Dispersion-corrected r²SCAN based double-hybrid functionals Open Access

Kohn–Sham (KS) density functional theory (DFT) is widely accepted as the work-horse of computational chemistry due to its excellent cost-accuracy ratio.^1,2 According to Perdew’s “Jacob’s Ladder” picture, density functionals can be categorized with respect to their physical ingredients.³ The highest, fifth rung is represented by the so-called double-hybrid (DH) functionals that include both an admixture of Hartree–Fock exchange (HFX) and a wavefunction-theory-based correlation contribution into the density functional formulation. The latter is usually computed by second-order perturbation theory (PT2), with the second-order Møller–Plesset theory (MP2) being the most prominent method of choice.^4–7 

A crucial part of the DH approach is the choice of the underlying exchange (X) and correlation (C) functionals. While, in principle, any reasonable XC functional can be combined in the DH scheme, the quality of the XC functionals can strongly influence the overall performance of the DH functional. In recent years, two functional design philosophies have emerged: (i) semi-empirical density functionals that typically include fitting to extensive data sets, such as B3LYP^8,9 or ωB97X-V,¹⁰ and (ii) non-empirical functionals that try to fulfill specific mathematical and physical constraints, such as TPSS¹¹ or PBE0.¹² The latter is inspired by the expectation of better transferability even though the superiority of any of these philosophies is the subject of ongoing debate.⁶ Nevertheless, recent functional development brought forth the successful, non-empirical SCAN family of density functionals.^13–15 Specifically, the regularized and restored SCAN meta-GGA functional (r²SCAN^16,17) and its London dispersion-corrected variants^18,19 have proven valuable additions. r²SCAN was also successfully used in hybrid functional frameworks (e.g., r²SCAN0)^20,21 and in the composite DFT scheme r²SCAN-3c.^22,23 The r²SCAN-QIDH double hybrid was already successfully used for the investigation of singlet–triplet excited-state inversion of two hydrocarbons,²⁴ and first rudimentary tests were already conducted with r²SCAN0-DH, r²SCAN-QIDH, and r²SCAN0-2 on the GMTKN55 database.^25,26 Accordingly, an extensive investigation of the r²SCAN functional in the double-hybrid scheme seems to be promising.

The philosophy of non-empirical functionals can also be transferred to double-hybrid functional construction by deriving the admixture parameters for HFX and MP2 contributions from the adiabatic connection formalism.^27,28 Pioneered by Sharkas, Toulouse, and Savin,²⁹ several parameter-free double-hybrid methods have been developed, including DFT0-DH,³⁰ DFT0-2,³¹ DFT-QIDH,^32,33 and DFT-CIDH.³⁴ With these schemes, four non-empirical r²SCAN-double-hybrids were generated, resulting in the r²SCAN0-DH, the r²SCAN-CIDH, the r²SCAN-QIDH, and the r²SCAN0-2 functionals.

In this work, we present four semi-empirical r²SCAN-based double-hybrids, named Pr²SCAN50, Pr²SCAN69, κPr²SCAN50, and ωPr²SCAN50, where we optimized the MP2 contribution empirically. Pr²SCAN50 is a 50% HFX global double-hybrid that targets overall robustness and follows the philosophy of the successful PWPB95. Pr²SCAN50 is further combined with the κ-MP2-regularization scheme recently proposed by Shee et al.³⁵ MP2-regularization is typically applied to overcome the divergent behavior of MP2 for small orbital energy gaps, thus adding another layer of robustness. This scheme has already been investigated in the context of double-hybrids and was found to also improve the performance for low amounts of HFX.³⁶ We further tested range-separated HFX in the ωPr²SCAN50 functional to improve the description of long-range exchange effects, which has also already been successfully employed for double-hybrids.^37–39 Finally, we also optimize a high-HFX variant Pr²SCAN69 (69% HFX) inspired by the success of other global double-hybrid functionals that perform well for main-group thermochemistry.^40,41

A general problem of common density functional approximations (DFAs) is the insufficient description of long-range correlation effects, manifesting in a systematic underestimation of London dispersion interactions.⁴² This cannot be completely corrected for by the inclusion of MP2 in the correlation part, and thus, further London dispersion corrections are necessary. Some of the most common dispersion corrections schemes are our D3⁴³ and D4^44–46 corrections as well as the variants of Vydrov and Van Voorhis VV10⁴⁷ model, including rVV10⁴⁸ and DFT-NL.^49,50 Specifically, the efficient DFT-D approach has proven reliable in countless quantum chemical applications and workflows.^51–53 Therefore, all optimized functionals presented in this work were combined with tailored London dispersion corrections (D4 and NL).

All constructed functionals were assessed using a comprehensive collection of 90 diverse benchmark sets, including the GMTKN55,²⁶ MOR41,⁵⁴ and ROST61⁵⁵ benchmark sets. Overall, 25 800 data points were evaluated, yielding a very reliable statistical assessment of the functional performance, and a comparison to the so far successful PWPB95-D4 functional is drawn.

Dispersion correction parameters were fitted in line with the proven original parametrization strategy against the S22 × 5,⁶⁴ NCIBLIND10,⁶⁵ and S66 × 8⁶⁶ (2022 revision by Martin et al.⁶⁷) benchmark sets for non-covalent interaction energies by least-squares minimization. For DFT-D4, a modified Levenberg–Marquardt algorithm is used, while DFT-NL was optimized via a simplex algorithm.^68,69 All parameters were fitted based on DFT results close to the complete basis set limit using the large def2-QZVPP⁷⁰ quadruple-ζ basis set. This approach was shown to be more specific to only correct London dispersion effects compared to approaches such as minimizing the WTMAD-2 on the GMTKN55 by Santra and Martin.^21,39,41 Such approaches can be susceptible to overfitting due to the inclusion of systems prone to various other functional error sources, such as the self-interaction error (SIE).

All quantum chemical calculations were performed with the ORCA 5.0.3 program package.^71,72 For regularized MP2 and range-separated ωr²SCAN calculations, a development version of ORCA 5.0 was used. Ahlrich’s quadruple-ζ basis set def2-QZVPP⁷⁰ (def2-QZVP for GMTKN55 for comparability and def2-QZVPD⁷³ for several GMTKN55 subsets) with matching effective core potentials (ECPs)^74,75 for heavy elements with Z > 36 was generally employed. The RIJCOSX⁷⁶ approximation was used for the self-consistent field (SCF) part and RI for the MP2 part to accelerate the calculations. Matching auxiliary basis sets were applied as implemented in ORCA (def2/J and def2-QZVPP/C). The DefGrid3 option was applied for the numerical integration grid as well as TightSCF convergence criteria as implemented in ORCA.

DFT-D4 dispersion corrections were calculated with the dftd4 3.4.0 standalone program, whereas for DFT-NL, the ORCA native, post-SCF implementation was used.

The adiabatic-connection-derived double-hybrids r²SCAN0-DH, r²SCAN-CIDH, r²SCAN-QIDH, and r²SCAN0-2 were combined with the D4 dispersion correction and evaluated as a starting point for further optimization. The corresponding parameters are presented in Table I. Their performance is discussed in Sec. VI.

The extensive GMTKN55 database is a compilation of 55 main-group thermochemistry benchmark sets covering five different categories. These are basic properties and reactions of small systems (basic properties), isomerizations and reactions of large systems (reactions), barrier heights (barriers), intermolecular noncovalent interactions (intermol. NCIs), and intramolecular noncovalent interactions (intramol. NCIs). The ROST61 dataset contains open-shell metal-organic reactions, while the MOR41 focuses on closed-shell metal-organic reactions, both including relatively large molecules containing up to 120 atoms. The TMBH benchmark set covers a range of transition-metal reactions, combining different catalytic pathways and species, whereas the MOBH35 focuses on transition-metal reaction barrier heights, offering insights into a variety of transition-metal-mediated reactions. These benchmark sets provide an insight into the performance of the presented functionals for metal-organic and transition-metal chemistry. By incorporating not only the GMTKN55 but also transition-metal sets into the optimization procedure, we aim at an overall better robustness and transition-metal thermochemistry performance of our functionals at the cost of slight performance losses on the GMTKN55. The final parameters for the double-hybrids and the ωr²SCAN range-separated hybrid are given in Table I.

We optimized the a_C of the 50% Fock exchange non-empirical double-hybrid (r²SCAN0-DH), yielding the Pr²SCAN50 functional. There are several arguments for a double-hybrid with around 50% HFX. First of all, lower amounts of HFX lead to a better implicit description of static correlation, which improves the quality of reference for MP2—but at the expense of a better description of dynamic correlation effects.⁵ It is also a reasonable compromise for the need for high amounts of Fock exchange for main-group chemistry without deteriorating results for transition-metal chemistry, where more static correlation is encountered. Therefore, especially transition-metal chemistry is prone to larger errors with high Fock exchange double-hybrids.

The performance of Pr²SCAN50-D4 as a function of a_C is depicted in Fig. 1, where the dotted line indicates the performance of the non-empirical r²SCAN0-DH-D4 for comparison. We found that the lowest WTMAD-2^fit of 2.52 kcal mol⁻¹ is achieved at a_C = 0.25, resulting in the Pr²SCAN50 functional. Pr²SCAN50-D4 yields an improved WTMAD-2^fit by 0.81 kcal mol⁻¹ over the non-empirical version. This value consists of 1.16 kcal mol⁻¹ on the GMTKN55 and 0.47 kcal mol⁻¹ on the transition-metal sets (WTMAD-$2TMfit$⁠). We observed that especially the transition-metal sets are very susceptible to the amount of MP2 correlation employed and that performance worsens rapidly above a_C = 0.25, whereas the GMTKN55 is less sensitive with a shallower minimum at a_C = 0.30.

We further optimized the a_C of the 69% Fock exchange non-empirical double-hybrid (r²SCAN-QIDH), yielding the Pr²SCAN69 functional. This was motivated by the relatively promising performance of the 69% HFX variant, r²SCAN-QIDH-D4, among the non-empirical double-hybrids and by the apparent sweet spot of around 69% Hartree–Fock exchange for main-group thermochemistry—as previously found by Martin et al. on the GMTKN55 database.

The performance of Pr²SCAN69-D4 as a function of a_C is depicted in Fig. 2, where the dotted line indicates the non-empirical r²SCAN-QIDH-D4 as a reference. The lowest WTMAD-2^fit of 2.19 kcal mol⁻¹ is achieved at a_C = 0.44, resulting in the Pr²SCAN69 functional. Pr²SCAN69-D4 yields an improved WTMAD-2^fit by 0.53 kcal mol⁻¹ over the non-empirical version. This value consists of 0.79 kcal mol⁻¹ on the GMTKN55, yielding a WTMAD-2_GMTKN55 of 2.81 kcal mol⁻¹ and 0.28 kcal mol⁻¹ on the transition-metal sets (WTMAD-$2TMfit$⁠).

For the 69% Pr²SCAN69, the same observation as with Pr²SCAN50-D4 is made, and the GMTKN55 profits from more MP2 correlation compared to the TMs, where a smaller a_C is favorable.

With the κ-regularization, the κ parameter has to be determined in addition to a_C. Therefore, we kept the optimal a_C = 0.25 value of the κPr²SCAN50 functional fixed and optimized only the κ value. Since Santra and Martin found that utilizing a (stronger) regularization is accompanied by a larger optimal amount of MP2 correlation a_C, we also optimized κ for a slightly larger a_C = 0.30 value. The resulting performance with a_C = 0.25 and a_C = 0.30 for three κ values is given in Table II, and details on the optimization can be found in Appendix C1 of the supplementary material.

On the GMTKN55 and the four transition-metal sets, a minor regularization for κPr²SCAN50-D4 with a_C = 0.25 and κ = 3.62 was beneficial for the overall performance. Main-group thermochemistry degrades quickly by employing a stronger regularization (smaller κ), whereas for transition-metal thermochemistry, a slightly stronger regularization yields better results. For a_C = 0.30, employing a slight regularization of κ = 2.75 allows to retain the very good GMTKN55 performance while simultaneously obtaining better results for transition-metal thermochemistry than with Pr²SCAN50-D4.

By employing regularized MP2, we expect an improvement, especially for systems that have small orbital energy gaps. The C60-ISO subset of the GMTKN55 has a small mean HOMO–LUMO gap of $<4$ eV with Pr²SCAN50, and therefore, regularization should have a large effect for this set. For both a_C = 0.25 and 0.30, we found a substantial improvement by utilizing the regularized MP2 scheme, as shown in Table II. For a_C = 0.25, the MAD decreases from 3.61 to 2.70 kcal mol⁻¹, and for a_C = 0.30, it decreases from 5.61 to 2.20 kcal mol⁻¹.

In line with our strategy, the parameters a_C = 0.30 and κ = 2.75 were chosen for the κPr²SCAN50 functional as they deliver the best performance with a WTMAD-2^fit of 2.45 kcal mol⁻¹.

The parametrization of κPr²SCAN69 was carried out analogous to κPr²SCAN50, testing a_C = 0.44 and a_C = 0.49, and the results can be found in Appendix C2 of the supplementary material. Contrary to κPr²SCAN50, employing a regularization for both tested 69% HFX variants does not provide an overall benefit for thermochemistry. For very small-gap systems, such as the C60-ISO, we still find regularization to be very beneficial. For a_C = 0.44, the MAD decreases from 3.97 to 2.30 kcal mol⁻¹, and for a_C = 0.50, it decreases from 5.43 to 2.08 kcal mol⁻¹.

As the unregularized Pr²SCAN69-D4 delivers the same or better overall results, κPr²SCAN69-D4 was discarded.

As a starting point for the range-separated double-hybrid, the plain hybrid was optimized (for details, see Appenndix D1 of the supplementary material). We found that a_X = 0.0 and ω = 0.3 bohr⁻¹ yielded a WTMAD-2^fit of 3.82 kcal mol⁻¹. In comparison to the well-performing global hybrid r²SCAN0-D4, this is an improvement of 0.24 kcal mol⁻¹ in the WTMAD-2^fit. This gain in performance over the r²SCAN0-D4 consists of 0.33 kcal mol⁻¹ on the GMTKN55 and of 0.14 kcal mol⁻¹ WTMAD-$2TMfit$ on the transition-metal sets.

The performance of ωr²SCAN-D4 and the NL variant across all benchmark sets is given in Appendix D1 of the supplementary material.

For the double-hybrid, the range-separation parameter ω was determined non-empirically by enforcing the exact treatment of the ground-state energy of the hydrogen atom³⁷ (see Appendix D2 of the supplementary material for details). It was shown by Alipour and Karimi and Brémond et al. that the parameter obtained this way yields results that are competitive with the ones obtained via conventional fitting or optimal tuning approaches.^87,88

With the non-empirically determined ω, the amount of MP2 correlation a_C was optimized with respect to the WTMAD-2^fit. The resulting a_C-scan is shown in Fig. 3. This yielded an optimal amount of MP2 correlation of a_C = 0.35 with ω = 0.2140 bohr⁻¹ for the final ωPr²SCAN50, which improved the WTMAD-2^fit by 0.16 kcal mol⁻¹ over Pr²SCAN50-D4. This improvement is made up of 0.28 kcal mol⁻¹ on the GMTKN55 and of 0.05 kcal mol⁻¹ WTMAD-$2TMfit$ on the transition-metal sets.

For the 69% range-separated double-hybrid, the range-separation parameter ω was also determined by enforcing the exact ground-state energy of the hydrogen atom. The optimization of the a_C-parameter is given in Appendix D3 of the supplementary material.

Employing a range-separation to Pr²SCAN69 with an already high amount of global HFX does not improve the overall performance over the global HFX variant, Pr²SCAN69-D4. Consequently, ωPr²SCAN69-D4 was discarded.

To evaluate the performance of the proposed functionals, the fit set is extended by 25 main-group and six transition-metal thermochemistry benchmark sets. A list of all sets used in addition to GMTKN55 is shown in Table III. In total, the presented functionals are evaluated for 90 different benchmark sets, totaling 25 800 data points. A comparison of the performance for selected functionals on these benchmark sets and a comparison to the well-established PWPB95-D4 double-hybrid⁸⁹ are given in Fig. 4. For this purpose, three additional statistical descriptors were employed, namely, the total WTMAD-2_total, which is the WTMAD-2 over all 90 benchmark sets; the transition-metal (TM) WTMAD-$2TMall$⁠, which is the WTMAD-2 for all 10 transition-metal chemistry sets, and the WTMAD-2_MG, which is calculated for all 25 main-group sets excluding the GMTKN55.

In the following, we will discuss these categories separately and focus on Pr²SCAN69, Pr²SCAN50, and ωPr²SCAN50 in combination with the D4 dispersion correction, since using the NL correction instead does not change the overall trends. Worth mentioning is, however, that Pr²SCAN69-NL and Pr²SCAN50-NL outperform their D4 counterpart, especially in main-group thermochemistry, while D4 always outperforms NL on the transition-metal thermochemistry sets. The comparison of D4 and NL over all tested benchmarks is given in Appendixes E2 and E3 of the supplementary material.

The results of all presented functionals for the GMTKN55 and for the respective subsets are given in Table IV and are visualized for PWPB95-D4, Pr²SCAN69-D4, Pr²SCAN50-D4, and ωPr²SCAN50-D4 in Fig. 5. For the GMTKN55 database, Pr²SCAN69-D4 (a_X = 0.69) is the best-performer with a very good WTMAD-2_GMTKN55 of 2.81 kcal mol⁻¹, followed by ωPr²SCAN50 (a_X = 0.5) with a WTMAD-2_GMTKN55 of 3.08 kcal mol⁻¹.

The global Pr²SCAN50-D4 and its regularized variant κPr²SCAN50-D4 with a WTMAD-2_GMTKN55 at around 3.3–3.4 kcal mol⁻¹ still perform very well. All of the mentioned functionals surpass the PWPB95-D4 with its WTMAD-2_GMTKN55 of 4.06 kcal mol⁻¹.

The decline in performance of Pr²SCAN50-D4 compared to Pr²SCAN69-D4 is mainly due to the slightly worse performance in basic properties, barriers, and intermolecular NCIs. The basic properties and barriers subsets are substantially influenced by the amount of Hartree–Fock exchange used. The decline in performance on the intermolecular NCIs by the 50% HFX functionals is likely attributable to the lower fraction of MP2 correlation used, compared to Pr²SCAN69-D4, since medium-range correlation might be covered better with larger MP2 contributions.

Employing range-separated Fock exchange with ωPr²SCAN50-D4 recovers some of the lost performance of Pr²SCAN50-D4 and κPr²SCAN50-D4 compared to Pr²SCAN69-D4 in the basic properties, reactions, and barrier subsets. Reactions of large systems and intramolecular NCIs are fairly equal among all variants with ωPr²SCAN50-D4 being the best performer on the reactions subsets and Pr²SCAN50-D4 on the intramolecular NCIs.

A key ingredient of double-hybrid functionals is the typically large (≥50%) admixture of Fock exchange (a_X). On the GMTKN55, high amounts of Fock exchange seem to be beneficial for double-hybrids as the performance of Pr²SCAN69-D4 is unreachable by our 50% HFX double-hybrid variants. This is in line with the observations of Martin et al., which identify the region around a_X = 0.62 and 0.72 as optimum for the performance on the GMTKN55 database.

The proposed 50% HFX functionals outperform PWPB95-D4 by at least 0.68 kcal mol⁻¹ WTMAD-2_GMTKN55 (Pr²SCAN50-D4) up to 0.98 kcal mol⁻¹ with ωPr²SCAN50-D4 on the whole GMTKN55. PWPB95-D4 is slightly better for basic properties and barriers while being greatly surpassed in all other categories.

The performance of Pr²SCAN69-D4, Pr²SCAN50-D4, κPr²SCAN50-D4, and ωPr²SCAN50-D4 is remarkable, considering that these are not or not heavily fitted to GMTKN55. In general, the aim of our presented functionals is robustness for a broad range of chemical systems. The so-called “mindless benchmark” (MB16-43) of the GMTKN55 is considered a valuable indicator in this respect. The MADs for a variety of state-of-the-art functionals together with the newly created double-hybrids are shown in Fig. 6. Here, the global Pr²SCAN69-D4 and Pr²SCAN50-D4 perform very well, with a MAD of 8.13 and 7.32 kcal mol⁻¹, respectively. The κPr²SCAN50-D4 and the ωPr²SCAN50-D4 functionals are slightly worse compared to their plain counterparts but still very good with MADs of 9.31 and 10.16 kcal mol⁻¹, respectively.

Even though the GMTKN55 provides a comprehensive and extensive number of data points, many other more specialized benchmark sets covering an extended chemical space are available (cf. Table III). Some examples are the LP14⁹⁹ (Lewis-pair interactions), CHAL336⁹³ (chalcogenide bonding interactions), IONPI19⁹⁷ (ion-π interactions), and AC12⁹¹ (singlet-triplet energy splittings) benchmark sets. A comprehensive figure for all these additional main-group thermochemistry benchmarks is given in Appendix E1 of the supplementary material.

In general, Pr²SCAN50-D4, κPr²SCAN50-D4, ωPr²SCAN50-D4, and Pr²SCAN69-D4 perform well, which is shown by the overall main-group thermochemistry WTMAD-2_MG (i.e., all 25 main-group benchmark sets without the GMTKN55). Pr²SCAN50-D4 achieves 0.96, κPr²SCAN50-D4 0.94, Pr²SCAN69-D4 0.80, and ωPr²SCAN50-D4 0.72 kcal mol⁻¹, i.e., all surpass the performance of PWPB95-D4 with 1.08 kcal mol⁻¹.

Regarding the performance on (intermolecular) non-covalent interactions, i.e., the sets from the non-covalent interactions atlas project, NCIBLIND10, S66 × 8 and S22 × 5, Pr²SCAN69-D4, and ωPr²SCAN50-D4 perform similarly with very distinct trends. The high amounts of global HFX as in Pr²SCAN69-D4 seem to be more beneficial for (ionic) hydrogen bonding and chalcogenide bonding interactions (IHB100 × 10,⁹⁶ HB375 × 10,⁹⁶ HB300SPX,⁹⁵ and CHAL336), whereas the range-separated, lower HFX ωPr²SCAN50-D4 seems to be beneficial for general London dispersion interaction, repulsive contacts, and σ–hole interactions (D442 × 10,⁹⁴ D1200,⁹⁴ R160 × 6,^101,102 R739 × 5,¹⁰³ and SH250 × 10¹⁰⁶).

The lower, global HFX Pr²SCAN50-D4 seems to be beneficial for ion–π interactions (IONPI19) and often performs on par with the other two, as with lewis-pair, σ–hole, and general London dispersion interactions (LP14, SH250 × 10, NCIBLIND10, S66 × 8, S22 × 5, and X40 × 10¹⁰⁸). The added regularization of κPr²SCAN50-D4 compared to the unregularized Pr²SCAN50-D4 does not change the performance much on intermolecular NCI sets.

The performance for conformer energies (i.e., intramolecular noncovalent interactions) in the extended benchmarks is similar to the performance trends on the intramolecular NCI subsets of GMTKN55. The sets included are the 37CONF8, ACONFL⁹² and MPCONF196. In all cases, Pr²SCAN69-D4 performs overall the best, while Pr²SCAN50-D4 performs equally on the ACONFL and ωPr²SCAN50-D4 even slightly better on the 37CONF8. The performance of κPr²SCAN50-D4 compared to the unregularized Pr²SCAN50-D4 is very similar again.

The behavior on the extended barriers and SIE benchmark sets is also similar to the barrier subsets of the GMTKN55. These sets include the barriers of the revBH9^104,105 set and the SIE8¹⁰⁷—the SIE4 × 4 from the GMTKN55 (see Appendix G41 of the supplementary material) will also be included. Generally, Pr²SCAN69-D4 is the overall best-performer, as high amounts of global HFX are efficient for correction of the self-interaction error. Consequently, none of the 50% HFX functionals can match the performance of the global high HFX Pr²SCAN69-D4, which is especially evident on the SIE4 × 4 and SIE8. Employing the range-separation does benefit performance on these sets, with ωPr²SCAN50-D4 nearly catching up to the global Pr²SCAN69-D4, indicating a reasonable reduction of the SIE compared to its global counterpart Pr²SCAN50-D4. Regularization, however, seems to reduce SIE and improves barriers in some cases, yet worsen in others. The SIE8 and SIE4 × 4 are worse, while the revBH9 is better for κPr²SCAN50-D4 compared to the unregularized Pr²SCAN50-D4.

A comprehensive description for the transition-metal thermochemistry benchmarks is given in Fig. 7, and the complete errors of all functionals are given in Table III.

Large amounts of Fock exchange can sometimes be favorable for main-group thermochemistry, as observed for GMTKN55, but can potentially be problematic for transition-metal chemistry where a higher degree of static correlation effects can be expected. Similar to main-group chemistry, the transition-metal thermochemistry of all four functionals—Pr²SCAN50, Pr²SCAN69, κPr²SCAN50, and ωPr²SCAN50—show similar performance, as shown in Fig. 4, and all functionals are within 0.11 kcal mol⁻¹ WTMAD-$2TMall$ of each other. As for the main-group sets, the high HFX Pr²SCAN69-D4 functional performs especially well for benchmark sets containing barrier heights, such as the LTMBH, TMBH, and the MOBH35, while slightly worse performance is observed for the coinage metal clusters of the CUAGAU-2 set and the organometallic reaction energies of the MOR41. In both of these sets, Pr²SCAN69-D4 is outperformed by Pr²SCAN50-D4 and ωPr²SCAN50-D4. The overall best-performer is Pr²SCAN69-D4 with a WTMAD-$2TMall$ of 1.98 kcal mol⁻¹, closely followed by ωPr²SCAN50-D4, Pr²SCAN50-D4, and κPr²SCAN50-D4 with 2.04, 2.08, and 2.09 kcal mol⁻¹, respectively. All presented variants outperform the PWPB95-D4 with its WTMAD-$2TMall$ of 2.24 kcal mol⁻¹. Surprisingly, the κPr²SCAN50 functional performs similarly to Pr²SCAN50 and is not able to benefit from the MP2 regularization. This might be explained by the overall higher (5%) fraction of MP2 correlation in κPr²SCAN50.

The yet good performance of the high HFX Pr²SCAN69-D4 functional for transition-metal chemistry is counter-intuitive as static correlation is expected to play a larger role in transition-metal chemistry than in main-group chemistry. Nevertheless, the 50% variants tend to outperform Pr²SCAN69-D4 on the ROST61, CUAGAU-2, and TMIP (ΔMAD_{Pr²SCAN50-D4} = −0.13, −0.55, and −0.97 kcal mol⁻¹, respectively) that include open-shell transition-metal complexes that are specifically prone to static correlation effects. Regarding the relatively small improvement, one has to consider that almost all transition-metal benchmark sets employed were designed to be well-behaved with respect to static correlation and employ single-reference methods to generate their benchmark data (with the exception of the TMIP set). Therefore, static correlation might be less of an issue here as it potentially is for real chemical applications, e.g., modeling catalytic cycles with 3d transition-metal complexes. Additionally, the effects of SIE can be large in transition-metal chemistry¹¹⁹ since more polar bonds occur. Therefore, high HFX double-hybrids can be favorable in situations with less static correlation but significant SIE, as typically observed for reaction barriers, covered by, e.g., the LTMBH and MOBH35 benchmark sets.

Additionally, the ωB97M-V functional¹²⁰ was also evaluated over all test sets and compared to the range-separated functional ωr²SCAN-D4 presented in this work (see Appendix E of the supplementary material for details). We find that ωB97M-V yields excellent results for the GMTKN55 with a WTMAD-2^GMTKN55 of 3.26 kcal mol⁻¹, which is 2.05 kcal mol⁻¹ better than the performance of ωr²SCAN-D4. However, ωB97M-V and ωr²SCAN-D4 perform equally well on the transition metal sets with WTMAD-$2TMall$ of 3.04 and 3.07 kcal mol⁻¹, respectively. On the main-group sets without the GMTKN55, ωB97M-V slightly outperforms ωr²SCAN-D4 with a WTMAD-2_MG of 0.70 kcal mol⁻¹ compared to 1.05 kcal mol⁻¹. Overall, we find a WTMAD-2^all for ωr²SCAN-D4 of 4.46 kcal mol⁻¹, while ωB97M-V achieves 2.99 kcal mol⁻¹. Therefore, ωr²SCAN-D4 can be employed as a robust alternative to ωB97M-V with the advantage of lower empiricism but with some drawbacks in the overall accuracy for thermochemistry.

We find a WTMAD-2^all of 4.02 for Pr²SCAN50-D4, 3.35 for Pr²SCAN69-D4, and 3.05 kcal mol⁻¹ for ωPr²SCAN-D4. All of those surpass the performance of the robust PWPB95-D4 with a value of 4.51 kcal mol⁻¹. The presented ωPr²SCAN-D4 delivers equal performance compared to the empirical non-DH ωB97M-V with its WTMAD-2^all of 2.99 kcal mol⁻¹. However, Pr²SCAN50-D4, ωPr²SCAN-D4, and Pr²SCAN69-D4 all outperform ωB97M-V by nearly 1.00 kcal mol⁻¹ on the transition metal sets with their WTMAD-$2TMall$ of 2.08, 2.04, and 1.98 kcal mol⁻¹, respectively. In addition to the comparison to the robust PWPB95-D4 functional, the well-known revDSD-PBEP86-D4 functional⁴⁰ (see Appendix E of the supplementary material) was tested and yielded a WTMAD-2^all of 2.65, which is almost matched by our less empirical ωPr²SCAN-D4. In revDSD-PBEP86-D4, the combination of exchange and correlation functionals was additionally specifically selected for ideal performance for thermochemistry on the GMTKN55, while the underlying exchange and correlation functionals in our work remain unchanged (with the exception of the range-separation). Therefore, revDSD-PBEP86-D4 includes an additional level of empiricism, which can decrease overall robustness as shown by the relatively bad performance on the MB16-43 in Fig. 6. In passing, we note that Pr²SCAN69-NL and Pr²SCAN50-NL perform 0.35 and 0.18 kcal mol⁻¹, respectively, better than its D4 variants on the WTMAD-2^all. For the other functionals, NL variants perform worse.

For the range-separated ωr²SCAN-D4 hybrid, we find a WTMAD-3_all of 1.01 kcal mol⁻¹, while ωB97M-V achieves a significantly better WTMAD-3 of 0.68 kcal mol⁻¹. For the double-hybrids, we obtain a WTMAD-3^all of 0.80 for Pr²SCAN50-D4, 0.77 for κPr²SCAN50-D4, and 0.66 kcal mol⁻¹ for both Pr²SCAN69-D4 and ωPr²SCAN50-D4. Each of those surpasses the performance of PWPB95-D4 with its 0.89 kcal mol⁻¹ WTMAD-3^all. The two best-performing DHs yield a smaller WTMAD-3 than ωB97M-V, while revDSD-PBEP86-D4 with a WTMAD-3^all of 0.59 is almost matched. The WTMAD-3 measure stresses that our presented double-hybrid functionals are robust and less empirical alternatives to ωB97M-V and revDSD-PBEP86-D4.

In this work, we present and assess dispersion-corrected double-hybrid functionals based on the parent r²SCAN meta-GGA functional for a variety of chemical systems with 25 800 (relative) energy data points (thermochemistry). In line with the philosophy of physically motivated low-empirical functionals, adiabatic-connection-derived double-hybrid variants of the 0-DH, CIDH, QIDH, and 0–2 types using spin-opposite-scaled MP2 were combined with the state-of-the-art London dispersion correction D4.

These were used as a starting point for further modification. It was found that the optimized quadratic integrand double-hybrid, with adjusted amount of MP2 correlation, Pr²SCAN69-D4/NL performs well for a variety of main-group and organometallic thermochemistry benchmark sets. Nevertheless, the main target of this work was to find a DH variant that does not rely on often problematic high amounts of Fock exchange (>60%). Inspired by the so far successful PWPB95 functional that employs 50% of HFX, the mediocre performing r²SCAN0-DH was used as a starting point to create a modified variant by adjusting the amount of MP2 correlation and using a tailored dispersion correction. The resulting Pr²SCAN50-D4/NL functional with 50% HFX and 25% MP2 clearly improved the performance over the parental 0-DH type functional, yielding a good WTMAD-2_GMTKN55 of 3.38 kcal mol⁻¹. Furthermore, PWPB95-D4 (WTMAD-2_GMTKN55 = 4.06 kcal mol⁻¹) is outperformed by 0.68 kcal mol⁻¹. Accordingly, Pr²SCAN50-D4 represents a promising and robust replacement for PWPB95-D4 in the class of global double-hybrid functionals.

We further combined Pr²SCAN50 with Head-Gordon’s MP2-regularization scheme employing the κ-regularizer (κPr²SCAN50-D4/NL), thus introducing another level of robustness regarding the shortcoming of MP2 diverging for small orbital energy gaps. Nevertheless, introducing MP2-regularization yields only minor improvements compared to the unregularized variant.

Finally, Pr²SCAN50 was extended by the introduction of range-separated Fock exchange using the ω-formalism. ωPr²SCAN50-D4 further improves on Pr²SCAN50-D4, reducing the GMTKN55 WTMAD-2 to a very good value of 3.08 kcal mol⁻¹ while also slightly improving on the good performance of Pr²SCAN50-D4 for organometallic thermochemistry. The range separation in the exchange part of the functional further improves the functionals flexibility in terms of specific enhancement in the framework of optimal tuning and its potential usage for excited state calculations and other properties.

Concluding, we assessed various r²SCAN-based double-hybrid functionals for main-group and organometallic thermochemistry and present a global 69% HFX Pr²SCAN69 as well as the novel low-HFX global double-hybrid Pr²SCAN50 and its κ-regularized (κPr²SCAN50) and range-separated (ωPr²SCAN50) variants that all perform very well and robust for a large variety of chemical systems. The new r²SCAN double-hybrid family represents a valuable addition to the field of double-hybrid functionals that can be recommended for general exploratory use.

See the supplementary material for complete statistical data (Appendix G), functional availability information, sample inputs for ORCA (Appendix F), and definition of all used statistical measures (Appendix A).

The German Science Foundation (DFG) is gratefully acknowledged for financial support (Grant No. 1927/16-1). Furthermore, S.G. and M.B. gratefully acknowledge the financial support of the Max Planck Society through the Max Planck fellow program.

The authors have no conflicts to disclose.

L.W. and H.N. contributed equally to this work.

Lukas Wittmann: Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). Hagen Neugebauer: Data curation (equal); Formal analysis (equal); Methodology (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). Stefan Grimme: Investigation (equal); Project administration (equal); Resources (lead); Supervision (equal); Validation (equal); Writing – original draft (equal); Writing – review & editing (equal). Markus Bursch: Conceptualization (lead); Formal analysis (equal); Investigation (lead); Methodology (equal); Project administration (lead); Supervision (lead); Writing – original draft (equal); Writing – review & editing (equal).

The data that support the findings of this study are available within the article and its supplementary material.