Magnetic exchange couplings evaluated with Rung 3.5 density functionals

Rung 3.5 exchange-correlation functionals are assessed for the calculation of magnetic exchange coupling parameters and atomic spin populations for a variety of inorganic and organic magnetic systems. Density functional theory calculations of exchange couplings sensitively depend on nonlocal contributions to the exchange-correlation functional. Semilocal functionals, Rungs 1-3 on "Jacob's Ladder" of density functional approximations, yield excessively delocalized electrons and overestimated absolute exchange couplings. Fourth-rung hybrid functionals admixing nonlocal exchange improve the results. We show that new "Rung 3.5" functionals give magnetic properties intermediate between semilocal and hybrid functionals, providing additional evidence that these functionals incorporate some desirable aspects of nonlocal exchange. Results for ferromagnetic complexes indicate areas for future improvement.     

Nonspherical model density matrices for Rung 3.5 density functionals

"Rung 3.5" exchange-correlation functionals for Kohn-Sham density functional theory depend linearly on the nonlocal one-particle density matrix of the noninteracting Kohn-Sham reference system. Rung 3.5 functionals also require a semilocal model for the one-particle density matrix. This work presents new model density matrices for Rung 3.5 functionals. The resulting functionals give reasonable predictions for total energies, molecular thermochemistry and kinetics, odd-electron bonds, and conjugated polymer bandgaps. Global-hybrid-like combinations of semilocal and Rung 3.5 exchange, and empirical density matrix models, also show promise.     

Rung 3.5 density functionals: Another step on Jacob's ladder

Applications of density functional theory (DFT) to computational chemistry and solid‐state physics rely on a “Jacob's Ladder” of progressively more complicated approximations to the many‐body exchange‐correlation (XC) density functional. Accurate, computationally tractable DFT calculations on large and periodic systems remain challenging for existing XC functionals. Simple XC functionals on the three lowest rungs of Jacob's Ladder are insufficiently accurate for many properties, while fourth‐rung hybrid functionals incorporating nonlocal information can be prohibitively expensive. This perspective presents our work toward a compromise, a new class of “Rung 3.5” functionals that incorporate a linear dependence on the nonlocal one‐particle density matrix. This work reviews these functionals' formal underpinning, numerical performance, and prospects for modeling solids and surfaces. © 2012 Wiley Periodicals, Inc.     

The AM05 density functional applied to solids

We show that the AM05 functional [Armiento and Mattsson, Phys. Rev. B 72, 085108 (2005)] has the same excellent performance for solids as the hybrid density functionals tested in Paier et al. [J. Chem. Phys. 124, 154709 (2006); 125, 249901 (2006)]. This confirms the original finding that AM05 performs exceptionally well for solids and surfaces. Hartree-Fock hybrid calculations are typically an order of magnitude slower than local or semilocal density functionals such as AM05, which is of a regular semilocal generalized gradient approximation form. The performance of AM05 is on average found to be superior to selecting the best of local density approximation and PBE for each solid. By comparing data from several different electronic-structure codes, we have determined that the numerical errors in this study are equal to or smaller than the corresponding experimental uncertainties.     

Analysis of the Heyd-Scuseria-Ernzerhof density functional parameter space

The Heyd-Scuseria-Ernzerhof (HSE) density functionals are popular for their ability to improve upon the accuracy of standard semilocal functionals such as Perdew-Burke-Ernzerhof (PBE), particularly for semiconductor band gaps. They also have a reduced computational cost compared to hybrid functionals, which results from the restriction of Fock exchange calculations to small inter-electron separations. These functionals are defined by an overall fraction of Fock exchange and a length scale for exchange screening. We systematically examine this two-parameter space to assess the performance of hybrid screened exchange (sX) functionals and to determine a balance between improving accuracy and reducing the screening length, which can further reduce computational costs. Three parameter choices emerge as useful: "sX-PBE" is an approximation to the sX-LDA screened exchange density functionals based on the local density approximation (LDA); "HSE12" minimizes the overall error over all tests performed; and "HSE12s" is a range-minimized functional that matches the overall accuracy of the existing HSE06 parameterization but reduces the Fock exchange length scale by half. Analysis of the error trends over parameter space produces useful guidance for future improvement of density functionals.     

Generalized nonlocal kinetic energy density functionals based on the von Weizsäcker functional

We generalize the ideas behind the procedure for the construction of kinetic energy density functionals with a nonlocal term based on the structure of the von Weizs?cker functional, and present several types of nonlocal terms. In all cases, the functionals are constructed such that they reproduce the linear response function of the homogeneous electron gas. These functionals are designed by rewriting the von Weizs?cker functional with the help of a parameter β that determines the power of the electron density in the expression, a strategy we have previously used in the generalization of Thomas-Fermi nonlocal functionals. Benchmark calculations in localized systems have been performed with these functionals to test both their relative errors and the quality of their local behavior. We have obtained competitive results when compared to semilocal and previous nonlocal functionals, the generalized nonlocal von Weizs?cker functionals giving very good results for the total kinetic energies and improving the local behavior of the kinetic energy density. In addition, all the functionals discussed in this paper, when using an adequate reference density, can be evaluated as a single integral in momentum space, resulting in a quasilinear scaling for the computational cost.     

The importance of middle-range Hartree-Fock-type exchange for hybrid density functionals

Hybrid functionals are responsible for much of the utility of modern Kohn-Sham density functional theory. When rigorously applied to solid-state metallic and small band gap systems, however, the slow decay of their nonlocal Hartree-Fock-type exchange makes hybrids computationally challenging and introduces unphysical effects. This can be remedied by using a range-separated hybrid which only keeps short-range nonlocal exchange, as in the functional of Heyd et al. [J. Chem. Phys. 118, 8207 (2003)]. On the other hand, many molecular properties require full long-range nonlocal exchange, which can also be included by means of a range-separated hybrid such as the recently introduced LC-omegaPBE functional [O. A. Vydrov and G. E. Scuseria, J. Chem. Phys. 125, 234109 (2006)]. In this paper, we show that a three-range hybrid which mainly includes middle-range Hartree-Fock-type exchange and neglects long- and short-range Hartree-Fock-type exchange yields excellent accuracy for thermochemistry, barrier heights, and band gaps, emphasizing that the middle-range part of the 1/r potential seems crucial to accurately model these properties.     

Why are time-dependent density functional theory excitations in solids equal to band structure energy gaps for semilocal functionals, and how does nonlocal Hartree-Fock-type exchange introduce excitonic effects?

We examine the time-dependent density functional theory (TD-DFT) equations for calculating excitation energies in solids with Gaussian orbitals and analytically show that for semilocal functionals, their lowest eigenvalue collapses to the minimum band orbital energy difference. With the introduction of nonlocal Hartree-Fock-type exchange (as in hybrid functionals), this result is no longer valid, and the lowest TD-DFT eigenvalue reflects the appearance of excitonic effects. Previously reported "charge-transfer" problems with semilocal TD-DFT excitations in molecules can be deduced from our analysis by taking the limit to infinite lattice constant.     

Removing critical errors for DFT applications to transition-metal nanoclusters: correct ground-state structures of Ru clusters

As ruthenium plays an important role in heterogeneous catalysis, understanding the structural and electronic properties of Ru clusters is crucial to advancement of technology. Because of its efficiency, density functional theory (DFT) calculations are often utilized in nanoscience, but careful validation is necessary. Recently, small, nonmetallic Ru(n) clusters were reported by Zhang et al. [J. Phys. Chem. B 2004, 108, 2140] to form unusual square and cubic ground-state structures within DFT by treating the exchange-correlation (XC) functional at the level of general-gradient-corrected approximation (GGA). For such clusters, we show that the calculated, energetically preferred structures are sensitive to which XC functional is used and whether relativistic effects are included. We find that a hybrid XC functional with partially exact exchange, such as PBE0, corrects the Ru2 magnetic moment, bond length, and dissociation energy in agreement with experiment and high-level quantum chemistry calculations and changes the Ru4 ground-state structure to a tetrahedron, instead of a square. The change in structural preference is explained by the corrections to the electronic structure of a Ru atom, where the relative position of majority spin s level is shifted with respect to e(g) levels. We also find that standard nonrelativistic DFT-GGA gives similar results to relativistic DFT-PBE0, i.e., relative shifting of s level, but not for the right reasons. Our results again stress the need to validate an XC functional before application to transition-metal nanoclusters.     

Parameterized local hybrid functionals from density-matrix similarity metrics

We recently proposed a real-space similarity metric comparing the Kohn-Sham one-particle density matrix to the local spin-density approximation model density matrix [Janesko and Scuseria, J. Chem. Phys. 127, 164117 (2007)]. This metric provides a useful ingredient for constructing local hybrid density functionals that locally mix exact exchange and semilocal density functional theory exchange. Here we present two lines of inquiry: An approximate similarity metric comparing exact versus generalized gradient approximation (GGA), exchange and parameterized mixing functions using these similarity metrics. This approach yields significantly improved thermochemistry, including GGA local hybrids whose thermochemical performance approaches GGA global hybrids.     

Hyper-generalized-gradient functionals constructed from the Lieb-Oxford bound: implementation via local hybrids and thermochemical assessment

In 2009 Odashima and Capelle (OC) showed a way to design a correlation-only density functional that satisfies a Lieb-Oxford bound on the correlation energy, without empirical parameters and even without additional theoretical parameters. However, they were only able to test a size-inconsistent version of it that employs total energies. Here, we show that their alternative size-consistent form that employs energy densities, when combined with exact or semilocal exchange, is a local hybrid (lh) functional. We test several variants of this nonempirical OC-lh functional on standard molecular test sets. Although no variant yields enthalpies of formation with the accuracy of the semilocal Tao-Perdew-Staroverov-Scuseria (TPSS) exchange-correlation, OC-lh correlation with exact exchange yields rather accurate energy barriers for chemical reactions. Our purpose here is not to advocate for a new density functional, but to explore a previously published idea. We also discuss the importance of near-self-consistency for fully nonlocal functionals.     

Reaction energy benchmarks of hydrocarbon combustion by Gaussian basis and plane wave basis approaches

The reaction energies of 275 elementary reactions from the hydrocarbon combustion model GRI-Mech 3.0 were evaluated by electronic structure calculations using both localized Gaussian basis and plane wave basis sets. In the Gaussian basis calculations, the d-polarization function on C, N, and O elements reduces the mean absolute deviation (MAD) from the experimental value by 53%, a significant improvement in computational accuracy. In the plane wave basis calculation using different exchange-correlation (XC) functionals, the MAD values were 0.316–0.426 eV when non-hybrid type XC functionals such as RPBE, PBE, PW91, revPBE, and PBEsol were used. On the other hand, hybrid functionals like B3LYP and HSE06 reduced the MAD values significantly down to 0.182 and 0.233 eV, respectively. The B3LYP results have 49% less MAD compared to the PBE results. These demonstrated the strong advantage of the hybrid functional for calculating gas-phase reaction energies. The present comprehensive benchmarks will be crucial for future microkinetics as well as machine learning studies on the catalytic reactions. © 2019 Wiley Periodicals, Inc.     

Assessment and validation of a screened Coulomb hybrid density functional

This paper presents a revised and improved version of the Heyd-Scuseria-Ernzerhof screened Coulomb hybrid functional. The performance of this functional is assessed on a variety of molecules for the prediction of enthalpies of formation, geometries, and vibrational frequencies, yielding results as good as or better than the successful PBE0 hybrid functional. Results for ionization potentials and electron affinities are of slightly lower quality but are still acceptable. The comprehensive test results presented here validate our assumption that the screened, short-range Hartree-Fock (HF) exchange exhibits all physically relevant properties of the full HF exchange. Thus, hybrids can be constructed which neglect the computationally demanding long-range part of HF exchange while still retaining the superior accuracy of hybrid functionals, compared to pure density functionals.     

Comparison of DFT methods for molecular orbital eigenvalue calculations

We report how closely the Kohn-Sham highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) eigenvalues of 11 density functional theory (DFT) functionals, respectively, correspond to the negative ionization potentials (-IPs) and electron affinities (EAs) of a test set of molecules. We also report how accurately the HOMO-LUMO gaps of these methods predict the lowest excitation energies using both time-independent and time-dependent DFT (TD-DFT). The 11 DFT functionals include the local spin density approximation (LSDA), five generalized gradient approximation (GGA) functionals, three hybrid GGA functionals, one hybrid functional, and one hybrid meta GGA functional. We find that the HOMO eigenvalues predicted by KMLYP, BH&HLYP, B3LYP, PW91, PBE, and BLYP predict the -IPs with average absolute errors of 0.73, 1.48, 3.10, 4.27, 4.33, and 4.41 eV, respectively. The LUMOs of all functionals fail to accurately predict the EAs. Although the GGA functionals inaccurately predict both the HOMO and LUMO eigenvalues, they predict the HOMO-LUMO gap relatively accurately (approximately 0.73 eV). On the other hand, the LUMO eigenvalues of the hybrid functionals fail to predict the EA to the extent that they include HF exchange, although increasing HF exchange improves the correspondence between the HOMO eigenvalue and -IP so that the HOMO-LUMO gaps are inaccurately predicted by hybrid DFT functionals. We find that TD-DFT with all functionals accurately predicts the HOMO-LUMO gaps. A linear correlation between the calculated HOMO eigenvalue and the experimental -IP and calculated HOMO-LUMO gap and experimental lowest excitation energy enables us to derive a simple correction formula.