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 N 4 or even N 3 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).

Double hybrid density functional theory arguably sits on the seamline between wavefunction methods and DFT: it represents a special case of Rung 5 on the "Jacob's Ladder" of John P. Perdew. For large and chemically diverse benchmarks such as GMTKN55, empirical double hybrid functionals with dispersion corrections can achieve accuracies approaching wavefunction methods at a cost not greatly dissimilar to hybrid DFT approaches, provided RI-MP2 and/ or another MP2 acceleration techniques are available in the electronic structure code. Only a half-dozen or fewer empirical parameters are required. For vibrational frequencies, accuracies intermediate between CCSD and CCSD(T) can be achieved, and performance for other properties is encouraging as well. Organometallic reactions can likewise be treated well, provided static correlation is not too strong. Further prospects are discussed, including range-separated and RPA-based approaches.These are not the final page numbers! �� quantum Monte Carlo results [18] for uniform electron gases. The LDA XC functional only depends on the electronic density and not on any derivatives nor any other entities.Rung two corresponds to GGAs or generalized gradient approximations, in which the reduced density gradient is introduced in the XC functional. Examples are the popular BP86 [19,20] and PBE [21] functionals.Rung three consists of meta-GGAs (mGGAs), which additionally involve higher density derivatives (or the kinetic energy density, which contains similar information to the density Laplacian). TPSS [22] is a popular meta-GGA, as is the recent SCAN (strongly constrained and appropriate normed [23] ).Rungs two and three are often collectively referred to as semi-local functionals.Orbital-dependent DFT [24] covers rungs four and five: on rung four, only occupied orbital-dependency is introduced, while on rung five the unoccupied orbitals make their appearance. The most important rung four functionals are the hybrids, which can be further subdivided into four subclasses: * global hybrid GGAs, such as the popular B3LYP [25,26] and PBE0 [27] hybrids, as well as B97-1. [28] (We note that Hartree-Fock theory itself is a special case, with 100 % exact exchange and null correlation.) * global hybrid meta-GGAs, such as M06, [29] M06-2X, [29] and BMK [30] * range-separated hybrid GGAs, such as CAM-B3LYP [31] and ωB97X-V [32] * range-separated hybrid meta-GGAs, such as ωB97M-V [33] This leaves rung five, of which we will presently consider one subcase, the double hybrids. More general reviews have been published earlier; [34][35][36] the present review will focus primarily on our own work, as well as the broader context.

We present a family of minimally empirical double-hybrid DFT functionals parametrized against the very large and diverse GMTKN55 benchmark. The very recently proposed wB97M(2) empirical double hybrid (with 16 empirical parameters) has the lowest WTMAD2 (weighted mean absolute deviation over GMTKN55) ever reported at 2.19 kcal/mol. However, our xrevDSD-PBEP86-D4 functional reaches a statistically equivalent WTMAD2=2.22 kcal/mol, using just a handful of empirical parameters, and the xrevDOD-PBEP86-D4 functional reaches 2.25 kcal/mol with just opposite-spin MP2 correlation, making it amenable to reduced-scaling algorithms. In general, the D4 empirical dispersion correction is clearly superior to D3BJ. If one eschews dispersion corrections of any kind, noDispSD-SCAN offers a viable alternative. Parametrization over the entire GMTKN55 dataset yields substantial improvement over the small training set previously employed in the DSD papers.

For the large and chemically diverse GMTKN55 benchmark suite, we have studied the performance of density-corrected density functional theory (HF-DFT), compared to self-consistent DFT, for several pure and hybrid GGA and meta-GGA exchange–correlation (XC) functionals (PBE, BLYP, TPSS, and SCAN) as a function of the percentage of HF exchange in the hybrid. The D4 empirical dispersion correction has been added throughout. For subsets dominated by dynamical correlation, HF-DFT is highly beneficial, particularly at low HF exchange percentages. This is especially true for noncovalent interactions where the electrostatic component is dominant, such as hydrogen and halogen bonds: for π-stacking, HF-DFT is detrimental. For subsets with significant nondynamical correlation (i.e., where a Hartree–Fock determinant is not a good zero-order wavefunction), HF-DFT may do more harm than good. While the self-consistent series show optima at or near 37.5% (i.e., 3/8) for all four XC functionals—consistent with Grimme’s proposal of the PBE38 functional—HF-B n LYP-D4, HF-PBE n -D4, and HF-TPSS n -D4 all exhibit minima nearer 25% (i.e., 1/4) as the use of HF orbitals greatly mitigates the error at 25% for barrier heights. Intriguingly, for HF-SCAN n -D4, the minimum is near 10%, but the weighted mean absolute error (WTMAD2) for GMTKN55 is only barely lower than that for HF-SCAN-D4 (i.e., where the post-HF step is a pure meta-GGA). The latter becomes an attractive option, only slightly more costly than pure Hartree–Fock, and devoid of adjustable parameters other than the three in the dispersion correction. Moreover, its WTMAD2 is only surpassed by the highly empirical M06-2X and by the combinatorially optimized empirical range-separated hybrids ωB97X-V and ωB97M-V.

Atomic partial charges are among the most commonly used interpretive tools in quantum chemistry. Dozens of different 'population analyses' are in use, which are best seen as proxies (indirect gauges) rather than measurements of a 'general ionicity'. For the GMTKN55 benchmark of nearly 2,500 maingroup molecules, which span a broad swathe of chemical space, some two dozen different charge distributions were evaluated at the PBE0 level near the 1-particle basis set limit. The correlation matrix between the different charge distributions exhibits a block structure; blocking is, broadly speaking, by charge distribution class. A principal component analysis on the entire dataset suggests that nearly all variation can be accounted for by just two 'principal components of ionicity': one has all the distributions going in sync, while the second corresponds mainly to Bader QTAIM vs. all others. A weaker third component corresponds to electrostatic charge models in opposition to the orbital-based ones. The single charge distributions that have the greatest statistical similarity to the first principal component are iterated Hirshfeld (Hirshfeld-I) and a minimal-basis projected modification of Bickelhaupt charges. If three individual variables, rather than three principal components, are to be identified that contain most of the information in the whole dataset, one representative for each of the three classes of Corminboeuf et al. is needed: one based on partitioning of the density (such as QTAIM), a second based on orbital partitioning (such as NPA), and a third based on the molecular electrostatic potential (such as HLY or CHELPG).