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We investigate the potential performance improvements of double-hybrid density functionals by replacing the standard opposite-spin-scaled MP2 (SOS-MP2) with the modified opposite-spin-scaled MP2 (MOS-MP2) in the nonlocal correlation component. Using the large and diverse GMTKN55 dataset, we find that MOS-double hybrids provide significantly better accuracy compared to SOS-MP2-based double hybrids when empirical dispersion correction is not employed. The non-covalent interaction subsets account for the majority of this improvement. Adding the DFT-D4 dispersion correction to MOS-type double hybrids does not provide any superior performance over conventional dispersion-corrected SOS-MP2-based double hybrids. Nevertheless, for nine tested transition metal sets, dispersion-corrected spin-component-scaled (SCS) double hybrids are still significantly better than any MOS-double hybrid functional.

Benchmark calculations on noncovalent interactions typically exclude correlation effects beyond valence CCSD(T) owing to their steep computational cost scaling. In this work, we consider their importance for water clusters, specifically, eight isomers of (H$_2$O)$_6$ and four Wales-Hodges isomers of (H$_2$O)$_{20}$. Higher order connected triples, $T_3$--(T), reduce dissociation energies of the latter by about 0.4 kcal/mol, but this is more than compensated by an increase of up to 0.85 kcal/mol due to connected quadruple excitations. In general, higher-order correlation effects favor more compact isomers over more `spread-out' ones. We also consider additional small effects for balance: scalar relativistics reduce binding in (H$_2$O)$_{20}$ by ca. --0.4 kcal/mol, which fortuitously is compensated by the ca. 0.55 kcal/mol diagonal Born-Oppenheimer correction. Core-valence correlation has the greatest impact, at ca. 1.3 kcal/mol for the icosamer., Comment: version updated from ResearchGate, now including 3-body contributions for water20

In two recent papers [A. D. Becke, J. Chem. Phys. 156, 214101 (2022) and 157, 234102 (2022)] we compared two Kohn-Sham density functionals based on physical modelling and theory with the best density-functional power-series fits in the literature. The best error statistics reported to date for a hybrid functional on the GMTKN55 chemical database of Goerigk, Grimme, and coworkers [Phys. Chem. Chem. Phys. 19, 32184 (2017)] were obtained. In the present work, additional second-order perturbation-theory terms are considered. The result is a 12-parameter double-hybrid (DH) density functional with the lowest GMTKN55 "WTMAD2" error yet seen for a DH functional. We call it "DH23"., Comment: 23 pages + supporting information, J. Chem. Phys., accepted

We have investigated the effect of F12 geminals on the basis set convergence of harmonic frequencies calculated using two representative double-hybrid density functionals, namely B2GP-PLYP and revDSD-PBEP86-D4. Like previously found for energetics [N. Mehta and J. M. L. Martin, \textit{J. Chem. Theor. Comput.} \textbf{18}, 5978--5991 (2022)] one sees an acceleration by two zeta steps, such that even the cc-pVDZ-F12 basis set is quite close to the complete basis set (CBS) limit. However, the basis set convergence problem is not as acute as for energetics, and compared to experimental harmonic frequencies, conventional orbital calculations with augmented triple zeta quality basis set are acceptably close to the CBS limit, and can be carried out using analytical second derivatives. An efficient implementation of double hybrid-F12 analytical derivatives would make the F12 approach attractive in the sense that even an $spd$ orbital basis set would be adequate. For the accurate revDSD-PBEP86-D4 functional, the role of differing local correlation terms (Perdew-Zunger 1981 vs. VWN5) in different electronic structure programs has been investigated: while optimal double hybrid parameters and performance statistics for energetics as well as frequencies differ slightly between the two implementations, these differences are insignificant for practical purposes., Comment: Can. J. Chem., minor revision (WATOC 2022 special issue)

We report an update and enhancement of the ACONFL (conformer energies of large alkanes [Ehlert, S.; Grimme, S.; Hansen, A. J. Phys. Chem. A 2022, 126, 3521-3535]) dataset. For the ACONF12 (n-dodecane) subset, we report basis set limit canonical CCSD(T) reference data obtained from MP2-F12/cc-pV{T,Q}Z-F12 extrapolation, [CCSD(F12*)-MP2-F12]/aug-cc-pVTZ-F12, and a (T) correction from conventional CCSD(T)/aug-cc-pV{D,T}Z calculations. Then we explored the performance of a variety of single and composite localized-orbital CCSD(T) approximations, ultimately finding an affordable LNO-CCSD(T)-based post-MP2 correction that agrees to 0.008 kcal/mol MAD (mean absolute deviation) with the revised canonical reference data. In tandem with canonical MP2-F12/CBS extrapolation, this was then used to re-evaluate the ACONF16 and ACONF20 subsets for n-hexadecane and n-icosane, respectively. A revised ACONFL set was thus obtained. It was then used to assess the performance of different localized-orbital coupled cluster approaches, such as PNO-LCCSD(T) as implemented in MOLPRO, DLPNO-CCSD (T1) as implemented in ORCA, and LNO-CCSD(T) as implemented in MRCC, at their various accuracy settings. A three-tier LNO-CCSD(T)-based composite scheme disagrees by only 0.02 kcal/mol from the revised ACONFL reference data. When extrapolated to the complete PNO space limit, DLPNO-CCSD(T1, Tight) and a composite method are the best picks among all the localized coupled cluster methods tested for the dodecane conformers. Dispersion-corrected dRPA-based double hybrids perform remarkably well for the ACONFL set. While the revised reference data do not affect any conclusions on the less accurate methods, they may upend orderings for more accurate methods with error statistics on the same order as the difference between reference datasets., Comment: Revised MS submitted to JPC A: level of theory for reference calculations upgraded; many other changes

The S66x8 noncovalent interactions benchmark has been re-evaluated at the "sterling silver" level, using explicitly correlated MP2-F12 near the complete basis set limit, CCSD(F12*)/aug-cc-pVTZ-F12, and a (T) correction from conventional CCSD(T)/sano-V{D,T}Z+ calculations. The revised reference value disagrees by 0.1 kcal/mol RMS with the original Hobza benchmark and its revision by Brauer et al, but by only 0.04 kcal/mol variety from the "bronze" level data in Kesharwani et al., Aust. J. Chem. 71, 238-248 (2018). We then used these to assess the performance of localized-orbital coupled cluster approaches with and without counterpoise corrections, such as PNO-LCCSD(T) as implemented in MOLPRO, DLPNO-CCSD (T1) as implemented in ORCA, and LNO-CCSD(T) as implemented in MRCC, for their respective "Normal", "Tight", and "very Tight" settings. We also considered composite approaches combining different basis sets and cutoffs. Furthermore, in order to isolate basis set convergence from domain truncation error, for the aug-cc-pVTZ basis set we compared PNO, DLPNO, and LNO approaches with canonical CCSD(T). We conclude that LNO-CCSD(T) with veryTight criteria performs very well for "raw" (CP-uncorrected), but struggles to reproduce counterpoise-corrected numbers even for veryVeryTight criteria: this means that accurate results can be obtained using either extrapolation from basis sets large enough to quench basis set superposition error (BSSE) such as aug-cc-pV{Q,5}Z, or using a composite scheme such as Tight{T,Q}+1.11[vvTight(T) - Tight(T)]. In contrast, PNO-LCCSD(T) works best with counterpoise, while performance with and without counterpoise is comparable for DLPNO-CCSD(T1). Among more economical methods, the highest accuracies are seen for dRPA75-D3BJ, {\omega}B97M-V, {\omega}B97M(2), revDSD-PBEP86-D4, and DFT(SAPT) with a TDEXX or ATDEXX kernel., Comment: Final published version with CC license

To better understand the thermochemical kinetics and mechanism of a specific chemical reaction, an accurate estimation of barrier heights (forward and reverse) and reaction energy are vital. Due to the large size of reactants and transition state structures involved in real-life mechanistic studies (e.g., enzymatically catalyzed reactions), DFT remains the workhorse for such calculations. In this paper, we have assessed the performance of 88 density functionals for modeling the reaction energies and barrier heights on a large and chemically diverse dataset (BH9) composed of 449 organic chemistry reactions. We have shown that range-separated hybrid functionals perform better than the global hybris for BH9 barrier heights and reaction energies. Except for the PBE-based range-separated nonempirical double hybrids, the exchange term's range separation helps improve the performance for barrier heights and reaction energies. The sixteen-parameter Berkeley double hybrid, {\omega}B97M(2), performs remarkably well for both properties. However, our minimally empirical range-separated double hybrid functionals offer marginally better accuracy than {\omega}B97M(2) for BH9 barrier heights and reaction energies., Comment: J. Phys. Chem. A, revised (Vincenzo Barone festschrift)

The S66 non-covalent interactions are studied through localized coupled-cluster methods and general LNO-CCSD(T)-based composite schemes. Very small RMS deviations (\leq 0.05 kcal/mol) for the low-cost composite approaches from the SILVER reference interaction energies of S66 indicate that we can safely avoid carrying out the largest basis set calculations with veryVeryTight thresholds, and apply instead additivity corrections in smaller basis sets. Interestingly, the counterpoise corrections do not have an appreciable effect on the composite schemes. These findings may prove useful for intermolecular and intramolecular NCIs of larger systems., Comment: 6 pages, AIP Conference Proceedings, in press (ICCMSE-2021)

While the inclusion of the nonlocal correlation in fifth rung "double hybrid" functionals is definitely beneficial, one might rightfully ask whether its evaluation in the basis of Kohn-Sham (KS) orbitals has additional value compared to the use of Hartree-Fock reference orbitals (in a type of multilevel scheme). We have investigated this question for a very large and chemically diverse dataset, GMTKN55. We conclude that KS reference orbitals are undoubtedly beneficial, but the benefit is not as large as one might intuitively expect., Comment: 4 pages, AIP Conference Proceedings, in press (ICCMSE-2021)

We propose here a DFT-based diagnostic for static correlation %TAEX[TPSS@HF - HF] which effectively measures how different the DFT and HF exchange energies for a given HF density are. This and %TAEcorr[TPSS] are two cost-effective a priori estimates for the adequacy of the importance of static correlation. %TAEX[TPSS@HF - HF] contains nearly the same information as the earlier A diagnostic, but may be more intuitive to understand. Principal component and variable clustering analysis of a large number of static correlation diagnostics reveals much of the variation is explained by just two components, and almost all of it by four; these are blocked by four variable clusters (single excitations; correlation entropy; double excitations; pragmatic energetics)., Comment: 5 pages, AIP Conference Proceedings, in press (ICCMSE-2021)

By adding a GLPT3 (third-order G\"orling-Levy perturbation theory, or KS-MP3) term E3 to the XYG7 form for a double hybrid, we are able to bring down WTMAD2 (weighted total mean absolute deviation) for the very large and chemically diverse GMTKN55 benchmark to an unprecedented 1.17 kcal/mol, competitive with much costlier composite wavefunction ab initio approaches. Intriguingly: (a) the introduction of E3 makes an empirical dispersion correction redundant; (b) GGA or mGGA semilocal correlation functionals offer no advantage over LDA in this framework; (c) if a dispersion correction is retained, then simple Slater exchange leads to no significant loss in accuracy. It is possible to create a 6-parameter functional with WTMAD2=1.42 that has no post-LDA DFT components and no dispersion correction in the final energy.

For revDSD double hybrids, the G\"orling-Levy second-order perturbation theory component is an Achilles' Heel when applied to systems with significant near-degeneracy ("static") correlation. We have explored its replacement by the direct random phase approximation (dRPA), inspired by the SCS-dRPA75 functional of K\'allay and coworkers. The addition to the final energy of both a D4 empirical dispersion correction, and of a semilocal correlation component lead, to significant improvements, with DSD-PBEdRPA75-D4 approaching the performance of revDSD-PBEP86-D4 and the Berkeley $\omega$B97M(2). This form appears to be fairly insensitive to the choice of semilocal functional, but does exhibit stronger basis set sensitivity than the PT2-based double hybrids (due to much larger prefactors for the nonlocal correlation). As an alternative, we explored adding an MP3-like correction term (in a medium-sized basis sets) to a range-separated $\omega$DSD-PBEP86-D4 double hybrid, and found it to have significantly lower WTMAD2 (weighted mean absolute deviation) for the large and chemically diverse GMTKN55 benchmark suite; the added computational cost can be mitigated through density fitting techniques., Comment: J. Phys. Chem. A 125, ASAP (2021) Creative Commons License

We have explored the use of range separation as a possible avenue for further improvement on our revDSD minimally empirical double hybrid functionals. Such $\omega$DSD functionals encompass the XYG3 type of double hybrid (i.e., xDSD) as a special case for $\omega$->0. As in our previous studies, the large and chemically diverse GMTKN55 benchmark suite was used for evaluation. Especially when using the D4 rather than D3BJ dispersion model, xDSD has a slight performance advantage in WTMAD2. As found previously, PBEP86 is the winning combination for the semilocal parts. xDSDn-PBEP86-D4 marginally outperforms the previous 'best in class' $\omega$B97M(2) Berkeley double hybrid, but without range separation and using fewer than half the number of empirical parameters. Range separation turns out to offer only marginal further improvements on GMTKN55 itself. While $\omega$B97M(2) still yields better performance for small-molecule thermochemistry, this is outweighed in WTMAD2 by superior performance of the new functionals for conformer equilibria. Results for two external test sets with pronounced static correlation effects may indicate that range-separated double hybrids are more resilient to such effects., Comment: J. Phys. Chem. A 125, ASAP (2021) Creative Commons License (1st of two papers back-to-back)

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, 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 -- particularly noncovalent interaction subsets -- HF-DFT is highly beneficial, particularly at low HF exchange percentages. For subsets with significant static 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-BnLYP-D4, HF-PBEn-D4, and HF-TPSSn-D4 all exhibit minima nearer 25% (i.e., 1/4). Intriguingly, for HF-SCANn-D4, the minimum is near 10%, but the weighted mean absolute error (WTMAD2) for GMTKN55 is only barely lower than that of 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 combinatorically optimized empirical range-separated hybrids wB97X-V and wB97M-V., Comment: Journal of Chemical Theory and Computation (2021)

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 "Jacobs 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., Comment: Israel Journal of Chemistry, EarlyView (2020); 19 pages, Open Access

Benchmarks that span a broad swath of chemical space, such as GMTKN55, are very useful for assessing progress in the quest for more universal DFT functionals. We find that the WTMAD2 metrics for a great number of functionals show a clear "Jacob's Ladder hierarchy"; that the "combinatorial" development strategy of Head-Gordon and coworkers generates "best on rung" performers; that the quality of the nonlocal dispersion correction becomes more important as functionals become more accurate for nondispersion properties; that fitting against small, unrepresentative benchmark sets leads to underperforming functionals; and that {\omega}B97M(2) is currently the best DFT functional of any kind, but that revDSD-D4 functionals are able to reach similar performance using fewer parameters, and that revDOD-D4 in addition permits reduced-scaling algorithms. If one seeks a range-separated hybrid (RSH) GGA that also performs well for optical excitation energies, CAM-QTP-01 may be a viable option. The D4 dispersion model, with its partial charge dependence, appears to be clearly superior to D3BJ and even possibly NL. Should one require a double hybrid without dispersion model, noDispSD-SCAN is a viable option. Performance for the MOBH35 transition metal benchmark is different: the best double hybrids are competitive but not superior to {\omega}B97M-V, which offers the best performance compromise for mixed main group-transition metal problems., Comment: 5 pages (ICCMSE-2019 conference proceedings), AIP Conference Proceedings, in press