hole density ${\ensuremath{\rho}}_{\mathrm{xc}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}, {\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}^{\ensuremath{'}})$ about an electron at $\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}$, as $|\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}|\ensuremath{\rightarrow}\ensuremath{\infty}$.
A local Kohn-Sham potential can be constructed explicitly either in the "exchange-only" density-functional theory of Talman, which does not constrain the density to its Hartree-Fock value, or in the "Hartree-Fock" density-functional theory of von Barth, which does. The Kohn-Sham orbital energies are essentially the same by either choice, as shown here for the beryllium atom. Our conclusion still stands that the exact Kohn-Sham orbital energies for tightly bound electrons are not physical removal energies.
We construct a generalized gradient approximation (GGA) for the density {ital n}{sub xc}({ital r},{ital r}+{ital u}) at position {ital r}+{ital u} of the exchange-correlation hole surrounding an electron at {ital r}, or more precisely for its system and spherical average {l_angle}{ital n}{sub xc}({ital u}){r_angle}=(4{pi}){sup {minus}1}{integral}{ital d}{Omega}{sub {ital u}}{ital N}{sup {minus}1}{integral}{ital d}{sup 3}{ital r} {ital n}({ital r}){ital n}{sub xc}({ital r},{ital r}+{ital u}). Starting from the second-order density gradient expansion, which involves the local spin densities {ital n}{sub {up_arrow}}({ital r}),{ital n}{sub {down_arrow}}({ital r}) and their gradients {nabla}{ital n}{sub {up_arrow}}({ital r}),{nabla}{ital n}{sub {down_arrow}}({ital r}), we cut off the spurious large-{ital u} contributions to restore those exact conditions on the hole that the local spin density (LSD) approximation respects. Our GGA hole recovers the Perdew-Wang 1991 and Perdew-Burke-Ernzerhof GGA{close_quote}s for the exchange-correlation energy, which therefore respect the same powerful hole constraints as LSD. When applied to real systems, our hole model provides a more detailed test of these energy functionals, and also predicts the observable electron-electron structure factor. {copyright} {ital 1996 The American Physical Society.}
