Hierarchy of Commonly Used DFT Methods for Predicting the Thermochemistry of Rh-Mediated Chemical Transformations.

The accuracy and reliability of 17 commonly used density functionals in conjunction with Poisson-Boltzmann finite solvation model were gauged for predicting the free energy of Rh(I)- and Rh(III)-mediated chemical transformations such as ligand exchange, hydride elimination, dihydrogen elimination, chloride affinity, and silyl hydride bond activation reactions. In total, six Rh-mediated reactions were examined, and the computed density functional theory results were then subjected to comparison with the experimentally reported values. For reaction A, involving replacement of N2 with η2-H2 over Rh(I), MPWB1K-D3, B3PW91, B3LYP, and BHandHYLP emerged to be the best functionals of all the tested methods in terms of their deviations ≤2 kcal mol-1 from experimental data. For reaction B, in which exchange of η2-C2H4 with N2 over Rh(I) takes place, MPWB1K-D3 and M06-2X-D3 functionals performed the best, while as for reaction C (hydride elimination reaction in Rh(III) complex), it is PBE functional that showed impressive performance. Similarly, for reaction D (H2 elimination reaction in Rh(III) complex), PBE0-D3 and PBE-D3 showed exceptional results compared to other functionals. For reaction E (H2O/Cl- exchange), the PBE0 again shows impressive performance as compared to other functionals. For reaction F (Si-H activation), M06-2X-D3, PBE0-D3, and MPWB1K-D3 functionals are undoubtedly the best functionals. Overall, PBE0-D3 and MPWB1K-D3 functionals were impressive in all cases with lowest mean unsigned errors (3.2 and 3.4 kcal mol-1, respectively) with respect to experimental Gibbs free energies. Thus, these two functionals are recommended for studying Rh-mediated chemical transformations.

Introduction

Rhodium is used in homogeneous catalysis for carrying a wide variety of chemical transformations.[1−11] Very often, these catalytic chemical processes involve transfer hydrogenation from the metal hydride bond to unsaturated substrates,[4,9,12−17] dehydrogenation of H2 from the metal center,[18] or interaction of alkenes with the metal center of the catalyst.[19] Moreover, the interaction of small molecules such as H2 and ethene with Rh complexes is very significant in CO2 hydrogenation and C–C coupling reactions, respectively.[7,9,20] In order to understand these transition-metal-mediated catalytic processes, density functional theory (DFT) has been extensively used over the last 20 years,[21−29] and it has revolutionized this field not only by predicting the reaction mechanism but also by paving a way for catalyst designing as well.[30−33] However, the evidence of accuracy of DFT methods remains less because of limited benchmarking studies for every elementary step of a transition-metal-mediated chemical reaction.[34] Thus, prior to applying DFT model to any catalytic process, its accuracy should be evaluated. Therefore, understanding the accuracy of commonly used DFT methods for predicting thermochemistry involving interactions with such small molecules will immensely help in better understanding of these catalytic processes.

Several benchmarking DFT studies of transition-metal-mediated chemical transformations have been proposed, which provide in-depth insights,[22,35−44] but in the case of Rh complexes, only small benchmarking studies are available in the literature. Of these, Hu and Boyd examined the free energy of CO dissociation from the Rh(CO)2(Cl2)− complex using the B3LYP density functional and was found to be within the 1–2 kcal mol–1 accuracy limit when compared with CCSD(T) results.[45] Wilson and Laury studied the performance of various density functionals for predicting gas-phase enthalpies for formation of small 4d transition-metal complexes.[46] The results were compared with relativistic pseudopotential correlation-consistent composite approach (rp-ccCA), and of the various DFT functionals used, the double-hybrid functionals, B2GP-PLYP and mPW2-PLYP, were found to be accurate with mean absolute deviations of 4.25 and 5.19 kcal mol–1, respectively, from the experimental values. Moreover, Chen et al. studied the H–H and C–H σ-bond activation coordinated with the Ru and Rh metals.[47] The good functionals for both Ru and Rh for predicting activation energies were reported to be MN12SX < CAM-B3LYP < M06-L < MN12L < M06 < ωB97X < B3LYP < LC-ωPBE based on their increasing mean unsigned deviations of less than 2 kcal/mol. However, for computing reaction energies, the PBE0 functional was proposed to be the best functional. Furthermore, Gusev also examined the accuracy of M06-L functional for predicting thermochemistry of ligand dissociation, ligand replacement, and cleavage in 4d and 5d metal complexes.[48] They concluded that the difference between the theory and experiment is ≤1 kcal mol–1 in several cases.

Despite all these studies, the detailed benchmarking studies assessing the accuracy of various density functionals for Rh-mediated catalytic reactions are limited, particularly those in which different reactions of a same metal complex are considered. Moreover, there may not be a single answer to every Rh-mediated transformation, and for every reaction, there must be a best functional which must be searched through benchmarking studies. Owing to the importance of Rh in homogeneous catalysis, we examined the accuracy of commonly used DFT methods for predicting free energy of binding small molecules such as η2-H2 and η2-C2H4, hydride elimination, H2 elimination, chloride affinity, and Si–H activation of Rh-mediated reactions as shown in Scheme . Both dispersion-corrected and dispersion-uncorrected functionals were employed comprising a total of 17 functionals, namely, B3LYP, B3LYP-D3, BHandHYLP, PBE, PBE-D3, PBE0-D3, PBE0, M06, M06-D3, M06-L, M06-2X-D3, MPW1K, MPWB1K-D3, B3PW91, B3PW91-D3, B97-D, and B97-D3. In order to assess the accuracy of these DFT methods for the above said reactions, we compared our computed results with their experimental values as reported in the literature.[49−52] The free-energy values for reactions A, B, C, and D are −0.71 ± 0.03, +1.08 ± 0.01, +50.4, and +0.44 kcal mol–1, respectively.[49,50] The values of standard-state reaction Gibbs free energy for reactions E and F were computed form the experimentally reported equilibrium constant values using the equation ΔG = −RT ln K.[51,52] The free energy of reactions A and B were also computed by Milstein and co-workers, but instead of using the Rh[Bu2PCH2(C6H3)CH2PBu2] pincer catalyst, they used the Rh[Bu2P(CH2)2(CH)(CH2)2PBu2] complex for their studies.[53] They reported the value of free energy for reactions A and B at the same temperature and pressure equal to −1.24 ± 0.04 and +1.57 ± 0.01, respectively, which is almost similar to the values reported by Fujita and co-workers.[49] Thus, these values of experimental free energies are reliable at the reported conditions of temperature and pressure. The values of reactions C and D have been computed at a temperature of 23 ± 2 °C. Similarly, for computing the equilibrium constant of reaction E[log K(H2O/Cl–) values], Enyedy et al. tested the number of Rh complexes by replacing various ligands around the metal center and obtained results within the range 1.5–3.0.[51] For reaction F, Sadow et al. determined the equilibrium constant at 297 K from four independent measurements with the value equal to 2.8 ± 0.3.[52] They also reported the standard-state enthalpy and entropy of the reaction with their values equal to −5.5 ± 0.2 kcal mol–1 and −16 ± 1 cal mol–1 K–1, respectively.[52]

Scheme 1

Reactions Studied along with Their Experimentally Reported Reaction Free Energies

Results and Discussion

Energetics and Structural Consideration of Complexes

The optimized geometrical parameters of Rh(I) and Rh(III) complexes between the metal and various ligands do not vary much from method to method. In Rh(I) PCP complex, the Rh–C and Rh–P bond distances computed with B3LYP-D3 method are 2.068 and 2.326 Å, respectively, while the same respective bond distances computed at M06-2X-D3 are 2.047 and 2.331 Å. For Rh(III) complex (cis-form complex, reaction D), the Rh–P and Rh–H distances computed at B3LYP-D3 are 2.347 and 1.597 Å, respectively. The same bond distances computed at M06-2X-D3 are 2.323 and 1.569 Å, respectively.

The transition-metal complexes containing phosphines with either iso-propyl (iPr), tert-butyl (tBu), or other types of groups can have a number of conformational and rotational isomers by the rotation of these groups with respect to each other. Neglecting these conformational changes can have huge impact on the accuracy of computed results in such complexes. Therefore, care must be taken while choosing the true ground-state structure before computing the free energy of these complexes. After carrying conformational analysis of Rh(I) PCP–pincer complex, the rotation of tBu groups of the phosphines does have many conformers. However, there are four major conformers represented as Form I, Form II, Form III, and Form IV shown in Scheme based on the position of one tBu group with respect to the other (Sym or Asym). Out of these four conformers, Form III is the most stable conformation which involves agostic interactions between the Rh and the two tBu groups instead of four, as predicted by Fujita and co-workers.[49] The minimum energy structure is close to Form III and it is used as a ground-state structure in our calculations, which was further confirmed from DFT as well as an X-ray crystallographic structure.[54] Moreover, the binding of small molecules such as H2, N2, C2H4, and so forth, with the metal center does not affect the structure of the complex upon binding. Likewise, the structure of [(H)2Rh(dmpe)2]+ for reactions C and D has to be trans-octahedral and distorted cis-octahedral in nature, respectively.[50] The geometry of [Rh(dmpe)2]+ is square-planar in nature; however, when dmpe (dimethyl phosphine ethyl) is replaced with dmpp (dimethyl phosphine propyl) groups, the geometry of the complex changes from square-planar to a distorted tetrahedral geometry.[55] The structure of Rh complexes for reactions E and F was directly taken from the X-ray crystallographic structures.[51,52]

Scheme 2

Conformational Isomers in Rh-Based PCP–Pincer Complex

Computed Free Energy of Reactions

Inspired by the use of Rh complexes in homogeneous catalysis, we examined the accuracy of 17 functionals for predicting the thermochemistry of reactions shown in Scheme . The experimentally reported reaction-free energy of these reactions was taken from the literature, determined by NMR and spectrophotometric equilibrium studies at 25 °C.[49−52] All these values are listed below each reaction in Scheme along with their reference. Reactions A and B are ligand-exchange reactions involving binding of η2-H2 and η2-C2H4 to the PCP–Rh pincer complex by replacing N2 in cyclohexane. Reaction C is a hydride elimination reaction in which the solvent molecule, CH3CN, binds itself to the Rh complex by eliminating the hydride ion, while reaction D is a dihydrogen elimination reaction. Reaction E is an exchange reaction between the coordinated H2O with chloride ion in aqueous media, while reaction F is Si–H activation reaction in benzene in which simultaneous loss of CO ligand from the complex takes place. The computed energetics and optimized geometrical coordinates of all the chemical species involved in these reactions are given in detail in the Supporting Information.

The computed TR/BS1 and TR/BS1/BS2 reaction free-energy values for reaction A are shown in Figure and their mean unsigned and signed errors (MUE and MSE) are listed in Table . As it is apparent from the TR/BS1 results, the best performance is shown by the hybrid-meta-GGA, MPWB1K-D3 functional with 44% HF exchange having minimum deviation (ΔGcal – ΔGexpt) from the experiment equal to +0.8 kcal mol–1 (the deviations of computed Gibbs free-energy values from experimental reported values are given in detail in Table S1 of the Supporting Information). This functional is best suited for thermochemical calculations as suggested by Truhlar.[68] The other hybrid-meta-GGA functional such as M06 (27% HF exchange) and M06-2X-D3 (54% HF exchange) also provide reasonably good results with their deviations (+2.3 and +1.8 kcal mol–1, respectively) from experimental value. It is noteworthy to mention here that the hybrid-meta-GGA functionals with high HF exchange percentage show better results than functionals with relatively low HF exchange percentage. Like other hybrid-meta-GGA functionals, the meta-GGA functional, M06-L, also gives satisfactory results and is deviating by +1.8 kcal mol–1 from the experiment. The hybrid functionals such as B3PW91 and B3LYP (20% HF exchange) also reproduce good results with deviations +1.4 and +1.5 kcal mol–1, respectively, which are close to that of the MPWB1K-D3 functional. The PBE0 hybrid functional, which contains 25% Fock exchange, also performs better with deviation equal to +0.8 kcal mol–1. The hybrid GGA, MPW1K, is the second best functional after MPWB1K-D3 with deviation equal to −1.3 kcal mol–1. The GGA functional such as PBE do not furnish good results either and the result deviates by +4.8 kcal mol–1 from the experiment. The B97-D3 functional produced inferior results deviating by +7.9 kcal mol–1 from the experiment and is not recommended for studying such kind of Rh(I)-mediated chemical transformations. The second protocol (TR/BS1/BS2) do not procreate good results except for BHandHYLP and MPW1K functionals, and such computational schemes are also discouraged for studying such reactions. This reaction is highly sensitive to dispersion correction and inclusion of it does not improve the accuracy of the results; instead, they reproduce poor results compared to uncorrected functionals. The probable reason for such poor performance might be due to the σ-complexation between the metal and H2. These effects of “D3” dispersion correction on the accuracy of results are more pronounced in hybrid functionals (B3LYP and B3PW91) than hybrid-meta-GGA functionals such as M06, which is understandable because of some inbuilt dispersion correction in the latter ones.

Figure 1

Accuracy of various DFT methods for predicting the free energy of reactions (A–F) using TR/BS1 and TR/BS1/BS2 protocols. All the given values are in kcal mol–1.

Table 1

Mean Unsigned and Signed Errors of All the Tested Functionals for the Specified Reactions with Respect to the Experimental Gibbs Free Energya

	TR/BS1	TR/BS1/BS2	TR/BS1	TR/BS1/BS2
	MUE	MUE	MSE	MSE
B3LYP	8.1	8.1	3.6	4.2
B3LYP-D3	6.1	6.3	0.2	0.8
B3PW91	6.9	6.9	1.4	2.1
B3PW91-D3	4.8	4.6	–2.6	–1.9
B97-D	6.7	6.9	–2.1	–1.5
B97-D3	8.1	8.2	–1.9	–1.3
BHandHYLP	5.5	5.7	–1.4	–0.8
M06	7.9	7.9	–0.3	0.3
M06-D3	6.7	7.0	–0.3	0.2
M06-2X-D3	4.7	4.7	–3.8	–3.0
M06-L	6.2	5.9	–1.3	–0.2
MPW1K	5.3	5.3	–0.3	0.5
MPWB1K-D3	3.4	3.0	–3.0	–2.2
PBE	5.8	5.8	3.4	3.3
PBE-D3	4.7	5.1	1.1	1.4
PBE0	4.0	4.4	1.4	1.8
PBE0-D3	3.2	3.4	–1.3	–0.8

All the given data are in kcal mol–1.

The results of good functionals are shown in bold.

Reaction B is similar to reaction A, which involves addition of η2-C2H4 to Rh(I) PCP–pincer complex instead of η2-H2. In this case, M06-2X-D3 is unimpeachably the best functional with +0.5 kcal mol–1 deviation from the experiment when TR/BS1 computational scheme is considered (Figure ). The hybrid-meta-GGA functional, MPWB1K-D3, also performs well for this type of reaction deviating by −1.7 kcal mol–1 from the experimentally reported value. It is noteworthy to mention here that among the hybrid-meta-GGA functionals, good results are shown by those functionals with higher HF exchange (e.g., M06-2X-D3 has 54% HF exchange, while as MPWB1K-D3 functional has 44% HF exchange). However, the hybrid functional, M06, overestimates the free energy by +5.2 kcal mol–1 from the experiment with just 27% HF exchange. The M06-L functional also produced reasonable results for reaction B as well with deviation equal to +4.0 kcal mol–1. Unlike reaction A, PBE and the hybrid DFT methods B3PW91 and B3LYP overwhelmingly deviate from the experiment (+8.6, +7.3, and +9.1 kcal mol–1, respectively). The BHandHYLP functional performs reasonably well compared to other hybrid functionals (+3.1 kcal mol–1). The PBE0 functional also like other hybrid functionals do not reproduce good results, and it deviates from the experiment by +4.9 kcal mol–1. The results are very sensitive to dispersion corrections for this type of reaction, and introducing “D3” dispersion correction significantly improves the results of all DFT methods except for the B97-D functional. For example, the PBE0-D3 functional deviates by just +2.4 kcal mol–1, which is a significant improvement from +4.9 kcal mol–1 without dispersion. It is noteworthy to mention here that the TR/BS1/BS2 protocol also improves the results.

For reaction C, out of all the functionals tested, the best performance is given by PBE (−1.5 kcal mol–1), while M06-L gives worse results as shown in Figure , and this method deviates by −16.2 kcal mol–1 from the experiment. The inclusion of “D3” dispersion correction does not improve the results; instead, its use decreases the accuracy of the results. The accuracy of the results heavily relies on the accuracy of computed electronic energy of hydride ion. The more the accurate the electronic energy is, the more is the accuracy of free energy of hydride ion and thus more is the accuracy of the results. Therefore, using TR/BS1/BS2 protocol enhances the improvement in results except for BHandHYLP. However, this improvement is small in the case of PBE and PBE-D3 functionals.

Reaction D involves the elimination of H2 from the distorted cis-octahedral complex and the computation of free energy of such type of reaction is difficult. Several key factors need to keep in mind before computing free energy of such type of reactions. First, such reactions involve the increase in number of molecules and thus total free energy is added by a factor of 1.89 kcal mol–1, considering the standard-state conversion from 1 atm to 1 mol. Second, because of the dissociation of one molecule to two, the significant errors can arise in the calculations, particularly basis set superposition errors, errors due to overestimation of entropic term, and the solvation errors of separate molecules formed during the reaction. As far as this reaction is concerned, the best functional is PBE0-D3 because of its minimum deviation (−0.3 kcal mol–1, TR/BS1 protocol) from the experiment. The PBE functional with and without “D3” dispersion correction also give good performance (−0.4 and −2.3 kcal mol–1, respectively) and is second best of all the functionals examined in this study. The MPWB1K-D3 functional again shows much better performance (+0.5 kcal mol–1) followed by MPW1K (−1.1 kcal mol–1). All the hybrid functionals performed reasonably well, particularly the D3-corrected functionals such as B3PW91-D3 (−1.9 kcal mol–1). However, B3LYP with and without dispersion reproduces worse results as shown in Figure . Notably, the M06-L functional does not provide satisfactory results (−4.7 kcal mol–1); however, it gives much better performance than M06 and M06-2X-D3 (−11.5 and −9.3 kcal mol–1, respectively). The interesting observation here is that increasing the HF exchange percentage in the hybrid functionals enhances the accuracy of results. For example, B3LYP and B3PW91 with 20% HF exchange are having deviations equal to −8.6 and −5.4 kcal mol–1, respectively. In contrast to this, BHandHYLP and PBE0 with 50 and 25% HF exchange are much closer to the experimental values with deviation equal to +1.4 and −2.5, respectively. The inclusion of D3 dispersion correction has a huge impact on the accuracy of results and its inclusion improves the results. The TR/BS1/BS2 protocol does not improve the accuracy of functionals used except for BHandHLYP and MPWB1K-D3 whose deviation decreases from +1.4 to +0.6 and +0.5 to −0.3 kcal mol–1, respectively.

In reaction E, Cl– replaces the H2O coordinated with the Rh(III) center, and among all the DFT methods tested, the PBE0 functional shows impressive performance (TR/BS1 protocol) with and without D3 dispersion correction having −0.3 kcal mol–1 deviation from the experiment. After PBE0, it is B3LYP and M06-L functionals which procreate good results with their respective deviations equal to −0.7 and 1.7 kcal mol–1 from the experiment. The BHandHYLP functional with 50% Hartree–Fock exchange is the worst functional of all the DFT methods tested (−11.5 kcal mol–1 deviation). The second computational protocol TR/BS1/BS2 enhances the accuracy of DFT results. Finally, including dispersion correction in DFT functionals makes the results worse as displayed in Figure .

For reaction F, the performance of hybrid functionals B3PW91, M06, and PBE0 is almost similar with their respective deviations +16.2, +15.4, and +10.6 kcal mol–1 (TR/BS1 protocol). The hybrid B3LYP functional is slightly on the higher side with deviation equal to +24.3 kcal mol–1. The M06-2X-D3 functional is impressive of all the functionals tested with lowest value of deviation (+0.4 kcal mol–1). The PBE0-D3 functional is the second best functional after M06-2X-D3 deviating by +1.1 kcal mol–1 from the experiment. The B97-D functional also produces good results with its low deviation from the experimental value (+2.0 kcal mol–1). The M06-L functional also do not reproduce good results and deviates by +8.8 kcal mol–1. Moreover, the MPWB1K-D3 functional again reproduces good results with deviation −2.5 kcal mol–1. It is interestingly notable that dispersion correction plays a crucial role in the performance of DFT functionals. The dispersion correction functionals are exceptionally best compared to the uncorrected ones, and the performance dramatically enhances after adding dispersion correction in the functionals with large effects in some cases as portrayed in Figure . Of all the functionals, the highest influence of introducing dispersion correction has been found in the case of B3LYP functional (difference of ∼15 kcal mol–1). The TR/BS1/BS2 computational protocol reduces the accuracy of functionals.

The experimental enthalpy ΔH° = −5.5 ± 0.2 kcal mol–1 and entropy ΔS° = −16 ± 1 cal mol–1 K–1 are also reported for reaction F.[52] Thus, we calculated errors in our computed enthalpy and entropy and their total contribution toward the overall free-energy error of the reaction. As listed in Table , the errors in enthalpy computed by functionals without dispersion correction are significantly higher compared to their entropy errors while the opposite is the case for functionals with dispersion correction. For example, the errors in enthalpy and entropy for B3LYP functional are +19.0 kcal mol–1 and −3.5 cal mol–1 K–1 while the same values for B3LYP-D3 functional are +4.3 kcal mol–1 and −16.4 cal mol–1 K–1, respectively. While studying the mechanism of alcohol-mediated Morita Baylis–Hillman reaction, Singleton and Plata predicted that entropy is not a major contributor to the large computational errors.[56] However, in our case, such statement is correct only for functionals without D3 dispersion correction except for PBE-D3 and M06-D3. The importance of using dispersion-corrected functionals for studying reaction F is depicted by the small cancellation of errors in best performing functionals such as B3PW91-D3, B97-D3, M06-2X-D3, MPWB1K-D3, and PBE0-D3. For example, the M06-2X-D3 functional is having enthalpy error equal to +0.5 kcal mol–1, while the entropy error is −5.4 cal mol–1 K–1, which becomes +2.1 kcal mol–1 at 297 K. Thus, the good performance of this functional (with deviation +0.4 kcal mol–1) and other dispersion-corrected functionals is mainly due to inclusion of dispersion correction and not only because of cancellation of errors.

Table 2

Errors in Computed Enthalpy and Entropy and Their Contribution toward Total Error in Free Energy for Reaction Fa

DFT method	ΔHcomp°	error	ΔScomp°	error	–TΔS°	ΔG°
B3LYP	+13.5	+19.0	–19.5	–3.5	+1.0	+20.0
B3LYP-D3	–1.2	+4.3	–32.4	–16.4	+4.9	+9.2
B3PW91	+10.3	+15.8	–7.4	+8.6	–2.6	+13.3
B3PW91-D3	–6.2	–0.7	–22.8	–6.8	+2.0	+1.3
B97-D	–0.5	+5.0	–8.2	+7.8	–2.3	+2.7
B97-D3	+2.6	+8.1	+1.6	+17.6	–5.2	+2.9
BHandHYLP	+2.0	+7.5	–16.1	–0.1	+0.01	+7.5
M06	+7.5	+13.0	–16.4	–0.4	+0.1	+13.2
M06-D3	+3.6	+9.1	–20.7	–4.7	+1.4	+10.5
M06-2X-D3	–5.0	+0.5	–21.4	–5.4	+1.6	+2.1
M06-L	+5.6	+11.1	–4.0	+12.0	–3.6	+7.6
MPW1K	+5.4	+10.9	–12.6	+3.4	–1.0	+9.9
MPWB1K-D3	–8.5	–3.0	+2.3	+18.3	–0.7	–3.6
PBE	+10.6	+16.1	–0.5	+15.5	–4.6	+11.4
PBE-D3	–4.3	+6.4	–15.8	+0.2	–0.1	+6.4
PBE0	+45.3	+11.0	–45.8	–29.8	+8.9	+19.9
PBE0-D3	+0.9	+1.2	–14.1	+1.9	–0.6	+0.6
mean error |error|		8.0		1.1		7.9

All the given values are in kcal mol–1 except ΔScomp°, which is in cal mol–1 K–1.

Comparison of Results with Literature

There are limited studies available in the literature regarding Rh(I) complexes. Thus, it would be appropriate to compare the results with some related studies. Of these, recent study by Chen and co-workers assessed the accuracy of DFT methods for H–H σ-bond heterolytic cleavage over Rh(III) complexes and they reported that M06-L, M06, B3LYP, and PBE0 are the best functionals for studying such reactions.[47] However, in our present case, the η2-H2 addition over Rh(I) PCP pincer complex is considered (reaction A) and the common finding in both the studies is that PBE0, B3LYP, and M06 perform reasonably well. It would be slightly more appropriate to compare their results with reaction D in which the homolytic cleavage of H–H over Rh(III) complex takes place (reverse reaction). According to their findings, the best performing functionals for reaction energy are PBE0 and B3LYP, especially the dispersion-corrected ones, which is similar to our computed results. Moreover, as per Chen et al., inclusion of dispersion correction is beneficial for most DFT functionals, which is again in accordance with our results. Concerning H–H σ-bond homolytic cleavage over Rh(I) complex, Neese and co-workers reported B3LYP to be the best functional, which is also in accordance with our computed results.[57]

Conclusions

In this study, we tested the accuracy of various density functionals with and without dispersion correction for predicting the free energy of η2-H2, η2-C2H4, and Cl– ligand-exchange reactions, hydride elimination reaction, H2 elimination reaction, and Si–H activation reaction mediated by Rh complexes, comprising total six reactions. In total, 17 density functionals and two computational schemes were employed, represented as TR/BS1 and TR/BS1/BS2. The computed results of all the functionals were compared with the free-energy values already available in the literature.

For reaction A, of all the density functionals tested, MPWB1K-D3 and PBE0 are outstanding best deviating from experimental data by +0.8 kcal mol–1. The hybrid functionals such as B3LYP, B3PW91, and BHandHYLP were also good with deviations lesser than 2 kcal mol–1. The hybrid-GGA, MPW1K, and the dispersion-corrected global-hybrid meta-GGA, M06-2X-D3, functionals also performed well (−1.3 and +1.8 kcal mol–1, respectively). For reaction B, the MPWB1K-D3 functional was impressive followed by M06-2X-D3, while it is the PBE functional that is most suited for reaction C. Similarly, for reaction D, PBE0-D3 and PBE (with and without dispersion correction) are outstanding functionals for studying Rh-mediated catalytic reactions with their deviations less than 1 kcal mol–1 when compared with experiment. For reaction E, PBE0, M06-L, and B3LYP showed impressive performance with deviation less than 2 kcal mol–1. Likewise, for reaction F, the M06-2X-D3, PBE0-D3, and MPWB1K functionals again reproduced undoubtedly the best results.

The inclusion of dispersion correction has huge impact on the accuracy of results and the functionals with dispersion correction increasing the accuracy of results for reactions B and D. However, for reaction A, significant errors arise and thus use of dispersion-corrected functionals should be avoided for studying Rh(I)-mediated reactions in which exchange of N2 with η2-H2 takes place. For reaction F, the accuracy of the results heavily relies on the dispersion correction of the functional used. Moreover, employing TR/BS1/BS2 computational scheme does not improve the accuracy of results either except for reaction C and thus the use of such schemes is also discouraged.

In all the reactions, the PBE0-D3 and MPWB1K-D3 functionals are undoubtedly the best based on their minimum MUE values (3.2 and 3.4 kcal mol–1, respectively). The other functionals are good in one reaction but perform badly in another, and thus, it is concluded here that MPWB1K-D3 and PBE0-D3 functionals and TR/BS1 computational scheme are preferred as far as Rh-mediated chemical transformations are concerned.

Computational Details

The geometry optimizations of all the complexes without any truncations were carried out by employing all 17 density functionals, namely, B3LYP,[58,59] B3LYP-D3,[58−61] BHandHYLP,[58−62] PBE,[63] PBE-D3,[63] PBE0-D3,[64] PBE0,[64] M06,[65] M06-D3,[65] M06-L,[66] M06-2X-D3,[67] MPW1K,[68] MPWB1K-D3,[69] B3PW91,[59,70] B3PW91-D3,[59,70] B97-D,[71] and B97-D3,[71] in conjunction with the 6-31+G(d,p)[72] basis set for all nonmetallic atoms and LACVP**+[73,74] pseudopotential and basis set for Rh (BS1). All the optimizations were carried out first in gas phase and then solvent effects were incorporated employing Poisson–Boltzmann finite element solvation model.[75,76] No significant geometry change was observed during such optimization scheme. The optimizations of all the complexes were carried out in cyclohexane (reactions A and B), acetonitrile (reactions C and D), water (reaction E), and benzene (reaction F) as reported in the literature. The ground-state structures were confirmed by analyzing the harmonic frequency data with no imaginary frequency computed on the above said optimized geometries. The reaction-free energies calculated by the above said method (without using any scaling factor) are termed as TR/BS1. The single-point energy of all the complexes was also computed using the 6–311++G(d,p) basis set for all nonmetallic atoms and LACVP**++ for Rh atom (BS2). The BS2 single-point energies of the complexes were calculated using maximum grid density and ultrafine SCF. The BS1 single-point energies in TR/BS1 were replaced by the BS2 single-point energies as per the equation, namely, GBS2 = GBS1 – EBS1 + EBS2 and such results are designated as TR/BS1/BS2.

The equation ΔGgas→sol = kTln(24.5) = 1.89 kcal mol–1 corresponds to a change in free energy on going from standard gas-phase (1 atm) to liquid-state (1 M) conversion. The total free energy of reaction D was corrected (added) by the same factor (1.89 kcal mol–1) as the increase in the number of moles during the course of reaction.[77] All the above said calculations were carried out in experimentally reported conditions using Jaguar program as implemented in the Schrodinger Suite of program.[78]