Statistical analysis of C–H activation by oxo complexes supports diverse thermodynamic control over reactivity

Abstract

Transition metal oxo species are key intermediates for the activation of strong C–H bonds. As such, there has been interest in understanding which structural or electronic parameters of metal oxo complexes determine their reactivity. Factors such as ground state thermodynamics, spin state, steric environment, oxygen radical character, and asynchronicity have all been cited as key contributors, yet there is no consensus on when each of these parameters is significant or the relative magnitude of their effects. Herein, we present a thorough statistical analysis of parameters that have been proposed to influence transition metal oxo mediated C–H activation. We used density functional theory (DFT) to compute parameters for transition metal oxo complexes and analyzed their ability to explain and predict an extensive data set of experimentally determined reaction barriers. We found that, in general, only thermodynamic parameters play a statistically significant role. Notably, however, there are independent and significant contributions from the oxidation potential and basicity of the oxo complexes which suggest a more complicated thermodynamic picture than what has been shown previously.

Statistical analysis of transition metal oxo mediated C–H activation indicates that thermodynamic factors dictate reactivity and that the energetics of proton and electron transfer have effects independent of the free energy of the reaction.

Introduction

The activation of C–H bonds through proton-coupled electron transfer (PCET) underpins a wide range of biological and synthetic processes. The applications of this reaction include drug metabolism by cytochrome P450 enzymes as well as synthetic methods for the preparation of fine chemicals.^1–3 In many cases PCET reactions are mediated by transition metal oxo intermediates generated within either protein-based or synthetic ligand scaffolds. In these reactions, both a proton and an electron are transferred to an oxo complex resulting in the net removal of a hydrogen atom from the organic substrate. The generality of this reaction combined with the ubiquity of C–H bonds in synthesis has led to considerable interest in determining what properties govern the PCET reactivity of transition metal oxo species.

A large body of work supports that the free energy of reaction (ΔG_PCET) is central to transition metal oxo mediated C–H activation and also offers a great deal of explanatory and predictive power.^4–7 Recently however, additional properties have been cited as important although it is not clear if any have a widespread effect on reactivity. Individual cases support the influence of O-centered spin density,⁸ spin state,^9–11 steric environment,^12–14 the free energies of proton and electron transfer (ΔG_PT and ΔG_ET),^15–20 or the asynchronicity (η) of the reaction,^21–24 but there is a lack of consensus regarding their generality and relative importance (Scheme 1).^8,25–27 Very few studies have explored these parameters outside of a narrow range of complexes,^{4,6,10,20,21,28} and none have statistically examined the significance of parameters other than ΔG_PCET on the reactivity of a broad set of metal oxo complexes.

Scheme 1. Investigated parameters of metal oxo species.

We previously found an atypical dependence on ΔG_PT in the concerted C–H activation reactivity of a terminal Co^III oxo complex which contrasts with the expected rate dependence on ΔG_PCET.¹⁵ Given the disparity of this result with the literature, we sought to understand the interplay of characteristics affecting a broad range of transition metal oxo mediated PCET reactions using multivariable linear free energy relationships (LFERs). These models can be used to relate experimentally determined data, such as reaction rates, to multiple predictor variables simultaneously. LFER models have recently been used as versatile tools to optimize organic methodology, predict reaction barrier heights, and investigate underlying mechanisms.^{29–33,116,117}

We have applied this analysis to examine trends in rates of PCET mediated C–H activation for a broad dataset of previously reported metal oxo complexes. This analysis enables a statistical examination of several hypotheses regarding what parameters of metal oxo species determine their PCET reactivity. Unsurprisingly, we observe that ΔG_PCET is the most important factor. However, we also observe a significant role for ΔG_PT and ΔG_ET beyond and independent of their contribution to ΔG_PCET. Furthermore, the other parameters investigated do not have broad significance. These results suggest that thermodynamic factors are generally the dominant contributors to transition metal oxo C–H activation reactivity, but also demonstrate that thermodynamic parameters beyond the commonly invoked ΔG_PCET are influential.

Results and discussion

We began our analysis by compiling an extensive data set of second order rate constants (k₂) for the oxidation of 9,10-dihydroanthracene (DHA) by thirty well-characterized metal oxo complexes.^{12,13,15,17,34–51} This diverse data set comprises complexes of five different transition metals, tetragonal and trigonal geometries, spin multiplicities from 1 to 5, charges from −1 to +3, and d-electron counts from 0 to 6. We manually divided the data set into a training set of seventeen metal oxo complexes and a test set of thirteen metal oxo complexes such that each set had a diverse mix of species. For our analysis, we calculated the values of parameters that have been hypothesized as important to metal oxo mediated PCET reactivity for each metal oxo complex. The investigated parameters include the steric environment (quantified by percent buried volume, %BV),⁵² the spin density on the oxygen atom (via intrinsic bond order (IBO) analysis),^53,54 the energetic cost of accessing a higher spin state if one lies closer to the product spin multiplicity than the reactant ground spin state (spin excitation), the thermodynamic free energies ΔG_PCET, ΔG_PT, and ΔG_ET, and the magnitude of the asynchronicity parameter (|η|).²¹ The detailed approaches used to determine the values for these parameters are provided in the methods section. Importantly, none of these parameters require transition state optimizations which are relatively difficult and less reliable than ground state calculations.^55,56 While this precludes a direct estimation of transition state effects, we expect to indirectly capture some of them. For instance, parameters such as ΔG_PCET and |η| have been shown to correlate with tunneling.^23,57

We examined the effect of each of these parameters on experimental reaction barriers by building multivariable free energy models via ordinary least squares regression of the barrier heights against the parameters. Each model consists of a set of coefficients (with variable units such that the product with the respective parameter gives units of kcal mol⁻¹) and an intercept (with units kcal mol⁻¹). These models were used to generate predicted reaction barriers for each data point, which could be compared with experimental reaction barriers to assess the utility of the model. Because ΔG_PCET has strong theoretical and experimental support for affecting reaction barrier heights,^4–6 we analyzed each parameter in combination with ΔG_PCET and compared the resulting model to regression against ΔG_PCET alone.

We evaluated each regression based on R², leave-one-out (LOO) R² (sometimes referred to as Q²), and a statistical F-test.^58–61R² is a goodness of fit measure which quantifies the amount of variation explained by a model. The predictive ability of a model is gauged with LOO R², in which each data point is left out and predicted by the remaining data points and the goodness of fit is then reevaluated. Critically, unlike regular R² this metric does not necessarily improve with an increase in parameters; overfitted models with too many parameters perform poorly with LOO R². For each R², a value close to 1 indicates a good fit. Finally, we report the p-value from an F-test on each model, which shows the probability the observed correlation arises from statistical noise. The lower this p-value is, the more significant a given parameter. Additionally, the calculation of p-values considers the number of parameters added to a model, so, as with LOO R², an F-test is not biased in favor of adding more parameters.

A summary of our findings is presented in Table 1. In line with previous reports, we find a strong correlation between the experimental reaction barriers and ΔG_PCET. This parameter alone explains 70% of the variation in reaction barriers within the training set (R² = 0.70) and has high predictive ability (LOO R² = 0.60). Interestingly, most other parameters do not significantly improve the model. While we do observe a small correlation with %BV steric metrics, the magnitude of the effect is too small to be statistically significant. Compared to the ΔG_PCET only model, spin-based parameters and |η| barely improve R² and perform similarly or worse in LOO cross-validation. While it is difficult to rule out the importance of these parameters in individual cases, an F-test indicates they do not have a statistically significant effect across our entire data set.

Statistical results of various models.

Parameter(s) Regressed with ΔGPCET	Training set on DHAa			All data for multiple substratesb
	R 2	LOOcR2	p-valued	R 2	LOOeR2
ΔGPCET only	0.70	0.60	<0.001f	0.45	0.36
%BV steric metrics	0.77	0.64	0.15	0.48	0.28
Oxo spin density	0.70	0.55	0.78	0.53	0.37
Spin excitation	0.71	0.50	0.49	0.50	0.39
|η|	0.73	0.53	0.22	0.50	0.30
ΔGPT, ΔGET	0.86	0.71	0.0082	0.64	0.50
			0.023g
			0.0038h

^aA subset of the reactions of 17 metal oxo complexes with DHA.

^bExcluding outlier metal oxo complexes (Ru oxos and oxo complexes of 13-TMC); substrates are DHA, 1,4-cyclohexadiene, xanthene, and fluorene.

^cLeave-one-out.

^dFrom an F-test where the null hypothesis is that only ΔG_PCET has an effect.

^eLeave-one-out, slightly modified such that all reactions for a given metal oxo are left out together.

^fFrom an F-test where the null hypothesis is that ΔG_PCET has no effect.

^gFrom an F-test where the null hypothesis is that ΔG_PT has no effect.

^hFrom an F-test where the null hypothesis is that ΔG_ET has no effect.

In contrast, addition of ΔG_PT and ΔG_ET does significantly improve the fit. For this {ΔG_PCET, ΔG_PT, ΔG_ET} model, R² increases from 0.70 to 0.86 and LOO R² increases from 0.60 to 0.71, indicating both better explanation of the available data and better predictive ability. An F-test gives p < 0.01 which suggests the observed effect is statistically significant. The equation from this fit is ΔG^‡ = 0.31ΔG_PCET + 0.07ΔG_PT + 0.12ΔG_ET − 0.26 (all coefficients unitless; free energies and intercept in kcal mol⁻¹). Typically, ΔG_PCET is a negative value while ΔG_PT and ΔG_ET are positive values. Thus, the coefficients' positive signs mean a more exergonic reaction will have a lower barrier while increases in ΔG_PT and ΔG_ET will raise the barrier. The larger coefficient of ΔG_PCET indicates the reaction barrier is most sensitive to this free energy. Satisfyingly, the ΔG_PCET coefficient agrees with experimental data: for metal oxo complexes that have a demonstrated trend of log(k_obs) vs. substrate bond dissociation free energy (BDFE), the average slope of ΔG^‡vs. substrate BDFE is ∼0.3 (see Table S1†), very similar to the 0.30 observed in our analysis. The intercept of −0.26 contains contributions to the average error not accounted for by the three free energies.

The significance of ΔG_PT and ΔG_ET is intriguing because the literature discussion of these values has often been framed in terms of how they contribute to ΔG_PCET rather than in terms of their intrinsic contribution to reaction barrier heights.^16–19,27 However, ΔG_PT and ΔG_ET as defined here are the energies to form the intermediates involved in stepwise reactivity – the protonated metal oxo with the deprotonated substrate, or the reduced metal oxo with the oxidized substrate (Scheme 1). Critically, ΔG_PT and ΔG_ET do not form a full thermodynamic cycle with ΔG_PCET and thus are fundamentally distinct. This fact is statistically supported by poor correlations between ΔG_PCET and ΔG_PT and between ΔG_PCET and ΔG_ET (−0.12 and 0.31, respectively, see Regression S6†). Finally, we find that ΔG_PT and ΔG_ET have importance independent of a contribution to ΔG_PCET as clearly demonstrated by the LOO R²s and F-tests. All of our analyses therefore suggest that the combination of ΔG_PT and ΔG_ET is an independent and significant contributor to C–H activation barrier heights.

While the observation of a dependence on ΔG_PT and ΔG_ET that arises from our linear regressions is principally empirical, it is consistent with prior theoretical models in the literature. The physical underpinning of this dependency on ΔG_PT and ΔG_ET is likely due to mixing of proton transfer and electron transfer intermediates into the concerted transition state despite these intermediates never being fully realized.^62–64 Within transition state theory, this can be envisioned by a More-O’Ferrall–Jencks plot in which the transition state lies not on a one-dimensional line connecting reactant and product but on a two-dimensional plane containing reactant, product, and intermediates.^65,66 In this case, the intermediates arise from proton transfer and electron transfer, and when either ΔG_PT or ΔG_ET lowers in energy the transition state can adopt structural and electronic components of these intermediates resulting in a lower barrier height. While the use of these classical structure–energy relationships to analyse PCET reactions has been questioned recently,^26,27 proton transfer and electron transfer states and their energies also have roles in nonadiabatic rate theories of PCET which treat proton transfer in a quantum mechanical fashion.^67,68 Therefore, the use of ΔG_PT and ΔG_ET to predict barrier heights of PCET is consistent with prior theoretical foundations.

Assigning a direct role for ΔG_PT and ΔG_ET is in line with recent computational studies of PCET transition states which invoke off-ΔG_PCET diagonal thermodynamic terms from Scheme 1, such as asynchronicity (η), as key contributors to DFT derived reaction barriers.^21–24 Asynchronicity is derived not from the sum of ΔG_PT and ΔG_ET, but rather their difference. Conversely, we instead find that the sum of ΔG_PT and ΔG_ET have a more significant effect than |η|. The reason for this discrepancy is unclear, but a possible explanation is that experimental noise prevents us from observing a comparatively more subtle trend between |η| and the experimental reaction barrier heights. Furthermore, the well-controlled nature of the series of complexes previously investigated for asynchronicity may have too little variation in (ΔG_PT + ΔG_ET) to manifest similarly to the effects we observe here.

Another way in which our data may not be amenable to investigating the effect of |η| is the variable reorganization energy of the metal oxo complexes examined here. |η| is specifically framed as an adjustment to the Marcus reorganization energy;²¹ therefore |η|'s effect may only be clear when reorganization is properly accounted for. While it is clear that the reorganization energy is important to PCET reactivity, there is no established way to compute it without computationally expensive transition state geometries.^11,67–69 We have made multiple attempts to derive reorganization or deformation parameters using the optimized metal oxo and metal hydroxide geometries and frequencies, but none of these parameters have statistically significant contributions to predicted reaction barriers with or without |η| (see ESI†). Therefore, a combination of noise in the experimental data and our inability to compute a reliable reorganization parameter could preclude us from observing an effect of |η| on the barrier heights. Nonetheless, previous studies as well as this current work offer increasing support that off-ΔG_PCET diagonal thermodynamic terms such as ΔG_PT and ΔG_ET have important effects on reactivity independent of ΔG_PCET.

The all-thermodynamic model we find here provides insights and possible alternative explanations for previously reported trends in PCET reactivity. In one study,¹² steric and spin state effects were invoked to explain the comparatively high reactivity of the S = 2 complex [Fe^IV(O)(TMG₂dien)(CH₃CN)]²⁺. A higher rate of C–H activation as compared to S = 2 [Fe^IV(O)(TMG₃tren)]²⁺ was ascribed to reduced steric hinderance in the TMG₂dien complex,¹³ and the higher rate of C–H activation as compared to the S = 1 complexes [Fe^IV(O)(N4Py)]²⁺ and [Fe^IV(O)(TMC)(CH₃CN)]²⁺ was ascribed to the S = 2 spin state in the TMG₂dien complex.^36,44 However, it was noted that the even faster reactivity of [Fe^IV(O)(Me₃NTB,CH₃CN)]²⁺, which is S = 1 and has a similar %BV profile to [Fe^IV(O)(TMG₂dien)(CH₃CN)]²⁺,³⁴ is not easily explained by either hypothesis. Our analysis suggests that the thermodynamic properties of these complexes may provide an alternative explanation in these comparisons (see Table S9†). The Me₃NTB complex has by far the most exergonic reaction with DHA (ΔG_PCET = −16 kcal mol⁻¹), followed by the TMG₂dien complex (ΔG_PCET = −9 kcal mol⁻¹), followed by the complexes of TMG₃tren, TMC, and N4Py (ΔG_PCET = −7, −6, and −6 kcal mol⁻¹, respectively). Thus, thermodynamic parameters would predict the Me₃NTB complex to have the lowest reaction barrier and fastest rate of reaction, with the TMG₂dien complex being the next most reactive, and the remaining complexes the least reactive as is observed experimentally.

In another study, it was observed that the rates of PCET reactions performed by [Fe^IV(O)(TMC)(X)]ⁿ⁺ decrease with more strongly donating axial ligands X.³⁶ Variation in ΔG_PCET was ruled out as a cause of this trend, as it was calculated to be similar for all complexes investigated. It was suggested that the accessibility of a high-spin state may explain this variation in the rates, as the energy of the quintet excited state decreased with stronger X ligands. However, our calculations indicate that while stronger axial donors increase ΔG_ET, ΔG_PT decreases more substantially (see Table S9†). In our model, these changes result in a net decrease in the reaction barrier, suggesting that despite a similar ΔG_PCET, the reactivity trend could be explained by thermodynamic effects. These analyses do not rule out that spin state or steric effects may be important in the previous studies, but suggest that thermodynamics may also play an important role.

The fit of the training data to {ΔG_PCET, ΔG_PT, ΔG_ET} and this model's performance on the test set is depicted graphically in Fig. 1. It is clear that the reaction barriers for most metal oxo complexes in the test set are well predicted. Nonetheless, several metal oxo complexes (given unique symbols in Fig. 1) deserve further discussion.

Fig. 1. Regression analysis of the experimental reaction barrier vs. ΔG_PCET, ΔG_PT, and ΔG_ET for various metal oxo complexes reacting with DHA. Specific cases discussed in the main text are given unique symbols. The grey line marks where predicted equals experimental. Predicted values are computed using the given formula.

The {ΔG_PCET, ΔG_PT, ΔG_ET} model behaves the poorest in predicting reaction barriers for the Fe^IV oxo and Co^IV oxo complexes of the ligand 13-TMC.^49,70 Essentially no barrier is predicted for these reactions which is not observed experimentally. This is due to a large negative calculated ΔG_PCET in both cases; in fact, these complexes are outliers even in the ΔG_PCET only fit (see Regression S1†). The cause of this discrepancy is not entirely clear. However, it appears to be systemic to the particular ligand scaffold rather than the identity of the metal center, which suggests this discrepancy could arise from ambiguity in the primary coordination sphere of these complexes. No structural characterization is reported for the Fe^IV complex, and while a short Co–O bond is identified by EXAFS for the Co^IV complex, it is difficult to conclusively determine the primary coordination sphere. Any discrepancy in coordination sphere would render our calculated parameters incorrect, potentially explaining their inability to predict the experimental reaction barriers.

The reaction barrier is overestimated for all Ru oxo complexes, and for three of them by more than two kcal mol⁻¹. As Ru is the only second row transition metal in our data set, we suspect this overestimation is due to a consistent difference between first and second row transition metals rather than Ru examples not following the same trends. For instance, it is possible that the Ru oxo complexes have relatively low structural reorganization energy or that relativistic effects influence the coefficients. It may also simply be a change in the systemic DFT error upon going to the second row. Regardless, regression of barriers from the kinetics of an individual Ru oxo complex reacting with several different substrates reveals there is a trend with ΔG_PCET, ΔG_PT, and ΔG_ET with similar coefficients to those obtained from the more general model with multiple different oxo complexes (see ESI†). This supports that the same trends in free energies are at play in the Ru oxo complexes.

Interestingly, the {ΔG_PCET, ΔG_PT, ΔG_ET} model only moderately underestimates the reaction barrier (by ∼2 kcal mol⁻¹) for a terminal Co^III oxo complex which has unusual trends in its reactivity with various substrates.¹⁵ Unlike most metal oxo complexes, the reactivity of this complex does not have a clear trend with ΔG_PCET; its kinetics are instead dominated by ΔG_PT. Therefore, its adherence to trends in {ΔG_PCET, ΔG_PT, ΔG_ET} as seen for the broad set of metal oxo complexes deserves further investigation. We regressed the experimental reaction barriers for the reactivity of this complex with several substrates against only ΔG_PT as well as against {ΔG_PT, ΔG_PCET} (Fig. 2). We find that the inclusion of ΔG_PCET significantly improves the model, increasing R² from 0.94 to 0.97 and LOO R² from 0.93 to 0.95 and having an F-test p-value of 0.02 (see ESI†). However, the relative weighting of the contribution from ΔG_PCET is quite different than for the broader set of complexes.

Fig. 2. Regression analysis for a Co^III oxo. The {ΔG_PT} fit is shown in black with predicted values computed using the indicated formula; R² = 0.94 and LOO R² = 0.93. The {ΔG_PT, ΔG_PCET} fit is shown in red with predicted values computed using the indicated formula; R² = 0.97 and LOO R² = 0.95. The negative barriers are due to overestimation of the entropy of association. The grey line marks where the predicted barrier matches the experimentally determined barrier.

In the broader set we observe that ΔG_PCET has a larger effect on the reaction barriers than either ΔG_PT or ΔG_ET, which is reflected in the larger coefficient for the ΔG_PCET term than for the ΔG_PT and ΔG_ET terms in the fit equation (Fig. 1). In contrast, ΔG_PT has a greater effect than ΔG_PCET on the reaction barriers for the Co^III oxo complex, again reflected in the magnitude of their coefficients (Fig. 2). Furthermore, the addition of ΔG_ET significantly improves the model for the broader set of metal oxo complexes (Table 1) but is insignificant for the series of substrates reacting with the Co^III oxo complex (p-value > 0.05, see Regressions S42 and S43†). Overall, this Co^III oxo complex is not so dissimilar from the broader set of metal oxo complexes in that the same thermodynamic free energies explain the reactivity of both. However, this individual case demonstrates a different weighting of parameters than those observed in the broad set.

Our analysis of the Co^III reactivity rests on the assumption that the coefficients of the model do not change appreciably from substrate to substrate. To test this assumption, we extended our analysis of the larger set of metal oxo complexes to include reactivity with 1,4-cyclohexadiene (CHD), fluorene, and xanthene in addition to DHA. We refit the model with reported data for the reactions between each substrate and all metal oxo complexes (excluding the previously discussed Ru and 13-TMC oxo complexes). As with our regressions for DHA alone, the inclusion of ΔG_PT and ΔG_ET notably improves the fit (Table 1, Fig. 3). Other parameters offer comparably little improvement to the fit and do not perform well by LOO cross validation. The equation for this model is ΔG^‡ = 0.23ΔG_PCET + 0.04ΔG_PT + 0.10ΔG_ET + 2.10, which is satisfyingly similar to the equation of the fit to DHA data alone, supporting the assumption that the coefficients of the model are not appreciably affected by the identity of the substrate.

Fig. 3. Regression analysis of the experimental reaction barrier to PCET mediated C–H activation vs. ΔG_PCET, ΔG_PT, and ΔG_ET for all non-outlier metal oxo complexes reacting with DHA, CHD, xanthene, and fluorene. The grey line marks where predicted equals experimental. Predicted values are computed using the given formula.

Conclusions

Overall, this thorough analysis of the reported C–H activation reactivity of transition metal oxo complexes demonstrates that ΔG_PCET, ΔG_PT, and ΔG_ET have a statistically significant correlation with the reaction barrier. Interestingly, no other parameters examined here, including steric environment and spin-based parameters, provide a significant improvement to a ΔG_PCET only model. This is in contrast to previous literature reports which implicate such factors in explaining metal oxo mediated PCET. The {ΔG_PCET, ΔG_PT, ΔG_ET} model predicts all but five of the reaction barrier heights for reactivity with DHA within 2 kcal mol⁻¹, and predicts most of these barrier heights within 1 kcal mol⁻¹. We find that the asynchronous reactivity of a Co^III oxo complex is also well-predicted, although a fit to just this complex alone reveals changes in the relative importance of ΔG_PCET, ΔG_PT, and ΔG_ET. The structural and electronic bases for such variation, and whether concomitant changes in selectivity can be leveraged, are exciting avenues for future research.

While the relative importance of these thermodynamic parameters can vary between specific cases, this study on a broad set of metal oxo complexes suggests that thermodynamic parameters provide the most general contribution to reaction barriers. Furthermore, while a strong dependence on ΔG_PCET is observed, as is expected based on literature precedent, significant and independent contributions from ΔG_PT and ΔG_ET are observed. This conclusion adds to the growing body of literature supporting the importance of thermodynamic parameters beyond ΔG_PCET.

Methods

Tabulation of experimental kinetics

In this study we used thirty reported k₂ values of metal oxo species reacting with DHA.^{12,13,15,17,34–51} For each of these oxo complexes, we tabulated various descriptors (metal, valency, d-count, coordination number, etc.), experimental parameters (M–O bond length and vibrational frequency, BDFE, pK_a, etc.), and the reported kinetics for reactions with various substrates (k₂, and, if reported, the statistical correction to this k₂, experimental ΔH^‡ and ΔS^‡, and the KIE) along with the conditions these data were reported in (temperature and solvent). A full tabulation is found in the ESI.† We excluded a few metal oxo species from our analysis despite having reported kinetic data for reactivity with DHA. The reasons for these exclusions were varied: several did not have a well-defined primary coordination sphere,^71–74 we were unable to calculate the reduced form of Mn^III oxo complexes without deprotonation of the hydrogen-bonding ligands,^17,20 one V^V oxo has too much experimental uncertainty in its k₂ value,⁷⁵ corrolazine complexes were too large to calculate their vibrational frequencies using our methods,^16,76 vibrational frequencies did not converge for the reduced form of two Ru^VI dioxo complexes,⁵⁰ we did not include third row complexes or complexes with ligand radicals,^37,77–79 and in one case saturation was reported at higher concentrations of DHA.⁸⁰ We also found several reports of metal oxo mediated C–H activation of substrates besides DHA^{14,39,70,81–94} and useful reviews.^95,96

All rate constants utilized here were reported as k₂ values with the exception of several rate constants used in the Co^III oxo reactivity analysis.¹⁵ In this case, for substrates which did not have a reported k₂, the pseudo-first order rate constant k_obs at 0.0125 M of substrate was divided by 0.0125 M to obtain an approximate k₂. We used all substrates with reported kinetic data in this analysis except for 1,1,3,3-tetraphenylpropene. This substrate reacts unusually slowly, which we believe to be due to large steric hindrance of the reacting C–H bond. The remaining substrates were sterically similar enough that there is no steric effect on their kinetics (see Regression S41†).

Determination of experimental barrier heights

Before determining barrier heights from experimental k₂ values, we first multiplied each k₂ by any reported stoichiometric and statistical adjustments so as to start from consistently unadjusted k₂ rate constants (experimental k₂ rate constants are often reported with statistical corrections to facilitate comparisons between substrates, either for the stoichiometry of the substrate's reactivity or for the number of benzylic C–H bonds). We assume that where no adjustment is noted in a paper, none has been made. The barriers of PCET reactivity were then determined from the unadjusted experimental k₂ values by solving the Eyring equation⁹⁷ and subtracting approximate expressions for the free energy of metal oxo-substrate association:

1

where h is Planck's constant, n_CH is 2 for DHA and CHD and 1 for fluorene and xanthene, n_O is the number of oxo ligands in the metal oxo complex, RT is the thermal energy, μ is the reduced mass of the metal oxo and the substrate, and C° is 1 M. Our n_CH adjusts for the typical stoichiometry of each substrate's oxidation (DHA and CHD tend to lose two H-atoms, fluorene and xanthene one H-atom); we do not adjust for the number of reactive H-atoms, as ring puckering of the substrates means that not all reactive C–H bonds are equivalent. For instance, DHA has four benzylic C–H bonds, with two lying equatorial to the central ring and two lying axial. It is unlikely that the equatorial and axial positions are equally reactive, and it is entirely possible that reactivity predominantly occurs at only one of the positions. It is therefore not necessarily true that DHA is four times as reactive as an otherwise similar substrate with only one benzylic C–H bond.

The second term in eqn (1) is an approximation for the free energy of association of the metal oxo and the substrate.⁹⁷ This adjustment allows us to compare kinetic data collected at different temperatures. As C–H bonds are poor hydrogen bond donors, we assume that the cost of association is purely entropic (or at least that enthalpic components vary minimally between different metal oxo complexes and substrates) and further assume this entropy cost is solely the loss of translational entropy. This neglects the loss of rotational entropy and the gain of low frequency metal oxo-substrate vibrational modes, but these effects will partially cancel. Regression with ΔG_PCET and RT does not fit DHA reaction barrier heights significantly better than a fit to ΔG_PCET alone (see Regression S10†), indicating that this adjustment satisfactorily accounts for the temperature dependence of the reaction barrier.

We do not take into account hydrogen bonding between the metal oxo complexes and protic solvents as we were unable to derive a suitably accurate correction. However, in the ESI† we demonstrate that our best attempt to do so does not change the main conclusions herein (see Table S11†).^28,98–100

Calculation of parameters

For each of these oxo complexes we calculated the values of parameters that have been proposed to influence metal oxo mediated PCET reactivity. Specifically, the parameters investigated were %BV steric metrics, spin density on the oxygen atom, available spin states, thermodynamic free energies ΔG_PCET, ΔG_PT, and ΔG_ET, and the magnitude of the asynchronicity parameter (|η|). Geometry optimization and frequency calculations were performed in ORCA using the def2 basis sets of Weigend and Ahlrichs and the O3LYP functional.^101–106 For the wider set of metal oxo complexes, all transition metals were given the def2-TZVPP basis set, all metal-bonded atoms and the transferring hydrogen atom the def2-TZVP basis set, and the remaining atoms the def2-SV(P) basis set. Substrates were calculated entirely with the def2-TZVP basis set. For calculations regarding the Co^III oxo and various substrates, Co, N, O, the carbene carbons of the ligand, and the carbon undergoing C–H activation was given def2-TZVPP; all other atoms were given def2-SVP. For both sets of calculations solvent effects were included as a polarizable continuum (CPCM with the dielectric constant of acetonitrile for the broader set of metal oxo complexes; COSMO with the dielectric constant of THF for the Co^III oxo with substrates). This is primarily to mitigate the effect of self-interaction error;¹⁰⁷ we assume that the solvent dielectric has little effect on the rate of PCET, as reported solvent effects on similar reactions are typically limited to hydrogen bonding.^28,100,108 The def2 ECP was used for ruthenium.¹⁰⁹ The resolution of identity approximation was used for coulomb integrals and the chain of spheres approximation for exchange integrals (with def2/J as the auxiliary basis). No change was made to ORCA's default grid settings. Free energies were derived from the electronic energies and vibrational calculations using the quasi-harmonic oscillator formulation of Grimme and coworkers.¹¹⁰

Unfortunately, several of our optimized structures have small imaginary frequencies (see ESI†), which we believe is due to numerical noise of CPCM solvation. Occasionally these frequencies lie below −100 cm⁻¹ but in each of these cases the mode is isolated to a soft dihedral motion, e.g. methyl rotation on an acetonitrile ligand. We used the absolute value of these frequencies when calculating the thermodynamic enthalpy and entropy values, believing that to be a better approximation for these modes than either nonexistence or a frequency of 0 cm⁻¹. We were unable to reoptimize these structures to remove the imaginary frequencies.

In many cases, the correct ground state multiplicity of a species was not immediately clear. In such cases we confirmed our initial assignment by running ten geometry optimization cycles on alternate spin states and confirming these alternate assignments were several kcal mol⁻¹ uphill of the assigned spin state. In a few cases were the energy was within 5 kcal mol⁻¹ and the optimization was not close to convergence, we fully optimized the alternate spin state. Whenever two spin states had nearly the same energy, we chose the higher spin state as the ground state due to the typically higher entropy of high spin states.

To quantify the steric environment around each metal oxo center or substrate reactive C–H bond, we calculated percent buried volume (%BV) steric metrics using the online SambVCA web application.⁵² We centered the calculation on the oxygen atom (for oxos) or the transferring hydrogen (for substrates), defined the negative z-axis as going through the metal center (for oxos) or the reacting carbon center (for substrates), and defined the xz plane as containing another atom bonded to the metal or carbon (the first such atom in the .xyz file). We had the center oxygen or hydrogen atom deleted from the calculation, included hydrogen atoms in the calculation, and left all other settings to their default value (using Bondi radii scaled by 1.17, a sphere radius of 3.5 angstroms, and a mesh setting of 0.10 angstroms). The application returns a total percent buried volume, as well as that for individual quadrants of the sphere. For metal oxo complexes, we used the total percent buried volume (%BV Tot) and the standard deviation of these four quadrants (%BV Dev) in our regressions in order to capture both overall steric bulk and how evenly distributed this bulk is around the metal oxo moiety. For substrates, we solely used %BV Tot. See the ESI† for a further discussion of steric parameters and their effect on reaction barrier heights.

To evaluate the effect of spin and spin state on reactivity, we used two parameters that have been discussed in the literature: spin density on the oxo ligand and the energy to excite to a higher spin state.^8,10 Atomic spin populations were determined via IBO analysis using the freely available IBO Viewer software.^53,54 We recorded the spin density on the metal and on oxygen for each metal oxo complex as well as how much spin both atoms gain upon PCET reduction; we also tabulated similar values for the IBO charges. In the regression analysis we solely used the spin density on the oxo ligand. The “Spin Excitation Energy” is the vertical energy from the ground spin state of the initial oxo complex to the lowest lying excited spin state that is within one spin multiplicity of the resulting metal hydroxide ground spin state. If the ground spin state is already one spin multiplicity greater or lower than the product hydroxide spin state, then the spin excitation energy is taken to be zero. For instance, in the case of a triplet Fe^IV oxo reacting to give a sextet Fe^III hydroxide the spin excitation energy is the energy of the quintet Fe^IV oxo relative to the triplet Fe^IV oxo at the ground state optimized geometry. This is the scenario for most Fe^IV oxos in the data set. But in the case of the two non-heme Fe^IV quintet oxos,^12,13 the spin excitation energy is zero because the ground spin state is already within one spin multiplicity of the sextet hydroxide product. Essentially, the spin excitation energy is the energy needed to reach a spin surface on which reduction to the metal hydroxide's ground spin state is spin allowed. While this simple metric ignores the nuances of two state reactivity theory (such as the spin inversion probability) it is relatively simple to compute and has precedent as a quantitative measure of PCET reactivity.^10,36

For each metal oxo-substrate combination assessed here, we tabulated the free energies of proton coupled electron transfer (ΔG_PCET, eqn (2)), proton transfer (ΔG_PT, eqn (3)), electron transfer (ΔG_ET, eqn (4)), and the asynchronicity as defined by Srnec and coworkers (η, eqn (5)):²¹

ΔG_PCET = G_M–OH + G_C· − G_{M O} − G_C–H 2

ΔG_PT = G_M–OH⁺ + G_C:⁻ − G_{M O} − G_C–H 3

ΔG_ET = G_M–O⁻ + G_C–H⁺ − G_{M O} − G_C–H 4

5

where G_{M O} is the calculated free energy of the oxo species, G_C–H is the calculated free energy of the substrate, and all other free energies are defined analogously. We also tabulated the absolute value of the asynchronicity (|η|), the average of ΔG_PT and ΔG_ET (ΔG_{CT Average}), and the analogous electronic energies (same notation, with G replaced with E).

Statistical analysis

All statistical analysis was performed in Python using the Numpy, Scipy, Pandas, Sklearn, and Matplotlib packages.^111–115 A script ran a prescribed set of regression models and reported statistics on each model. All regressions were performed with ordinary least squares. Prior to fitting any regression, we separated the data into a test set and a training set of metal oxo species. While we show both test and training sets for each regression in the ESI,† we initially did not plot the test set or calculate statistics with it. We solely used the training set in the earlier stages of our analysis, where we determined which parameters improved a fit to ΔG_PCET only and which did not. We then examined if the good fits to the training set extrapolated well to the test set. We had to make a few changes to the initial division of the training and test sets, however. Initially, the Co^IV oxo was included in the training set,⁴⁹ but that was interfering with the fit to ΔG_PCET. We moved it to the test set, and to provide insight into this poor fit we calculated the Fe^IV oxo complex of 13-TMC and added it to the test set.⁷⁰ We also saw, visually, that the inclusion of a Ru^IV oxo in the training set was interfering with the addition of other parameters and moved all Ru oxo complexes to the test set.^{38,43,45,50,51}

The simplest metrics reported from these models are the mean square error (MSE) and the goodness of fit R².^58–61 These both give an indication of how well a model fits the available data but are prone to overfitting; more complicated models can only improve these metrics, regardless of whether or not the model is actually better.

We also evaluated each model with cross validation (CV) metrics, which can become worse upon overfitting. In K-fold cross validation, the training data is further subdivided into K subsets, and each subset is predicted by the K − 1 remaining subsets.^59,61 When K is the number of data points, i.e. each data point being predicted by the rest of the data points, this is known as leave-one-out (LOO) cross validation. These predicted data points can be used to calculate the MSE and R² as above. The MSE from LOO cross validation is an approximately unbiased estimate of the expected error of a test set; however, it has high variability from training set to training set because each prediction uses nearly every point in a given training set. By repeatedly subdividing into larger groups and averaging the resultant K-fold MSEs, one obtains a pessimistic but less variable estimate of the expected test error. As we see similar trends for both LOO and 5-fold CV, we only report LOO R² in the main text but show all metrics in the ESI.†

Another way to determine the significance of the model is to use a statistical F-test.^58,60 This allows one to compare an unrestricted model with a more restricted one (fewer parameters used as regressors, or no parameters regressed, or restrictions placed on the relationship between coefficients, etc.). In the language of hypothesis testing, the null hypothesis is that the unrestricted model offers no improvement on the restricted model and the alternate hypothesis is that there is an improvement. When both models are fit to the data, the unrestricted model will have less total squared error than the restricted model. Assuming said error of each data point is normally distributed (or that there is enough data such that the error is approximately normally distributed), that the average error is zero, and that the model is properly formulated, it is possible to determine the probability that this reduction in total squared error is spurious. This probability is known as the p-value. The test relies on a well-defined number of degrees of freedom in both the restricted and unrestricted model to draw out what the statistical distribution of total squared error ought to be.

For regressions on multiple substrates at once, the unequal weighting of different metal oxo complexes (depending on how many substrates are reported for them) renders these statistical metrics unreliable.⁶¹ We ameliorate this issue for LOO cross validation by leaving out all reaction barriers for a given metal oxo complex together rather than one at a time. That is, we leave one metal oxo complex out and predict its reaction barrier heights based on all other metal oxos' reaction barrier heights rather than leave one reaction barrier height out and predict this barrier based off all other barriers. We accordingly only report LOO CV metrics for this set of regressions.

Conflicts of interest

There are no conflicts to declare.