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. 2022 Sep 20;7(39):34858–34867. doi: 10.1021/acsomega.2c03252

Rapid and Accurate Estimation of Activation Free Energy in Hydrogen Atom Transfer-Based C–H Activation Reactions: From Empirical Model to Artificial Neural Networks

Siqi Ma †, Shipeng Wang †, Jiawei Cao †, Fengjiao Liu †,‡,*
PMCID: PMC9535641  PMID: 36211072

Abstract

graphic file with name ao2c03252_0015.jpg

A well-performing machine learning (ML) model is obtained by using proper descriptors and artificial neural network (ANN) algorithms, which can quickly and accurately predict activation free energy in hydrogen atom transfer (HAT)-based sp3 C–H activation. Density functional theory calculations (UωB97X-D) are used to establish the reaction system data sets of methoxyl (CH3O·), trifluoroethoxyl (CF3CH2O·), tert-butoxyl (tBuO·), and cumyloxyl (CumO·) radicals. The simplified Roberts’ equation proposed in our recent study works here [R2 = 0.84, mean absolute error (MAE) = 0.85 kcal/mol]. Its performance is comparable with univariate Mulliken-type electronegativity (χ) with the ANN model. The ANN model with bond dissociation free energy, χ, α-unsaturation, and Nolan buried volume (%Vburied) successively improves R2 and MAE to 0.93 and 0.54 kcal/mol, respectively. It reproduces the test sets of trichloroethoxyl (CCl3CH2O·) with R2 = 0.87 and MAE = 0.89 kcal/mol and accurately predicts the relative experimental barrier of the HAT reactions with CumO· and the site selectivity of CH3O·.

Introduction

With the increasing demand of sustainable development, use of the sp3 C–H bond function in chemical synthesis has received great attention because it can provide a practical solution to upgrade abundant hydrocarbon raw materials into valuable products.1−11 The hydrogen atom transfer (HAT)-based method provides an effective strategy for hydrocarbon activation and has been widely used.2,12−20 However, the prediction of reactivity is still a challenge. The reactivity is closely related to the C–H bond strength and the choice of H-extracting radicals.21−32 Understanding the reaction mechanisms and predicting the reactivity should assist a more reasonable and effective design of C–H activation reactions.33−38

The activation free energy is a key parameter for the mechanism, reaction rate, and selectivity of chemical reactions. The prediction of the activation free energy helps to achieve a comprehensive understanding of chemical reactions and to construct chemical reaction diagrams, thus accelerating catalyst design. However, the activation free energy estimation is time-consuming, both experimentally and computationally, leading to a bottleneck effect on the rapid catalyst design and characterization.

In experimental work, the reaction rate constant is determined by running multiple experiments at different temperatures, and then the activation free energy can be obtained by application of the Eyring equation. In computational science, quantum chemistry methods can evaluate the activation free energy by identifying the transition states along a given reaction path. It is inevitable to spend a lot of computational cost to deal with complex chemical processes involving many reactions.39,40 In this situation, it is very valuable to achieve rapid and accurate prediction of activation free energies in C–H activation reactions.

As an empirical model, the Bell–Evans–Polanyi (BEP) correlation is widely used to quickly estimate the activation energy.41 It correlates the activation energy with reaction energy through ΔE‡ = γΔEr × n + ξ. Tedder found the existence of a BEP correlation between the rate constant and the C–H bond strength for H-abstraction reactions from alkanes by various radicals (CH3·, CF3·, Br·, etc.).42 Mayer et al. showed the generality of BEP correlations in the HAT of CrO2Cl2 and MnO4–.43 Shaik et al. used density functional theory (DFT) to study the BEP relationship of enzyme cytochrome P450 oxidizing a series of C–H bonds and established a prediction model.44 However, as the organic system becomes more and more complex, people cannot observe the simple linear relationship between the activation energy and the bond energy.45−48 In addition to bond energy or reaction energy, Roberts pointed out that polar effects, steric effects, and unsaturation should be considered when discussing various HAT reactions in the liquid and gas phases. In addition, his study has emphasized and quantified the polar effects.49 Tedder proposed that the polar effects in the HAT of CH3· and CF3· will cause a deviation from the BEP correlation.42 Our previous study used DFT calculations to explore the selectivity of dimethyldioxirane (DMDO) in the C–H oxidation of various compounds, and it revealed a different BEP correlation between the C–H bonds in saturated and unsaturated compounds.46 This bimodal BEP correlation was also found in HAT from sp3 C–H bonds to the cumyloxyl radical by Bietti et al. All these precedents urged us to think about the influence of nonlinear characteristics and factors other than thermodynamics in HAT.

Multiple linear regression (MLR) is one strategy to consider the effect of several variables on activation energy. Roberts and Steel proposed an improved form of the Evans–Polanyi equation (Roberts’ equation) involving reaction thermodynamics, radical electronegativity differences, and conjugate delocalization of the unpaired electron and structural factors, which work well for a variety of HAT reactions with different H-extracting radicals.49 Our recent work simplified Roberts’ equation with reaction energy, radical electronegativity differences, and unsaturation and led to good results for the HAT reactions of 26 sp3 C–H bonds by alkoxyl radicals, with R2 = 0.89 and mean absolute error (MAE) and root mean square error (RMSE) values of 0.9 and 1.1 kcal/mol, respectively (compared with the DFT calculations).50

On the other hand, the significant advancement of machine learning (ML) technology provides a new strategy for solving various chemical problems. The artificial neural network (ANN), as one of the most popular ML methods, is a data-driven adaptive method. Due to its parallel structure and the ability to simulate arbitrary functions, it can be regarded as a kind of multiple nonlinear regression that can handle complex nonlinear problems.51 By establishing the potential correlations from the data, the ANN can solve problems that involve unknown relationships. Nowadays, the ANN is actively used in various chemical fields, such as quantum chemistry, virtual screening of molecular materials, and synthetic pathways of chemical substances.52,53 While our study was ongoing, Hong et al. have provided an ML approach for reactivity prediction in photoredox-mediated HAT catalysis.54

In the present study, we aim to (1) apply the ANN model to the HAT-based sp3 C–H activation to achieve the goal of rapid and accurate prediction of the activation free energy, (2) assess the performance of the simplified Roberts’ equation proposed in our recent study,50 and (3) compare the performance between MLR and ANN with the same physical variables. The substrate C–H bonds have been expanded (Figure 1). We added a series of hydrogen donor substrates to build the model on the basis of our previous work46 [including allylic, benzylic, formylic, and α-heteroatom (O, N, S) group]. The activated C–H bond is highlighted in red. In addition to methoxyl (CH3O·), trifluoroethoxyl (CF3CH2O·), and tert-butoxy (tBuO·), cumyloxyl (CumO·) was also used as the H-extracting radical. Alkoxyl radicals are easily available and can be highly efficient for the abstraction of H atoms. They are powerful tools for the catalytic activation of hydrocarbons under mild reaction conditions. Zuo et al. achieved the C–H functionalization of short-chain alkanes (methane, ethane, etc.) utilizing cerium salts and alcohols under mild reaction conditions with CH3O·, CCl3CH2O·, and CF3CH2O· being used as the H-extracting radicals.55 Bietti carried out systematic kinetic studies of HAT reactions by CumO·.48 These experimental data are helpful to verify the theoretical model.

Figure 1.

Figure 1

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C–H bonds studied in this work.

Method

Database

We established data sets for the alkane sp3 C–H activation by various alkoxyl radicals: CH3O·, CF3CH2O·, tBuO·, and CumO· (Tables S1–S4). The data sets were used to build the model. The C–Hs listed in Figure 1 were studied. The substrates and H-extracting radicals were selected to consider polar effects, steric effects, and α-unsaturation effects. Figure 2 shows an example of the HAT step by the alkoxyl radical.55 All calculations were carried out with Gaussian 16.56 The geometries of minima and transition states were optimized using UωB97X-D57 with the 6-31G (d)58 basis set in the gas phase. Vibrational frequency analyses confirmed the nature of the structures as either local energy minima or first-order saddle points (transition states). Single point energies with a more extensive basis set were obtained with UωB97X-D/6-311++G (d,p)59 on the optimized geometries. The solvent effect of CH3CN on the reaction was estimated by using the SMD model.60

Figure 2.

Figure 2

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Scheme of alkoxyl radical-mediated HAT activation with methane.

Figure 3 shows an example of the data sets. The “sub” and “rad” qualifiers represent the descriptors of the substrate and H-extracting radical, respectively. The bond dissociation free energies (BDFEs) of the substrate C–H and alkoxyl radical O–H were both taken into consideration. The polar effect was quantified by the Mulliken-type electronegativity. The Mulliken-type electronegativity of the X· radical (χx) is defined in eq 1, where IE and EA are the X· vertical ionization energy and vertical electron affinity, respectively.

graphic file with name ao2c03252_m001.jpg 1

Figure 3.

Figure 3

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HAT reaction space representation using vectors.

ΔχAB (ΔχAB = χA – χB, A–H + B· → A· + BH) reflects the electronegativity difference between the substrate and H-extracting radical. The unsaturation effect refers to whether an unsaturated group adjacent to the investigated C–H is present. The “0” and “1” qualifiers are used for “saturated” C–H and “unsaturated” C–H bonds, respectively. The Nolan buried volume (%Vburied)61 is used to consider the steric effect (see Section S8 for the detailed calculation).

ANN Model

Figure 4 is an overview of the ANN model. Activation barriers and descriptors were all obtained by DFT methods. The ANN model was trained on the basis of appropriate descriptors.

Figure 4.

Figure 4

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Process of the ANN model predicting activation free energy.

The back-propagation ANN model62,63 was applied to predict the activation free energy. Through K-fold cross-validation (k = 3),64 the topology of the ANN model was finally determined with three hidden neurons and three hidden layers (see Table 1). The activation function was a tanh function (Inline graphic) that allows the network to map any nonlinear process. The second-order optimization method was an implementation of the Levenberg–Marquardt (LM) algorithm.65−67

Table 1. Main Parameters of ANN Models.

ANN
hidden layer sizes 6
hidden unit 3
activation tanh
tolerance 0.001
training function Levenberg–Marquardt
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Results and Discussion

Tables S1–S4 list the activation free energy (ΔG‡), the reaction free energy (ΔGr×n), the BDFE, the Mulliken electronegativity (χ), and the Nolan buried volume (%Vburied) for HAT reactions promoted by CH3O·, CF3CH2O·, tBuO·, and CumO·, respectively. The substrate C–H BDFE ranges from 71.2 to 96.0 kcal/mol. The ΔG‡ values range from 11.3 to 20.3 kcal/mol for CH3O·, range from 10.6 to 20.1 kcal/mol for tBuO·, range from 9.7 to 21.1 kcal/mol for CumO·, and range from 8.7 to 20.1 kcal/mol for CF3CH2O·. Overall, CF3CH2O· has a greater reactivity than CH3O·, tBuO·, and CumO·.

Empirical Model

In HAT, the BEP correlation can be expressed as shown in eq 2, where ΔG‡ is the activation free energy, ΔGr×n is the reaction free energy, BDFEsub and BDFErad are the BDFE of the substrate C–H and alkoxyl radical O–H, respectively, and γ and ξ are obtained from linear regression analysis.

graphic file with name ao2c03252_m003.jpg 2

Figure 5 shows the linear relationship of ΔG‡ vs ΔGr×n for the reaction data sets of 60 sp3 C–Hs (Figure 1) with four alkoxyl radicals (CH3O·, CF3CH2O·, tBuO·, and CumO·). The traditional BEP correlation obviously does not work here. Although the correlation is rough, the “saturated” and “unsaturated” C–Hs tend to be divided into two categories (Figure 5) as found in our previous study.46 The effects of the unsaturation and of the radical nature are responsible for the inadequacy of the BEP correlation.

Figure 5.

Figure 5

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Activation free energy for HAT as a function of ΔGr×n.

Figure 6a–d shows the scatter diagram of ΔG‡ vs ΔGr×n for each alkoxyl radical. In comparison with CH3O·, the higher electronegativity CF3CH2O· radical is more reactive and yields a worse linear correlation. The steric effect also influences the BEP correlation, as is evident from a comparison of the differences between the ΔG‡ vs ΔGr×n scatter plots for CH3O·, tBuO·, and CumO·. Due to the influence of different H-extracting radicals, the BEP correlation for the whole series is poor, as shown in Figure 5.

Figure 6.

Figure 6

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Activation free energy for the HAT reaction as a function of ΔGr×n for the different alkoxyl radicals: (a) CH3O·, (b) tBuO·, (c) CF3CH2O·, and (d) CumO·.

Figure 7 shows the scatter diagram of ΔG‡ vs Δχ2 (Δχ = χ_sub – χ_rad). This correlation is better than that of ΔG‡ vs ΔGr×n, and the unsaturation effect is less marked, to the point that the “saturated” and “unsaturated” C–Hs can be classified into one category.

Figure 7.

Figure 7

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Activation free energy for the HAT reaction as a function of Δχ2 for different alkoxyl radicals: (a) CH3O·, (b) tBuO·, (c) CF3CH2O·, (d) CumO·, and (e) all four alkoxyl radicals (CH3O·, CF3CH2O·, tBuO·, and CumO·).

The results of Figure 7 imply that Δχ2 plays a more important role than ΔGr×n in predicting the activation free energies with univariate linear regression. The importance of Δχ2 is further proved by the random forest (RF) algorithm.68 RF can provide a measure of the feature importance based on the mean decrease in impurity (MDI), and the impurity is calculated by the split criterion of the decision trees (entropy).68Figure 8 shows the feature importance of the unsaturation effect, Δχ2, and ΔGr×n analyzed by the RF algorithm. As shown in Figure 8, Δχ2 plays the most important role, followed by ΔGr×n and the least important is the effect of unsaturation. It is worth mentioning, however, that the feature importance analysis based on MDI is biased to high cardinality features (typically numerical features, e.g., Δχ2) and probably underestimates low cardinality features (binary features, e.g., the unsaturation effect).69 Therefore, the impact of descriptors needs to be further analyzed. Though Δχ2 shows a good correlation with ΔG‡, this is insufficient to accurately predict the activation free energy, especially using a simple linear relationship.

Figure 8.

Figure 8

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Feature importance of each descriptor analyzed by RF.

Then, the simplified Roberts’ relationship developed in our recent work50 was applied (eq 3).

graphic file with name ao2c03252_m004.jpg 3

This expression contains the reaction energy (ΔGr×n, ΔGr×n = BDFE_sub – BDFE_rad), the unsaturation term (d), and the Mulliken-type electronegativity difference (Δχ, Δχ = χ_sub – χ_rad) between the substrate and H-extracting radical. From an MLR analysis of our data sets, the coefficients for this simplified Roberts’ relationship are shown in eq 4.

graphic file with name ao2c03252_m005.jpg 4

The correlation between ΔG‡DFT and the ΔG‡Predict obtained by this multivariate linear relationship, shown in Figure 9, is good (R2 = 0.84, MAE = 0.85, and RMSE = 1.05).

Figure 9.

Figure 9

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DFT-computed vs predicted barriers using eq 4 for the reaction data sets of 60 sp3 C–Hs (Figure 1) with four alkoxyl radicals (CH3O·, CF3CH2O·, tBuO·, and CumO·).

ANN Model

The ANN model was trained on the reaction data sets of 60 sp3 C–Hs (Figure 1) with four alkoxyl radicals (CH3O·, CF3CH2O·, tBuO·, and CumO·) by using different descriptors. Figure 10a,b shows the activation free energies obtained by the ANN model (with only the BDFE and χ as descriptors, respectively) versus the DFT method. The BDFE can hardly be used to predict the activation free energy, as shown by R2 = 0.62, MAE = 1.28, and RMSE = 1.62. The χ still works better than BDFE when used as the unique descriptor in the ANN model. This is reflected by R2 = 0.85, MAE = 0.79, and RMSE = 1.03. It is worth noting that the performance of the single χ descriptor with the ANN model is comparable to the MLR of eq 4. Subsequently, we tried to add the descriptors. The combination of BDFE_sub, BDFE_rad, χ_sub, χ_rad, and α-unsaturation successively makes the R2 improve to 0.92, accompanied by MAE = 0.60 and RMSE = 0.75, as shown in Figure 10c. After adding the %Vburied, the ANN model performs best (R2 = 0.93, MAE = 0.54, and RMSE = 0.68), see Figure 10d.

Figure 10.

Figure 10

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DFT-computed barriers vs ANN-predicted barriers for the reaction data sets of 60 sp3 C–Hs (Figure 1) with four alkoxyl radicals (CH3O·, CF3CH2O·, tBuO·, and CumO·) by different descriptors (a) BDFE_sub and BDFE_rad, (b) χ_sub and χ_rad, and (c) BDFE_sub, BDFE_rad, χ_sub, χ_rad, and α-unsaturation. (d) BDFE_sub, BDFE_rad, χ_sub, χ_rad, α-unsaturation, %Vburied_sub, and %Vburied_rad. The “sub” and “rad” qualifiers represent the descriptors of the substrate and H-extracting radical, respectively.

In order to further evaluate the predictive ability of the model, we used the reaction data of CCl3CH2O· as the test set (Table S5). The R2 and MAE values obtained from the ANN model without %Vburied are 0.80 and 1.10 kcal/mol, showing that the ANN model trained on the data set composed of CH3O·, CF3CH2O·, tBuO·, and CumO· can map the influence of the polarity changes of CCl3CH2O· (Figure 11a). By adding the %Vburied term, the R2 and MAE values improve to 0.87 and 0.79 kcal/mol, respectively, showing the importance of the steric effect (Figure 11b).

Figure 11.

Figure 11

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DFT-computed vs ANN-predicted barriers for the test set of 60 sp3 C–Hs (Figure 1) with CCl3CH2O· by different descriptors: (a) BDFE_sub, BDFE_rad, χ_sub, χ_rad, and α-unsaturation. (b) BDFE_sub, BDFE_rad, χ_sub, χ_rad, α-unsaturation, %Vburied_sub, and %Vburied_rad.

Furthermore, we predicted the experimental results by using the trained model. According to Bietti’s research on the HAT reaction of CumO·, we collected another 45 sp3 C–Hs from his study and used the ANN model trained on the DFT computational data to make predictions. To eliminate the final prediction bias caused by the errors of the DFT calculation and the experiment, the relative activation free energy was used here. Rate constants and relative activation free energies are provided in the Supporting Information (Table S6). Figure 12 compares the activation free energies predicted by the ANN model with the relative activation free energies of Bietti’s experiments. The prediction results show that the ANN model containing %Vburied can better map the influence of the CumO· steric effects and make more accurate predictions (R2 = 0.70, MAE = 0.65, and RMSE = 0.80).

Figure 12.

Figure 12

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Experiment vs ANN-predicted barriers for the test set of 45 additional sp3 C–Hs with CumO· by different descriptors (a) BDFE_sub, BDFE_rad, χ_sub, χ_rad, and α-unsaturation. (b) BDFE_sub, BDFE_rad, χ_sub, χ_rad, α-unsaturation, %Vburied_sub, and %Vburied_rad.

We also tried to predict the results of the CH3O· selectivity from Zuo’s reports (Table S7).70 The BDFE, χ, α-unsaturation, and %Vburied descriptors were used. Figure 13 includes the relative ratio of site selectivity and the activation free energies obtained by the ANN prediction and by the DFT calculation. The numbers marked in red indicate that the prediction error is over 2 kcal/mol, and the blue backgrounds indicate the main activation site predicted by the ANN model. Among the six substrates not included in the training set, 2,3-dimethylbutane has been widely used as a standard substrate for evaluating selectivity in C–H bond functionalization. The α-tertiary carbon can be formed with good selectivity (ratio 97:3) when CH3O· was used as a HAT reagent. Comparing with the DFT-calculated and ANN-predicted activation free energies, the reaction site is the same. The tertiary C–H bond of adamantane can be functionalized and predicted as well. N-Hexane has three different types of C–H bonds in terms of steric hindrance and bond strength. In this case, CH3O· has high selectivity for the weaker methylene C–H bonds, and the predicted activation free energy at this site is also the lowest. In 2,4-dimethylpentane, CH3O· is used to obtain methine functionalization. Although the activation free energy predicted by the ANN model is lower, the selectivity is consistent with the experiment.

Figure 13.

Figure 13

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Evaluation of the ability to predict HAT reaction sites of CH3O·.

Conclusions

We have strived to achieve a rapid and accurate estimation of activation free energies in HAT-based C–H activation reactions with both an empirical method and the ANN model. First, we established a data set of 300 HAT reactions on the basis of DFT calculations. By simply analyzing the data set, we found that unsaturation effects are responsible for the poor performance of the BEP relationship, while the correlation between ΔG‡ and Δχ2 is better than that between ΔG‡ and ΔGr×n. The simplified Roberts’ equation proposed in our recent study also works here. Then, we used an ML method to establish a reactivity and selectivity prediction model based on appropriate descriptors. As a unique descriptor used in the ANN model, χ works better than the BDFE. Its performance is comparable with that of the simplified Roberts’ equation. The introduction of %Vburied can improve the generalization ability between different H-extracting radical and make more accurate predictions, which shows the importance of steric effects. The combination of BDFE, χ, α-unsaturation, and %Vburied successively makes the R2, MAE, and RMSE improve to 0.93, 0.54, and 0.68, respectively. The ANN model reproduces the experimental CumO· relative activation free energies and CH3O· selectivities with good accuracy.

Acknowledgments

We are grateful to the Shanghai Sailing Program (No. 20YF1416000) for financial support of this research.

Supporting Information Available

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acsomega.2c03252.

  • Bietti’s and Zuo’s experimental results and the computational results of activation free energies and descriptors for HAT (PDF)

The authors declare no competing financial interest.

Notes

The code for the ANN model, the csv file of data sets, and the cartesian coordinates of DFT-computed structures are freely available on GitHub.71 The instructions on how to use the model are also provided.

Supplementary Material

ao2c03252_si_001.pdf (523.6KB, pdf)

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