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The random phase approximation (RPA) stands on the top rung of the ladder of ground-state density functional approximations. The simple or direct RPA has been found to predict accurately many isoelectronic energy differences. A nonempirical local or semilocal correction to this direct RPA leaves isoelectronic energy differences almost unchanged, while improving total energies, ionization energies, etc., but fails to correct the RPA underestimation of molecular atomization energies. Direct RPA and its semilocal correction may miss part of the middle-range multicenter nonlocality of the correlation energy in a molecule. Here we propose a fully nonlocal, hybrid-functional-like addition to the semilocal correction. The added full nonlocality is important in molecules, but not in atoms. Under uniform-density scaling, this fully nonlocal correction scales like the second-order-exchange contribution to the correlation energy, an important part of the correction to direct RPA, and like the semilocal correction itself. For the atomization energies of ten molecules, and with the help of one fit parameter, it performs much better than the elaborate second-order screened exchange correction. [ABSTRACT FROM AUTHOR]

The Perdew-Burke-Ernzerhof (PBE) generalized gradient approximation for the exchange-correlation energy functional has two nonempirical constructions, based on satisfaction of universal exact constraints on the hole density or on the energy. We show here that, by identifying one possible free parameter in exchange and a second in correlation, we can continue to satisfy these constraints while diminishing the gradient dependence almost to zero (i.e., almost recovering the local spin density approximation or LSDA). This points out the important role played by the Perdew-Wang 1991 nonempirical hole construction in shaping PBE and later constructions. Only the undiminished PBE is good for atoms and molecules, for reasons we present, but a somewhat diminished PBE could be useful for solids; in particular, the surface energies of solids could be improved. Even for atoms and molecules, a strongly diminished PBE works well when combined with a scaled-down self-interaction correction (although perhaps not significantly better than LSDA). This shows that the undiminished gradient dependence of PBE and related functionals works somewhat like a scaled-down self-interaction correction to LSDA. [ABSTRACT FROM AUTHOR]

The common density functionals for the exchange-correlation energy make serious self-interaction errors in the molecular dissociation limit when real or spurious noninteger electron numbers N are found on the dissociation products. An “M-electron self-interaction-free” functional for positive integer M is one that produces a realistic linear variation of total energy with N in the range of M-1≤Na=M, and so can avoid these errors. This desideratum is a natural generalization to all M of the more familiar one of one-electron self-interaction freedom. The intent of this paper is not to advocate for any functional, but to understand what is required for a functional to be M-electron self-interaction-free and thus correct even for highly stretched bonds. The original Perdew-Zunger self-interaction correction (SIC) and our scaled-down variant of it are exactly one- and nearly two-electron self-interaction-free, but only the former is nearly so for atoms with M>2. Thus all these SIC’s produce an exact binding energy curve for H2+, and an accurate one for He2+, but only the unscaled Perdew-Zunger SIC produces an accurate one for Ne2+, where there are more than two electrons on each fragment Ne+0.5. We also discuss LiH+, which is relatively free from self-interaction errors. We suggest that the ability of the original and unscaled Perdew-Zunger SIC to be nearly M-electron self-interaction-free for atoms of all M stems in part from its formal resemblance to the Hartree-Fock theory, with which it shares a sum rule on the exchange-correlation hole of an open system. [ABSTRACT FROM AUTHOR]

Semilocal density functional approximations for the exchange-correlation energy can improperly dissociate a neutral molecule XY (Y≠X) to fractionally charged fragments X+q...Y-q with an energy significantly lower than X0...Y0. For example, NaCl can dissociate to Na+0.4...Cl-0.4. Generally, q is positive when the lowest-unoccupied orbital energy of atom Y0 lies below the highest-occupied orbital energy of atom X0. The first 24 open sp-shell atoms of the Periodic Table can form 276 distinct unlike pairs XY, and in the local spin density approximation 174 of these display fractional-charge dissociation. Finding these lowest-energy solutions with standard quantum chemistry codes, however, requires special care. Self-interaction-corrected (SIC) semilocal approximations are exact for one-electron systems and also reduce the spurious fractional charge q. The original SIC of Perdew and Zunger typically reduces q to 0. A scaled-down SIC with better equilibrium properties sometimes fails to reduce q all the way to 0. The desideratum of “many-electron self-interaction freedom” is introduced as a generalization of the one-electron concept. [ABSTRACT FROM AUTHOR]

Semilocal density functional approximations (DFAs) for the exchange-correlation energy suffer from self-interaction error, which is believed to be the cause of many of the failures of common DFAs, such as poor description of charge transfer and transition states of chemical reactions. The standard self-interaction correction (SIC) of Perdew and Zunger mends some of these failures but spoils such essential properties as thermochemistry and equilibrium bond lengths. The Perdew-Zunger SIC seems to overcorrect many-electron systems. In this paper, we propose a modified SIC, which is scaled down in many-electron regions. The new SIC has an improved performance for many molecular properties, including total energies, atomization energies, barrier heights of chemical reactions, ionization potentials, electron affinities, and bond lengths. The local spin-density approximation (LSDA) benefits from SIC more than higher-level functionals do. The scaled-down SIC has only one adjustable parameter. Rationalization of the optimal value of this parameter enables us to construct an almost-nonempirical version of the scaled-down SIC-LSDA, which is significantly better than uncorrected LSDA and even better than the uncorrected generalized gradient approximation. We present an analysis of the formal properties of the scaled-down SIC and define possible directions for further improvements. In particular, we find that exactness for all one-electron densities does not guarantee correct asymptotics for the exchange-correlation potential of a many-electron system. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

While most molecules and solids are spin-unpolarized, most chemically-active atoms are partly spin-polarized. As a result, the errors of the spin-dependence of a density functional are much more troublesome for atomization energies than they are for typical reaction or formation energies. This observation explains why the atomization energy errors of approximate functionals do not correlate with their other errors, and why the errors of atomization energies for a given functional can be radically reduced by fitting the energies of the atoms. We present an illustrative example from the recent nonempirical construction of the SCAN meta-generalized gradient approximation. [ABSTRACT FROM AUTHOR]

For accurate thermochemical tests of electronic structure theory, accurate true anharmonic zero-point vibrational energies ZPVEtrue are needed. We discuss several possibilities to extract this information for molecules from density functional or wave function calculations and/or available experimental data: (1) Empirical universal scaling of density-functional-calculated harmonic ZPVEharms, where we find that polyatomics require smaller scaling factors than diatomics. (2) Direct density-functional calculation by anharmonic second-order perturbation theory PT2. (3) Weighted averages of harmonic ZPVEharm and fundamental ZPVEfund (from fundamental vibrational transition frequencies), with weights (3/4, 1/4) for diatomics and (5/8,3/8) for polyatomics. (4) Experimental correction of the PT2 harmonic contribution, i.e., the estimate ZPV + (ZPV − ZPV) for ZPVEtrue. The (5/8,3/8) average of method 3 and the additive correction of method 4 have been proposed here. For our database of experimental ZPVEtrue, consisting of 27 diatomics and 8 polyatomics, we find that methods 1 and 2, applied to the popular B3LYP and the nonempirical PBE and TPSS functionals and their one-parameter hybrids, yield polyatomic errors on the order of 0.1 kcal/mol. Larger errors are expected for molecules larger than those in our database. Method 3 yields errors on the order of 0.02 kcal/mol, but requires very accurate (e.g., experimental, coupled cluster, or best-performing density functional) input harmonic ZPVEharm. Method 4 is the best-founded one that meets the requirements of high accuracy and practicality, requiring as experimental input only the highly accurate and widely available ZPV and producing errors on the order of 0.05 kcal/mol that are relatively independent of functional and basis set. As a part of our study, we also test the ability of the density functionals to predict accurate equilibrium bond lengths and angles for a data set of 21 mostly polyatomic molecules (since all calculated ZPVEs are evaluated at the correspondingly calculated molecular geometries). [ABSTRACT FROM AUTHOR]

In the assessment of density functional approximations for the exchange‐correlation energy, great weight is usually given to the accuracy of molecular atomization energies, or the enthalpies of formation constructed from calculated atomization energies. Here we show that a recent nonempirical functional, the meta‐generalized gradient approximation of Tao, Perdew, Staroverov, and Scuseria, achieves remarkably accurate atomization energies even for the larger organic molecules of the G3‐3 test set. But we also present strong evidence that most of the error of previous nonempirical functionals resides in the energy of the free atom and so cancels out of typical reaction energies. Finally, we suggest that enthalpies of formation calculated without any reference to the free atoms would provide a fairer assessment of the performance of approximate density functionals. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2005 [ABSTRACT FROM AUTHOR]

The common density functionals for the exchange-correlation energy make serious self-interaction errors in the molecular dissociation limit when real or spurious noninteger electron numbers N are found on the dissociation products. An “M-electron self-interaction-free” functional for positive integer M is one that produces a realistic linear variation of total energy with N in the range of M-1≤Na=M, and so can avoid these errors. This desideratum is a natural generalization to all M of the more familiar one of one-electron self-interaction freedom. The intent of this paper is not to advocate for any functional, but to understand what is required for a functional to be M-electron self-interaction-free and thus correct even for highly stretched bonds. The original Perdew-Zunger self-interaction correction (SIC) and our scaled-down variant of it are exactly one- and nearly two-electron self-interaction-free, but only the former is nearly so for atoms with M>2. Thus all these SIC’s produce an exact binding energy curve for H2+, and an accurate one for He2+, but only the unscaled Perdew-Zunger SIC produces an accurate one for Ne2+, where there are more than two electrons on each fragment Ne+0.5. We also discuss LiH+, which is relatively free from self-interaction errors. We suggest that the ability of the original and unscaled Perdew-Zunger SIC to be nearly M-electron self-interaction-free for atoms of all M stems in part from its formal resemblance to the Hartree-Fock theory, with which it shares a sum rule on the exchange-correlation hole of an open system. [ABSTRACT FROM AUTHOR]

The Tao-Perdew-Staroverov-Scuseria (TPSS) meta-generalized-gradient-approximation (MGGA) and its revised version, the revTPSS, are implemented self-consistently within the framework of the projector-augmented-wave (PAW) method, using a plane wave basis set. Both TPSS and revTPSS yield accurate atomization energies for the molecules in the AE6 set, better than those of the standard Perdew-Burke-Ernzerhof (PBE) generalized-gradient-approximation. For lattice constants and bulk moduli of 20 diverse solids, revTPSS performs much better than PBE, and on average as well as PBEsol and Armiento-Mattsson (AM05), GGAs designed for solids. The latter two overestimate the atomization energies for molecules to an unacceptable degree. However, the revTPSS presents only a slight improvement over PBEsol for the prediction of cohesive energies for solids, and some deterioration with respect to PBE. We also study the magnetic properties of Fe, for which both TPSS and revTPSS predict the right ground-state solid phase, the ferromagnetic body-centered-cubic (bcc) structure, with an accurate magnetic moment. [ABSTRACT FROM AUTHOR]

Computationally efficient semilocal approximations of density functional theory at the level of the local spin density approximation (LSDA) or generalized gradient approximation (GGA) poorly describe weak interactions. We show improved descriptions for weak bonds (without loss of accuracy for strong ones) from a newly developed semilocal meta-GGA (MGGA), by applying it to molecules, surfaces, and solids. We argue that this improvement comes from using the right MGGA dimensionless ingredient to recognize all types of orbital overlap. [ABSTRACT FROM AUTHOR]

The van der Waals coefficients between quasispherical nanostructures can be modeled accurately and analytically by those of classical solid spheres (for nanoclusters) or spherical shells (for fullerenes) of uniform valence electron density, with the true static dipole polarizability. Here, we derive analytically and confirm numerically from this model the size dependencies of the van der Waals coefficients of all orders, showing, for example, that the asymptotic dependence for C6 is the expected n2 for pairs of nanoclusters An-An, each containing n atoms, but n2.75 for pairs of single-walled fullerenes Cn-Cn. Large fullerenes are argued to have much larger polarizabilities and dispersion coefficients than those predicted by either the standard atom pair-potential model or widely used nonlocal van der Waals correlation energy functionals. [ABSTRACT FROM AUTHOR]