Nonempirical meta-GGA Research Articles

Balancing the Contributions to the Gradient Expansion: Accurate Binding and Band Gaps with a Nonempirical Meta-GGA.

The gradient expansion has been a long-standing guide rail in density-functional theory. We here demonstrate that for exchange-correlation approximations that depend on the gradient of the density and the kinetic energy density, i.e., for meta-generalized gradient approximations (meta-GGAs), there is a so far unexploited degree of freedom in the gradient expansion that allows to shift the relative weight of gradient and kinetic energy contributions. As the dependence on the kinetic energy density determines the derivative discontinuity, this allows to construct meta-GGAs that adhere to the known exact constraints, yet have new properties. We demonstrate this with the construction of a meta-GGA that describes both electronic bonds and band gaps with remarkable accuracy.

Framework for Laplacian-Level Noninteracting Free-Energy Density Functionals.

A framework for orbital-free Laplacian-level meta-generalized-gradient approximation (meta-GGA) for the noninteracting free-energy-density functionals based upon analysis of the fourth-order gradient expansion is developed. The framework presented here provides a new tool for developing advanced orbital-free thermal functionals at the meta-GGA level of theory. A nonempirical meta-GGA functional, which in the slowly varying density limit correctly reduces to the finite-temperature fourth-order gradient expansion for the noninteracting free energy, is constructed. Application to warm dense helium demonstrates that the developed meta-GGA functional drastically increases the accuracy of orbital-free-density functional theory simulations at temperatures below 40 eV as compared to the lower Thomas-Fermi and GGA rungs.

Minimally Empirical Double-Hybrid Functionals Trained against the GMTKN55 Database: revDSD-PBEP86-D4, revDOD-PBE-D4, and DOD-SCAN-D4.

We present a family of minimally empirical double-hybrid DFT functionals parametrized against the very large and diverse GMTKN55 benchmark. The very recently proposed ωB97M(2) empirical double hybrid (with 16 adjustable parameters) has the lowest WTMAD2 (weighted mean absolute deviation over GMTKN55) ever reported at 2.19 kcal/mol. However, refits of the DSD-BLYP and DSD-PBEP86 spin-component-scaled, dispersion-corrected double hybrids can achieve WTMAD2 values as low as 2.33 with the very recent D4 dispersion correction (2.42 kcal/mol with the D3(BJ) dispersion term) using just a handful of adjustable parameters. If we use full DFT correlation in the initial orbital evaluation, the xrevDSD-PBEP86-D4 functional reaches WTMAD2 = 2.23 kcal/mol, statistically indistinguishable from ωB97M(2) but using just four nonarbitrary adjustable parameters (and three semiarbitrary ones). The changes from the original DSD parametrizations are primarily due to noncovalent interaction energies for large systems, which were undersampled in the original parametrization set. With the new parametrization, same-spin correlation can be eliminated at minimal cost in performance, which permits revDOD-PBEP86-D4 and revDOD-PBE-D4 functionals that scale as N4 or even N3 with the size of the system. Dependence of WTMAD2 for DSD functionals on the percentage of HF exchange is roughly quadratic; it is sufficiently weak that any reasonable value in the 64% to 72% range can be chosen semiarbitrarily. DSD-SCAN and DOD-SCAN double hybrids involving the SCAN nonempirical meta-GGA as the semilocal component have also been considered and offer a good alternative if one wishes to eliminate either the empirical dispersion correction or the same-spin correlation component. noDispSD-SCAN66 achieves WTMAD2 = 3.0 kcal/mol, compared to 2.7 kcal/mol for DOD-SCAN66-D4. However, the best performance without dispersion corrections (WTMAD2 = 2.8 kcal/mol) is reached by revωB97X-2, a slight reparametrization of the Chai–Head-Gordon range-separated double hybrid. Finally, in the context of double-hybrid functionals, the very recent D4 dispersion correction is clearly superior over D3(BJ).

Nonempirical Meta-Generalized Gradient Approximations for Modeling Chemisorption at Metal Surfaces.

We assess the accuracy of popular nonempirical GGAs (PBE, PBEsol, RPBE) and meta-GGAs (TPSS, revTPSS, and SCAN) for describing chemisorption reactions at metal surfaces. Except for RPBE, all the functionals tend to overbind the adsorbate significantly. We then propose a nonempirical meta-GGA, denoted as RTPSS, that is based on RPBE in the same way that TPSS is based on PBE. The RTPSS functional remedies the overbinding problem and improves the description of chemisorption energies. As an example of an application of RTPSS, we study the adsorption of CO on Cu surfaces (a notably difficult problem for semilocal functionals) and find that RTPSS is the only tested functional that predicts accurate chemisorption energies and the preferred adsorption site of CO. Although RTPSS gives an accurate description of chemisorption, nonlocal correlation may be necessary to describe physisorption if long-range van der Waals interactions are involved (however, this is true for semilocal functionals in general). We suggest that RTPSS can be a useful meta-GGA for studying chemisorption processes and mechanisms of heterogeneous catalysis.

B88 exchange functional recovering the local spin density linear response

Following the strategy described in the landmark paper generalized gradient approximation (GGA) made simple (Perdew et al. in Phys Rev Lett 77(18):3865, 1996), we have constructed a non-empirical Becke 1988 enhancement factor (neB). The local spin density linear response constraint was imposed on the Becke 1988 enhancement factor (B88) leading to a parameter value of 0.00336, which is 20 % lower than the original fitted value 0.0042 and, therefore, decreases the gradient contribution to the exchange energy with respect to the original B88 functional. The results on typical test sets, in particular for predicting the heats of formation, showed that global hybrid neBPBE0 delivers a mean average error of 4.59 kcal/mol, which competes with results obtained with non-empirical meta-GGA functionals, making it an attractive simple hybrid functional to be used for molecular calculations for this rung of Perdew’s classification of functionals.

Construction of a general semilocal exchange-correlation hole model: Application to nonempirical meta-GGA functionals

Using a reverse-engineering method, we construct a meta-generalized gradient approximation (meta-GGA) angle-averaged exchange-correlation (XC) hole model which has a general applicability. It satisfies known exact hole constraints and can exactly recover the exchange-correlation energy density of any reasonable meta-GGA exchange-correlation energy functional satisfying a minimal set of exact properties. The hole model is applied to several nonempirical meta-GGA functionals: the Tao-Perdew-Staroverov-Scuseria (TPSS), the revised TPSS (revTPSS), and the recently Balanced LOCalization (BLOC) meta-GGA [L. A. Constantin, E. Fabiano, and F. Della Sala, J. Chem. Theory Comput. 9, 2256 (2013)]. The empirical M06-L meta-GGA functional is also considered. Real-space analyses of atoms and ions as well as wave-vector analyses of jellium surface energies show that the meta-GGA hole models, in particular the BLOC one, are very realistic and can reproduce many features of benchmark XC holes. In addition, the BLOC hole model can be used to estimate with good accuracy the Coulomb hole radius of small atoms and ions. Thus, the proposed meta-GGA hole models provide a valuable tool to validate in detail existing meta-GGA functionals, and can be further used in the development of density functional theory methods beyond the semilocal level of theory.

A new meta-GGA exchange functional based on an improved constraint-based GGA

We report the performance of a non-empirical meta-GGA that comes from converting our simple VT{8,4} GGA. That GGA satisfies the large dimensionless reduced gradient limit, obeys the Lieb–Oxford bound, and reduces to the exact second-order gradient expansion approximation in the slowly varying limit. Validation studies of meta-VT{8,4} for several properties using well-known test sets shows a modest improvement with respect to revTPSS. Compared with the heavily parameterized M06-L, the heats of formation of meta-VT{8,4} are substantially better but reaction barrier heights are considerably worse. This suggests the opportunity for additional constraints and the need for better correlation functionals.

Density functional theory study of the mechanism for Ni(NHC) 2 catalyzed dehydrogenation of ammonia–borane for chemical hydrogen storage

The mechanism for catalytic dehydrogenation of ammonia–borane (AB = H 3N–BH 3), a promising candidate for chemical hydrogen storage, by the Ni(NHC) 2 complexes was studied by using density functional theory at the non-empirical meta-GGA level, Tao–Perdew–Staroverov–Scuseria (TPSS) functional, with all-electron correlation-consistent polarized valence double-zeta (cc-pVDZ) basis set. The mechanism for both the first and the second H 2 release from AB was studied for the first time. Several unusual aspects of this catalytic mechanism were revealed through our calculations. First, the first H 2 release begins with proton transfer from nitrogen to the Ni bound carbene carbon, forming a new C–H bond, instead of the previously hypothesized direct B–H or N–H bond activation. Second, this new C–H bond is activated by the metal, transferring the H to Ni, then forming the H 2 molecule by transferring another H from B to Ni, rather than β-H transfer. Third, the second H 2 release from H 2N–BH 2 begins with the breaking of a 3-center, 2-electron Ni–H–B bridging structure with the assistance of the unsaturated carbene carbon atom to form a B–C bond. Fourth, a nearly rhombic N 2B 2H 6 structure is formed to help the regeneration of the catalyst Ni(NHC) 2. These reaction pathways explain the importance of NHC ligands in this catalytic process and yield lower energy barriers than those mechanisms that begin with N–H or B–H activations catalyzed by the metal atoms. The predicted reaction mechanism which features unexpected ligand participation points the way to finding new catalysts with higher efficiency, as partial unsaturation of the M–L bond may be essential for low energy H transfers.

Density Functional Theory Study of the Reaction Mechanism for Competitive Carbon−Hydrogen and Carbon−Halogen Bond Activations Catalyzed by Transition Metal Complexes

Carbon-hydrogen and carbon-halogen bond activations between halobenzenes and metal centers were studied by density functional theory with the nonempirical meta-GGA Tao-Perdew-Staroverov-Scuseria functional and an all-electron correlation-consistent polarized valence double-zeta basis set. Our calculations demonstrate that the hydrogen on the metal center and halogen in halobenzene could exchange directly through a kite-shaped transition state. Transition states with this structure were previously predicted to have high energy barriers (J. Am. Chem. Soc. 2005, 127, 279), and this prediction misled others in proposing a mechanism for their recent experimental study (J. Am. Chem. Soc. 2006, 128, 3303). Furthermore, other halo-carbon activation pathways were found in the detailed mechanism for the competitive reactions between cationic titanium hydride complex [Cp*((t)Bu(3)P=N)TiH](+) and chlorobenzene under different pressure of H(2). These pathways include the ortho-C-H and Ti-H bond activations for the formation and release of H(2) and the indirect C-Cl bond activation via beta-halogen elimination for the movement of the C(6)H(4) ring and the formation of a C-N bond in the observed final product. A new stable isomer of the observed product with a similar total energy and an unexpected bridging between the Cp* ring and the metal center by a phenyl ring is also predicted.

Meta-generalized gradient approximation for the exchange-correlation hole with an application to the jellium surface energy

We propose a model for the angle- and system-averaged exchange-correlation hole of a many-electron system. This hole analyzes the exchange-correlation energy into contributions of various distances $u$ from an electron. The model is ``reverse-engineered'' (derived from and not used to derive a density functional). It satisfies known exact hole constraints, including ones that can only be satisfied by a meta-generalized gradient approximation or meta-GGA. It incorporates the exchange-correlation energy density of the Tao-Perdew-Staroverov-Scuseria (TPSS) nonempirical meta-GGA. The hole model is tested for atoms and applied to jellium surfaces. The Fourier transform $(u\ensuremath{\rightarrow}k)$ of the hole is needed for wave-vector interpolation of the jellium surface energy from an exact small-$k$ or large-$u$ asymptote. We find essentially the same surface energies (close to the uncorrected TPSS values) whether we apply the wave-vector interpolation correction to the local spin density approximation, the GGA or the meta-GGA. These and other considerations suggest that these surface energies are accurate. Moreover, we find that the uncorrected TPSS surface energies have a realistic wave-vector analysis. Our TPSS hole model can be used to build the hole model for a TPSS-based global hybrid functional, or for a hyper-GGA that uses full exact exchange.

Energy storage capacity of polymeric nitrogen

We present electronic structure calculations on the single-bonded cubic gauche form of polymeric nitrogen and predict its energy storage capacity using density functional theory, Gaussian-type orbitals, and periodic boundary conditions. We have used several exchange correlation functionals including the local spin density approximation, generalized gradient approximation (GGA), a non-empirical meta-GGA, and a screened exchange hybrid functional. Our results indicate that solid polymeric nitrogen has a higher energy storage capacity than previously reported.

Structures and electronic properties of platinum nitride by density functional theory

We present the results of ground state electronic structure calculations of the zinc-blende and rocksalt phases of binary platinum nitride (PtN) using density functional theory. Several exchange-correlation functionals including the local spin density approximations, generalized gradient approximations (GGA), a nonempirical meta-GGA, and a screened Coulomb hybrid functional have been employed. We use Gaussian type orbitals within the framework of periodic boundary conditions. Our results confirm earlier findings, in that the zinc-blende structure of PtN is energetically more stable than the rocksalt structure. The predicted energy difference between the two phases is rather small with the more elaborate functionals. Both phases are predicted to be metallic and extended Pt $d\text{\ensuremath{-}}\mathrm{N}$ $p$ hybridizations are found in both of them. We have also calculated the phase transition pressure between both phases. The bulk modulus of the zinc-blend phase of PtN is significantly higher than that of bulk platinum.

Test of a nonempirical density functional: Short-range part of the van der Waals interaction in rare-gas dimers

It is known that the nonempirical generalized gradient approximation (GGA) of Perdew, Burke, and Ernzerhof (PBE) provides a much more realistic description of the short-range part of the van der Waals (vdW) interaction than does the local spin density (LSD) approximation. In the present work, the ability of the higher-level nonempirical meta-GGA of Tao, Perdew, Staroverov, and Scuseria (TPSS) to describe vdW interaction is tested self-consistently in ten rare-gas dimers with Z< or =36. The one-parameter hybrid version (TPSSh) of the TPSS exchange-correlation functional is also included in this test. Calculations show that both TPSS and TPSSh functionals correctly yield vdW bonds in these dimers and significantly improve the prediction of bond lengths, binding energies, and harmonic vibrational frequencies over LSD. The rather close agreement of TPSS with PBE for these dimers confirms a principle of the TPSS construction: preservation of the PBE large-gradient behavior. More importantly, it suggests that TPSS can serve as a platform on which to construct a still-higher level of nonempirical functionals. Compared with the PBE GGA, TPSS, and TPSSh yield a slightly weaker binding. As for normally bonded molecules, TPSSh yields the most accurate vibrational frequencies. The typically too-long bond lengths and too-small binding energies of TPSS meta-GGA suggest the need for some long-range vdW interaction correction even in this class of systems. The effect of basis-set superposition error on the calculated properties of these vdW systems is investigated. We also show that the relatively strong anharmonic effects in the rare-gas dimers are described remarkably well by the Morse potential.

Tests of a ladder of density functionals for bulk solids and surfaces

The local spin-density approximation (LSDA) and the generalized gradient approximation (GGA) of Perdew, Burke, and Ernzerhof (PBE) are fully non-empirical realizations of the first two rungs of ``Jacob's ladder'' of exchange-correlation density functionals. The recently proposed non-empirical meta-GGA of Tao, Perdew, Staroverov, and Scuseria (TPSS), featuring the kinetic energy density as an additional local ingredient, completes the third rung. A hierarchy of these functionals, complemented by the meta-GGA of Perdew, Kurth, Zupan, and Blaha (PKZB), is tested in self-consistent Gaussian-type orbital calculations of equilibrium lattice constants, bulk moduli, and cohesive energies for 18 solids, and in studies of the jellium surface energy. The ascent of the ladder generally results in better performance, although most of the improvement for bulk solids occurs in the transition from LSDA to PBE. For the jellium surface energy, PBE is less accurate than LSDA, but PKZB and TPSS are more accurate. We support the idea that most of the error of these functionals for bulk solids arises in the description of core--valence interaction, by demonstrating that it can be removed through adjustment of the corresponding term in the equation of state. Overall, TPSS gives the best description of solids and surfaces, as it was found to do for molecules in earlier work.