The results for the complete set are presented in Figure 3 in a highly condensed form as mean absolute deviation (MAD) from the reference values, and more details are given in the Supporting Information. Over all 164 points in the test set, the best functional is PBE0-D3 with an MAD of 1.1 kcal mol−1, followed by PW6B95-D3, the corresponding double hybrid PWPB95-D3, and B3LYP-D3 (each 1.9 kcal mol−1). These results are in line with the findings of Quintal et al.47
Figure 3. MAD over the complete set compared with CCSD(T)/CBS. The functionals are ordered according to Jacob’s ladder and Fock exchange. The bars are capped if the MAD is >5.5 kcal mol−1.
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The performances of the GGAs BP86-D3, PBE-D3, and BLYP-D3 are similar, with respective MADs of 4.1, 3.9, and 3.6 kcal mol−1. B97-D3 is outstanding here with a value of 2.9 kcal mol−1, which is closer to those of the hybrids. M06L performs surprisingly well, with an MAD of 2.2 kcal mol−1. The other meta-GGAs TPSS and a re-optimized version, namely oTPSS, have similar MADs to those of most of the other GGAs (3.8 and 3.9 kcal mol−1, respectively).
Compared with the other hybrid functionals BHLYP-D3 performs much worse. The main problem is the description of the van der Waals (vdW)-type reactant complex, which is much too high in energy (up to 10 kcal mol−1 in some cases, see Supporting Information). The hybrids are ordered with respect to the amount of Fock exchange included, from which an optimum of ∼25 % as in PBE0 can be deduced.
In the class of the meta-hybrid functionals, the M0X approaches partially fail. The problem is related to the Fock exchange contribution as for the other hybrids. This is apparent by the corresponding MADs, which drop from 7.0 kcal mol−1 (M06-HF, 100 % Fock exchange), over 6.3 kcal mol−1 (M06-2X, 54 % Fock exchange) to 4.9 kcal mol−1 (M06, 27 % Fock exchange). A similar behavior as with M06 is obtained for BMK-D3, with an MAD of 4.6 kcal mol−1. This result is in line with the findings of Quintal et al.48 A partial failure of the ‘high-Fock-exchange’ M06 functionals for transition-metal systems had already been noted by the developers of these methods, and is therefore not unexpected. The best meta-hybrid is PW6B95-D3 (1.9 kcal mol−1), which has a similar admixture of Fock exchange (28 %) as PBE0-D3.
In the group of double hybrids, PWPB95-D3 shows the best performance with an MAD of 1.9 kcal mol−1, and it is second best of all tested methods, followed by B2PLYP-D3 (2.8 kcal mol−1), the re-parameterized version B2GPPLYP-D3 (3.6 kcal mol−1), and DSD-BLYP (4.5 kcal mol−1). Notably, the DHDF with the highest amount of Fock exchange performs worst, and the functional with the lowest, best. Another interesting observation is made by comparing B2PLYP-D3 and PWPB95-D3, as both functionals have a similar amount of Fock exchange with 53 % and 50 %, respectively. This also holds for the perturbative part with 27 % and 26.9 %, respectively. However, the behavior of PWPB95-D3 is significantly (almost 1 kcal mol−1) better than B2PLYP-D3. The reason is that the perturbative part in PWPB95 contains only the opposite-spin (OS) component of the second-order correlation energy. In Figure 3 different variants of MP2 are also shown, and a comparison of the performance of MP2 (full OS part, full same-spin (SS) contribution) and SOS-MP2 (no SS part) underlines the unfavorable effect of the SS correlation if evaluated perturbatively (MAD(SOS-MP2)=9.3 kcal mol−1, MAD(MP2)=14.9 kcal mol−1). This also holds true for DSD-BLYP, as it uses SCS-MP2-type correlation (46 % OS, 37 % SS) instead of normal MP2. However, the main issue of this functional is the high Fock exchange admixture (69 %). The wave-function-based methods overall show poorer behavior than DFT average performers. One reason for this is the nickel subset, which is very demanding for single-reference wave-function methods such as MP2. However, this is explained in greater detail below. Hartree–Fock, together with the D3 dispersion correction, performs slightly better (MAD=13.9 kcal mol−1) than MP2. The effect of the D3 correction is discussed separately.
Figures 4 and 5 are similar to Figure 3, but they contain only the barriers or only the reaction energies, respectively. Based on these results it can be concluded that the main problem for most of the tested DFs is the description of the latter. Thereby, the largest difference between the MADs of the thermochemistry and the barriers (ΔMADthermo–kin) appears for the M06 functionals which additionally depends on the Fock exchange admixture (ΔMADthermo–kin between 2.0 kcal mol−1 for M06 and 6.3 kcal mol−1 for M06-HF). Functionals that are not strongly affected are, for example, PBE0-D3 with a difference of only 0.4 kcal mol−1. For the tested perturbation methods, a small ΔMADthermo–kin value is observed for SOS-MP2 and SCS-MP2 (1.2 and 0.4 kcal mol−1, respectively), whereas a greater difference is noted for HF-D3 (4.1 kcal mol−1).
Figure 4. MAD over the complete set without the dissociation and reaction energy compared with CCSD(T)/CBS. The functionals are ordered according to Jacob’s ladder and Fock exchange. The bars are capped if the MAD is >5.5 kcal mol−1.
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Figure 5. MAD over the complete set without the barriers compared with CCSD(T)/CBS. The functionals are ordered according to Jacob’s ladder and Fock exchange. The bars are capped if the MAD is >5.5 kcal mol−1.
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A more detailed picture is shown in Figure 6, in which the performance of selected methods for each subset is presented. The best DF is PBE0-D3, which in two out of the four subsets shows MADs <1 kcal mol−1 (which is typically considered as ‘chemical accuracy’). From the other functionals, only DHDFs yield an MAD of <1 kcal mol−1 for the palladium test set.
Figure 6. MAD for each subset for selected methods with respect to CCSD(T)/CBS. The functionals are ordered according to Jacob’s ladder and Fock exchange. The bars are capped if the MAD is >9 kcal mol−1.
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Furthermore, a few of the tested methods perform rather differently for each subset, which is discussed using ΔMADw–b values, that is, the difference between the worst and the best MADs. One extreme example is BP86-D3, which has an ΔMADw–b value of 3.3 kcal mol−1 between the best subset (MAD (PdCl−)=2.3 kcal mol−1) and the worst (MAD(Pd)=5.6 kcal mol−1). Another functional with a large difference is M06, which has an ΔMADw–b value of 3.1 kcal mol−1. This difference in performance is observed for the double hybrids as well. The reason here are the results for the nickel subset. The double hybrids yield bad results here because these functionals are more sensitive to cases with difficult electronic structures due to the use of the perturbative correlation. The results for the MP2 variants support this view. The worst results are provided by MP2, which has large MADs for each of the subsets: MAD(Pd)=4.9 kcal mol−1, MAD(PdCl−)=7.3 kcal mol−1, MAD(PdCl2)=10.0 kcal mol−1, MAD(Ni)=42.5 kcal mol−1. The scaled variants SCS-MP2 and SOS-MP2 perform much better, and for the nickel subset the MAD is decreased to 32.9 and 28.1 kcal mol−1, respectively. Improvements are also noted for the PdCl2 subset (7.6 and 7.0 kcal mol−1, respectively).
Based on these results the different behavior of the DHDFs, especially for the nickel subset, can be rationalized. PWPB95 has the lowest MAD for this case (4.7 kcal mol−1), as it incorporates only a small part of SOS-MP2 (26.9 %). B2PLYP includes a similar amount of perturbative correlation (27 %) but contains normal MP2-type correlation, and the MAD increases to 8.3 kcal mol−1. The re-parameterized version B2GPPLYP contains 36 % MP2 correlation, resulting in a further increased MAD of 11.2 kcal mol−1.
On the other hand, there are DFs that perform similarly for each subset, indicating greater robustness. The best performer is B3LYP-D3, with a ΔMADw–b value of 0.5 kcal mol−1 followed by PBE0-D3 and PW6B95-D3, with respective ΔMADw–b values of 0.8 and 0.9 kcal mol−1.
Effect of the dispersion correction
In this section we investigate the influence of the dispersion correction on the MADs for the barriers and the reaction energies separately. The results are shown in Figure 7. The Minnesota functionals are excluded because they might require dispersion corrections only for very large complexes. For the discussion we use the absolute value of the difference between the MAD of the dispersion corrected and the uncorrected functional (ΔMAD|D3−noD3|).
Figure 7. MAD for the thermochemistry (i.e., De, ΔEreac) and kinetics (i.e., ΔEforw, ΔEback) of the complete set, with and without dispersion correction, with respect to CCSD(T)/CBS. The functionals are ordered according to Jacob’s ladder, and the bars are capped if the MAD is >9 kcal mol−1.
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In their investigations, Lai et al.51 reported that the dispersion correction has no influence on the forward barriers. This is corroborated in our investigation for the backward barriers as well. A significant effect is only discovered for TPSSh (ΔMAD|D3−noD3|=0.2 kcal mol−1) and HF (ΔMAD|D3−noD3|= 0.4 kcal mol−1).
However, we obtained large effects on the reaction energies in some cases. The biggest change in MAD is found for HF (ΔMAD|D3−noD3|=4.6 kcal mol−1). This is understandable because in HF the Coulomb correlation is entirely missing, which is partly included in the medium-range part of the D3 correction. For the DFs the biggest influence is found for B97-d (ΔMAD|D3−noD3|=1.5 kcal mol−1), BHLYP (ΔMAD|D3−noD3|=1.4 kcal mol−1), B3LYP (ΔMAD|D3−noD3|=1.7 kcal mol−1), and TPSSh (ΔMAD|D3−noD3|=1.6 kcal mol−1). As can be observed from Figure 7, in most cases the MAD is lowered by adding a dispersion correction.
In this section we discuss two reactions as examples in greater detail. The first is the oxidative addition of Pd into the CCl bond of chloromethane shown in Figure 2. Based on the CCSD(T) results, the reactants form a complex (RC) with a stabilization energy of 14.0 kcal mol−1. Then the Pd atom inserts into the CCl bond via a transition state (TS) which is 1.8 kcal mol−1 higher in energy than the free reactants. The product (P) is formed directly from the TS and is 32.3 kcal mol−1 more stable than the reactants.
Several methods are included in Figure 2 for comparison. For a clearer picture the results are split into two plots. On the left side two hybrid functionals (namely B3LYP-D3 and PW6B95-D3), two double-hybrid functionals (B2PLYP-D3 and PWPB95-D3), and two wave-function-based methods (SCS-MP2 and SOS-MP2) are presented. The plots suggest that the DFs give a qualitatively and quantitatively balanced representation of the CCSD(T)/CBS energy surface, and only PW6B95-D3 has some problems in the description of the backward barrier (ΔEback). The two wave-function methods also seem to have problems with this reaction. SCS-MP2 overestimates the forward barrier, and SOS-MP2 yields a reactant complex which is too high in energy. However, the barriers are qualitatively correct.
In the right-hand part of Figure 2, the results of all tested Minnesota functionals are plotted. The only functional that qualitatively reproduces the reference values is the meta-GGA M06L, although we note that ΔEforw is too small. For the M06 series the reactant complex is increasingly destabilized with increasing admixture of Fock exchange. Also notable, for M06 both barriers are underestimated, but with increasing amount of Fock exchange this turns into an overestimation. This picture for M06, M06-2X, and M06-HF is complicated and cannot be rationalized easily. Overall, the Minnesota functionals perform worse than many standard DFs and also less well than one might have expected.
The second example is the oxidative insertion of Ni into the CC bond of ethane (Figure 8). The CCSD(T)/CBS values indicate strongly exothermic complex formation (RC), which is 29.3 kcal mol−1 lower in energy than the free reactants. The forward barrier leading is 10.4 kcal mol−1, and the product is 45.9 kcal mol−1 lower in energy than the reactants.
The same methods as in Figure 2 are presented in Figure 8. The data reveal that only the hybrid functionals are able to reproduce the CCSD(T)/CBS values quantitatively. However, a small underestimation of the reactant complex and the forward barrier is observed. Additionally, in the case of B3LYP-D3 the value of ΔEback is slightly too large.
The DHDFs yield qualitatively correct results. Thereby, PWPB95-D3 performs much better than B2PLYP-D3. However, in both cases De and ΔEback are too small, and in fact for B2PLYP-D3 ΔEreac is nearly zero. The reason is the poor behavior of the perturbative part as explained above. SCS-MP2 and SOS-MP2 show similar errors. The forward barrier is reasonable, while the RC energy is strongly overestimated, and the backward barrier is negative.
On the right side of Figure 8, the results for the Minnesota functionals are shown. They mostly give a qualitatively correct picture for this reaction, with M06 performing best. However, both barriers are underestimated. The meta-GGA M06L is the second-best functional, although it clearly underestimates the reactant complex and overshoots the backward barrier. By increasing the amount of Fock exchange, a similar behavior is observed for the M06 series as for the first example (Figure 2).
Performance of double-hybrid functionals
In the last section we present the results of a detailed evaluation of newly developed DHDFs compared with well-known variants. The dispersion correction is neglected in all cases for consistency. The methods in Table 2 are ordered according to the Fock exchange admixture for analysis purposes. The most extreme variants are PBE0-2, with 79.37 % of Fock exchange and 50 % of MP2 correlation, and PBE0-DH (50 % of Fock exchange and 12.5 % of MP2 correlation). Additionally, spin-component-scaled parts were used in three functionals, namely DSD-BLYP (using an SCS-MP2 variant), PTPSS, and PWPB95 (both using SOS-MP2).
Table 2. All tested DHDFs ordered with respect to the amount of Fock exchange (FE).
|Functional||FE [%]||MP2 corr [%]||Functional||FE [%]||MP2 corr [%]|
|DSD-BLYP||69||37 SS, 46 OS||B2PLYP||53||27|
The results of the tested DHDFs for the complete set are presented in Figure 9. With an MAD of 2.1 kcal mol−1, PWPB95 is the best DHDF followed by PBE0-DH (2.5 kcal mol−1), PTPSS, and mPW2PLYP (2.9 kcal mol−1 each). B2PLYP is slightly worse, with an MAD of 3.1 kcal mol−1. Because the DHDFs are ordered with respect to the amount of Fock exchange, Figure 9 shows that DHDFs with the lowest amount perform best (50–55 %). With increasing amounts of Fock exchange, the MAD increases. The largest MADs are obtained for 1DH-PBE (6.1 kcal mol−1) and PBE0-2 (5.8 kcal mol−1).
Figure 9. MAD over the complete set for all DHDFs compared with CCSD(T)/CBS. The functionals are ordered according to the amount of Fock exchange included. DHDFs with SCS-type MP2 correlation are shown as open bars.
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However, the perturbative part also plays an important role. A large contribution of MP2 correlation is problematic in electronically complicated cases like the nickel subset (see Figure 6) as already noted. This is supported by the data in Figure 9. Additionally, DHDFs with spin-component-scaled correlation were tested as well, namely DSD-BLYP (37 % SS, 46 % OS), PTPSS, and PWPB95 (both using the opposite-spin part only (PTPSS: 37.5 %, PWPB95: 26.9 %)). Compared with other double-hybrid functionals with very high amounts of Fock exchange (>65 %) DSD-BLYP performs better, with an MAD of 4.5 kcal mol−1 due to the incorporation of SCS-MP2-type correlation. Only B2GPPLYP has a lower MAD, but this is related to the lower amount of perturbative correlation relative to DSD-BLYP. PWPB95 shows the best performance for all tested double hybrids and seems to be more robust for electronically complicated cases due to the use of the OS term.