Thermochemical unification of molecular descriptors to predict radical hydrogen abstraction with low computational cost

Abstract

Chemistry describes transformation of matter with reaction equations and corresponding rate constants. However, accurate rate constants are not always easy to get. Here we focus on radical oxidation reactions. Analysis of over 500 published rate constants of hydroxyl radicals led us to hypothesize that a modified linear free-energy relationship (LFER) could be used to predict rate constants speedily, reliably and accurately. LFERs correlate the Gibbs activation-energy with the Gibbs energy of reaction. We calculated the latter as the sum of one-electron transfer and, if appropriate, proton transfer. We parametrized specific transition state effects to orbital delocalizability and the polarity of the reactant. The calculation time for 500 reactions is less than 8 hours on a standard desktop-PC. Rate constants were also calculated for hydrogen and methyl radicals; these controls show that the predictions are applicable to a broader set of oxidizing radicals. An accuracy of 30–40% (standard deviation) with reference to reported experimental values was found suitable for the screening of complex chemical systems for possibly relevant reactions. In particular, potentially relevant reactions can be singled out and scrutinized in detail when prioritizing chemicals for environmental risk assessment.

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1. Introduction

Free radicals are produced (photo)chemically in living cells, surface water, electrochemical devices and many other systems. A significant number of free radicals are reactive and oxidize molecules via diverse reaction pathways. They are important in the context of electrochemistry,¹ environmental science² and biology,³ and there is an interest to understand reaction pathways and to quantify the rates of reaction. Experimental methods to determine the rate constant, k_r, of a chemical reaction are in general laborious.⁴ Besides the cost of instrumentation and lab space, measuring a single rate constant with proper controls and data evaluation easily takes a skilled experimenter 3 days.

With 100 000+ chemicals to be assessed,^5,6 there is a growing need to supplement experiments with computational methods to reduce costs and increase the sample variation and throughput.^7–9 With this large number of chemicals, not all the reaction rate constants and pathways can be calculated with ultimate precision. Instead, we need ‘cheap’ calculus methods which are accurate and precise enough to (1) determine the rate constants of the bulk of reactions and (2) identify those reactions that require further, more detailed and expensive scrutiny.

Prediction of k_r is possible via, for example, ‘machine learning’ approaches which use a variety of algorithms and easily retrievable topological, electrostatic, and energetic parameters.² The approaches yield relationships of mixed quality that depend on the type of radical and substrate considered. Many machine learning methods are ‘black box’ approaches, not supporting facile interpretation in terms of reaction mechanisms. In addition, parameters can be ill-defined for molecules that are complex or outside the model's domain of applicability.^10–12 A different line of research aims to predict k_r mechanistically. ‘Ab initio’ methods base calculations of k_r on an accurate and precise determination of the free energy of activation ΔG^‡. The rate constant, k_r, of a chemical reaction is described by the Eyring-equation as¹³

ΔG^‡ is the free energy of activation, R the universal gas constant, T the absolute temperature and q a statistical factor. The ab initio methods involve the exploration of free energy changes along the reaction coordinate but the time-dependent three-dimensional interactions involved take much computing power for a full description.¹⁴ Simplifications might use static configurations for reactants or transition states, but require foreknowledge on the reaction pathway. Moreover, the statistical factor q (eqn (1)) has to be described separately; q differs between non-spherical ‘complex’ reactants.¹⁵

Simplifications of the Eyring-equation are free-energy relationships, made famous by the work of Hammett and Taft.^16,17 For reactions with a closely similar rate-determining step, LFER stipulates that differences in Gibbs energy of activation, or the ratio of rate constants k_r1/k_r2 for two substrates 1 and 2, are proportional to the difference in the reaction free energy change ΔG_r, eqn (2) and Fig. 1:

where σ_LFER is the substituent (fitting) constants.^16,18 This approach has been successful because it uses semi-empirical parameters, hence requires less effort as compared to ab initio methods. Thermodynamic values, such as the Gibbs energies of reaction ΔG_r, are often available from literature or comparatively easy to estimate computationally and are, therefore, ‘cheap’.

Fig. 1 Linear free energy relationships establish that, in a first approximation, the difference in activation energies Δ(ΔG^‡) (dotted lines) is proportional (σ_LFER as the proportionally constant) to the differences in Gibbs energies Δ(ΔG). (dashed lines) between reactions, e.g. r1 (blue curve) and r2 (yellow curve). The concept¹⁶ offers the prospect of efficient computational methods to predict ΔG^‡ and, by extension, k_r. The rectangle symbol at the intercept between green and blue curves refers to Fig. 2.

LFER predictions are possible for reactions of chemicals with conventional, ‘non-exotic’, substituents¹⁹ and when σ_LFER values are available (Fig. 1). Many LFERs developed over the years predict Δ(ΔG^‡) with reasonable accuracy. Comparison of Δ(ΔG_r) with experimental data allowed to approximate σ_LFER for various radicals (e.g. O₂˙⁻/HOO˙ and aryl radicals^20,21). Curiously, for those radicals, σ_LFER was identical (i.e. within 2 standard deviations). Given these findings,^18,20–22 we hypothesized σ_LFER to be a constant factor for a broader set of radicals. This implies that all radical-induced one-electron transfers have a similar rate-determining step.

The prediction of Δ(ΔG^‡) via LFER is a successful, albeit crude estimation, eqn (2). Changing solvents, i.e. different dielectric constants of the environment, is for instance known to alter product distribution and reaction kinetics.²³ This is direct evidence that charge polarization in the transition state (TS) can affect the Gibbs energies of activation for a specific transition state, and thus promote/inhibit a specific reaction pathway. In other words, there are additional, usually smaller, contributions to the activation energy, Δ(ΔG^‡_TS), that we need to consider,²⁴eqn (3):

Δ(ΔG^‡) = σ_LFERΔ(ΔG_r) + Δ(ΔG^‡_TS) (3)

Thus, the practical problem is the development of computationally efficient calculus of the quantum chemical influences on the transition states hidden in the term Δ(ΔG^‡_TS), Fig. 2:

Fig. 2 Enlarged version of the black rectangle in Fig. 1. Depending on the nature and environment of the reaction site, a specific pathway (p1 or p2, indicated by purple and red curves, respectively) may be favoured. This is indicated by a ‘perturbation’ Δ(ΔG^‡_TS), of the ‘normal’ LFER behaviour (LFER depicted in Fig. 1).

We therefore chose to calculate rate constants with a modified LFER approach that combines the ‘cheap’ calculus of the free-energy of reaction with selective ‘ab initio’-style additions. The selection reflects an analysis of over 500 published reactions of hydroxyl radicals. The resulting factors governing the rate constants originally caused us to embrace LFER terminology for the explanation of our work.

In our study, we investigated the details in transition states for one-electron oxidations, i.e. reactions that involve the net transfer of a single electron and, possibly, a single proton. We explain our selection of easily accessible parameters to successfully calculate and predict Δ(ΔG^‡_TS). With it, we estimate rate constants for one-electron transfer, or radical oxidation, reactions.

2. Methods

Linear free-energy relationships (LFERs) assume that Δ(ΔG_r) relates to k_r1/k_r2 under the condition of an ‘identical’ mechanism, the same number of reacting sites and similar geometry. Eqn (2) allows for calculation of Δ(ΔG^‡) and k_r1/k_r2 based on knowledge of Δ(ΔG_r).

For a successful use of LFER, all parameters for the basic eqn (2), prominently the proportionality constant σ_LFER, need to be derived. Our parametrization process, together with the introduction of all the modifications (Δ(ΔG^‡_TS) and the statistical factor q, eqn (1) and (3), respectively), is described step by step in the Sections 2.1–2.3.

The parametrization was developed with known reactions of HO˙ radicals, a case with ample experimental data (ESI,† S1), and opting for a model with a minimal set of parameters that would allow for ‘cheap’ and reliable calculation of unknown rate constants. We chose to calculate the Gibbs energy of reaction, ΔG_r, as per reactive site, which ensures the same number of reacting sites in the LFER (i.e. one). The total reactivity of the molecule is then calculated as the sum of the reactivities of all individual R–H bonds.

Subsequently, the approach as outlined above was validated with data from reactions of HOO˙, H˙ and CH₃˙ (see Results section). In the validation process, the derived value for σ_LFER and the calculated values for Δ(ΔG_r) and q were used to predict k_r1/k_r2, which in turn was compared to experimental data.

2.1. Gibbs energy calculations: ΔG_r

Eqn (2) predicts that prominent influences on Δ(ΔG^‡) are the Gibbs energy of the reaction Δ(ΔG_r) and the scaling factor σ_LFER. Empirically, we used values for Δ(ΔG^‡) for one-electron oxidation reactions (e.g. blue and yellow in Fig. 1) to quantify the relevance of Δ(ΔG_r). Thereby, we also determined σ_LFER. As we calculated the values for Δ(ΔG^‡) per reaction site, they do not include the statistical factor (Section 2.3).

Perturbation theory leads to a formula to calculate ΔG_r based on ‘frontier orbital’ interaction and charge transfer.²⁵ In its simplified form (S3, ESI†), the reaction free energy change for oxidation reactions is described as:

ΔG_r = ΔG_ET,ox − ΔG_CT = ΔG_ET,ox − ΔG_PT (4)

with possible driving forces being oxidative electron transfer (ET,ox) and additional “charge transfer” (CT), e.g. proton-transfer (PT). The reactions considered do not imply a change in net charge. Therefore, eqn (4) reflects negligible changes in the Gibbs-energy of solvation.

Any parametrization routine starts with the parameters of large influence and subsequently proceeds those with smaller impact. The most prominent contributions to the activation energy, ΔG^‡, are correlated to ET and PT, i.e. their contributions are most influential. By definition, no reactions occur without electrons. Hence, we began with the calculation of the contribution of ET.

2.1.1. Transfer of the electron. The ΔG for ET relates to interactions between the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO) of the closed-shell molecule R⁻ and the singly occupied molecular orbital (SOMO) of the radical X˙, the so-called “frontier orbitals”.²⁶ Outer-sphere ET is frontier-controlled (energetically and spatially) and does not involve charge-transfer complexes.^27–29 We describe oxidative ET via the energy difference in the frontier orbitals, E_SOMO − E_HOMO as:

For computational details and associated uncertainties, we refer to the ESI† (S2). E_HOMO (pure electron transfer) reflects the lower limit of the rate constant (Fig. S3A, ESI† and Fig. 6A). The majority of radical redox reactions are several orders of magnitude faster than that. In other words, the assumption of pure electron transfer often underestimates how favorable the reaction is, i.e. it overestimates the Gibbs activation energy of reaction.

2.1.2. Transfer of the proton. Compared to outer-sphere ET, hydrogen abstraction (HA) involves additionally the transfer of a positive charge not the result of the frontier electron (eqn (5) and eqn (S1), ESI†). If a formal transfer of both an electron and a proton takes place, i.e. with proton-coupled electron transfer (PCET) or hydrogen atom transfer (HAT), energies of proton dissociation (pK_a-values) need to be included. The overall result of the reaction does not involve charge transfer or even charge separation.

The homolytic bond dissociation enthalpy relates to the pK_a of R–H and the electron affinity of the radical R˙.^30–32 This applies to X–H as well. The enthalpy change of a reaction between R–H and X˙ involves an analysis of the relative bond strengths (blue rectangles in Fig. 3). Given the proton dissociation constants pK_a(R–H) and pK_a(X–H) of the conjugate hydrogen donor X–H of the attacking radical X˙, respectively, eqn (6):

E_pK = 2.3RT [pK_a(R–H) − pK_a(X–H)] (6a)

Frontier orbital (E_HOMO, eqn (5)) and charge transfer parameters (pK_a) can, therefore, be combined into ΔG_r:If the pK_a's of R–H and X–H are equal, we return to eqn (5) (pure ET). We obtained pK_a values from the literature and, when unavailable, by regression with atomic charges (S2, ESI†). The thermodynamic cycle in Fig. 3 reflects graphically the calculation of ΔG_r values for HA:

Fig. 3 Thermochemical cycle for H-abstraction. The reaction driving force for both sequential (left) or concerted (center) mechanisms is described using proton transfer (pK_a, green arrows) and electron transfer (red arrow). Electron transfer in absence of protolysis (black). The colors of arrows correspond to the data in Fig. S3 (ESI†). Depending on the energies of proton/electron-transfer, an ionic or a non-ionic transition state is preferred, and the mechanism is sequential or concerted.

2.2. Quantum chemical corrections

In the Hammett/Taft equation (eqn (2)), σ_LFER is introduced to compare between congeneric chemicals, i.e. chemicals with ‘the same’ reaction mechanism, which we know is not the case for the reactions we include here: an electron transfer is not a hydrogen abstraction. In order to apply the LFER assumption, we need a correction for different mechanisms (purple and red in Fig. 2). Therefore, we chose to include the parameter Δ(ΔG^‡_TS) into eqn (3) with the assumption that Δ(ΔG^‡_TS) is distinctly smaller than σ_LFERΔ(ΔG_r).

2.2.1. Transition state stabilization: Δ(ΔG^‡_TS). In order to obtain ‘experimental’ values for Δ(ΔG^‡_TS) we compared our values for σ_LFERΔ(ΔG_r) obtained above with Δ(ΔG^‡), in eqn (7a):with E_pK = 2.3RT[pK_a(R–H) − pK_a(X–H)] (Section 2.2.2). Remember that Δ(ΔG^‡) relates back to Section 2.1, eqn (7b):where c_HOMO² and summation terms (eqn (8b)) characterize the statistical factor q (Section 2.3).

We calibrated Δ(ΔG^‡_TS) on ET and then introduced a correction for the involvement of protons, i.e. ‘the polarization of the bond’ in PCET and HAT. The so obtained range for Δ(ΔG^‡_TS) involving reactions with OH˙ was ∼13 kJ mol⁻¹.

2.2.2. Parametrization of Δ(ΔG^‡_TS). Contributions for the TS required refinement. For the calculus, we parametrized Δ(ΔG^‡_TS) using a computationally inexpensive method. The idea is to find a value that is proportional to the ‘experimental’ one and to derive the proportionality constant (here ‘z’, see below). Assuming Δ(ΔG^‡_TS) ≪ σ_LFERΔ(ΔG_r), its accuracy does not need to be as high as the latter, allowing for simplifications.

Charge polarization and solvent interactions in the TS depend on the mechanism (e.g. HAT, PCET), which depends on the type of H-donor involved. We distinguished types via atom-specific Fukui delocalizability indices.³³ For computational details and associated uncertainties, we refer to S2 (ESI†). The index, computed for the R–H electron pair, can denote whether the H-atom has neighboring π-electrons (π) or not (σ). π-Transition states associate to allylic hydrogens (sp² hybridized centers) and σ-transition states to non-allylic hydrogens (sp³ hybridized centers), Table T2 (ESI†).

To parametrize specific TS effects, we introduced a global scaling parameter z. We calculated the correction for the transition state stabilization with z_σ = 0 for π-transition states and z_π = 0 for σ-transition states, eqn (8):

Δ(ΔG^‡_TS,σ) = −z_σ (8a)

Δ(ΔG^‡_TS,π) = −z_πμ (8b)

In eqn (7), μ is the local dipole moment surrounding R–H and z was left for radical-specific empirical fitting. The slope of the lines in Fig. 4 is the scaling parameter z, denoting the strength of the proportionality between Δ(ΔG^‡_TS) and the local dipole moment within the molecule. Note that if either z_σ = μ = 0 or z_π = 0, Δ(ΔG^‡_TS) ≈ 0 (Fig. 4). In this case, ΔG^‡ is directly proportional to ΔG (which follows from eqn (2)). As the dipole moment is tied to solvation free energies, the proportionality characterizes charge polarization in the TS.

Fig. 4 Δ(ΔG^‡_TS) for HA by OH˙ (y-axis) versus the local dipole moment μ (x-axis). We calculated Δ(ΔG^‡_TS) viaeqn (8), by taking the difference between 2 values: with μ, and with μ = 0, i.e. Δ(ΔG^‡_TS) = Δ(ΔG^‡) – Δ(ΔG^‡(μ = 0)). Δ(ΔG^‡_TS) characterizes the offsets between the LFERs in Fig. S8E (ESI†). Diamonds and squares denote π- and σ-type hydrogens, respectively solid lines denote data fits. We used the fit to parametrize the contribution by μ to ΔG^‡. Molecules illustrate the combined impact of σ, π and μ on Δ(ΔG^‡_TS).

2.3. The statistical factor q

In the Eyring equation (eqn (1)) q is a statistical factor related to the reaction probability. Following eqn (8), we treated q as a product function:

q = q_Σ(π,σ)q_{CHOMO(π,σ)²} (9)

q prominently includes the number of abstraction sites Σ(π,σ), Fig. S9-2 (ESI†). For compounds with only equivalent H atoms, i.e. symmetric molecules, we can use simple corrections based on symmetry.¹⁹ We derived the expected values for k_r depending on σΔG_r and the reaction pathway (TS effects, in Section 2.2). We filtered the data used for parametrization based on symmetry (see Table T2, ESI†) and then express q_Σ(π,σ) for the resulting compounds as:wherein the denominator is the expected rate constant for a single R–H bond (π = σ = 1). We can see that the values for q_Σ(π,σ) vary by a factor 10, Fig. 5A. We thus used the number of ‘chemically equal’ hydrogen atoms Σ(π,σ) to account for this, Fig. 5A:

Fig. 5 (A) The statistical factor q (on the y-axis), calculated using experimental data and eqn (8), is the number of equivalent H atoms, wherein π and σ denote the presence of either resonance stabilized or non-resonance stabilized mechanisms (which do not have neighboring π-electrons), respectively. The molecule illustrates π and σ: there are 3 equivalent σ hydrogens; 1 π hydrogen. (B) The statistical factor q (on the y-axes), calculated using experimental data and eqn (8). q correlates with the squared atomic orbital coefficient. The dashed line denotes theoretical relationships for symmetric molecules.

q can also be understood as the accessibility of abstraction sites, i.e. it comprises also the ‘effectiveness’ of collisions, the so-called quantum-mechanical ‘transmission’. By extension, the overlap between frontier orbitals is characteristic of the space in which the electron can move within the TS.^26,34 Thus, we can express q_{CHOMO(π,σ)²} as function of squared AO coefficients, c_HOMO² (Fig. 5B):

wherein the numerator is the expected rate constant for the molecule reflecting similar-sized orbitals (c_{HOMO(π,σ)²} = 1). We then see that the values for q_{CHOMO(π,σ)²} vary by almost a factor 10, Fig. 5B.

We obtained atomic orbital (AO) coefficients (denoted as c_HOMO, eqn (S1), ESI†) from quantum-mechanical calculations, denoting the ‘size’ or ‘shape’ of orbitals (Fig. 5B). Computational details and associated uncertainties are provided in S2 (ESI†). We thereby used the data for congeneric H-donors, to take into account the dependence of q_{CHOMO(π,σ)²} on AO coefficients, Fig. 5B.

3. Results

3.1. Linear free energy relationships

Within the (crude) model of LFER only (eqn (2)), ΔG_r can predict ∼95% of the k_r values (N ≈ 500) within ±1 order of magnitude of the expected value (Fig. S3B, ESI,†p < 1 × 10⁻⁵), i.e. for these 95%, the estimation of the activation energy ΔG^‡ was accurate within 6 kJ mol⁻¹.

Using symmetry-filtered data (Section 2.1), we found σ_LFER = log(k_r)/ΔG_r = 2.3 eV⁻¹, i.e. the rate constant doubles if the Gibbs energy of the reaction decreases by 13 kJ mol⁻¹. We obtained σ_LFER values (±2SD) of 2.4(±0.2) and 2.1(±0.2) p(k_r)/eV for H˙ (Fig. 8) and 2.6 (±0.6) p(k_r)/eV for CH₃˙ (Fig. S12, ESI†). These values do no statistically differ from those for OH˙ or OOH˙/O₂˙^− 21 (within two SDs).

3.2. Transition state stabilization: Δ(ΔG^‡_TS)

For π-type systems, our calculated Δ(ΔG^‡) shows a systematic deviation from the known rate data for water that is dependent on the dipole moment μ. In other words, hydrogen abstraction from non-polar hydrogen donors shows lower Δ(ΔG^‡_TS) compared to HA from polar and zwitterionic π-type hydrogen donors, Fig. 5. Parametrization of this effect resulted in a satisfactory fit of the rate constants, Fig. 6C.

Fig. 6 Rate constants (k_r) for reaction with OH˙ (x), versus reaction energies (y). (A) k_rversus frontier orbital energies; (B) ΔG^‡versus ΔG_r; (C) ΔG^‡ − Δ(ΔG^‡_TS) versus ΔG_r. Only symmetric compounds (with equivalent R–H groups) are taken into account. Energies were converted using 100 kJ mol⁻¹ ≙ 1.04 eV and 0 kJ ≙ −8 eV ≈ E_SOMO(OH˙). Lines are data fits. Error bars indicate uncertainty from predicted pK_a (Chemaxon program³⁵). Full parametrization in Fig. S8 and Table T2 (ESI†).

As a validation, we used the parametrized result to predict k_r. for OOH˙, H˙ and CH₃˙. Aqueous-phase abstraction by H˙ of a π-type H atom from a zwitterionic compound is ≈200 times slower (≈2.3 in log units, Fig. 7) than for a σ-type H atom. The difference in terms of Δ(ΔG^‡) is ≈13 kJ mol⁻¹ (≈100 kJ mol⁻¹ in terms of Δ(ΔG_r)). The values match those for Δ(ΔG^‡_TS) for HA involving OH˙ (Fig. S8E, ESI†) and CH₃˙ (Fig. S12A, ESI†). However, propene (purple star in Fig. 7) did not fit our prediction (see Discussion, Section 4.3.1). S4 (ESI†) provides additional controls.

Fig. 7 The product of the rate constant (k_r) and the statistical factor q, logarithmically transformed, for reaction with H˙(y) versus ΔG_r(x). Dashed lines denote the LFERs. Red diamonds are π-type zwitterionic R–H and blue circles are σ-type R–H. Error bars indicate uncertainty introduced upon prediction of pK_a.³⁵ Only compounds with equivalent X–H groups are taken into account. The purple star is propene (outlier, discussed in Section 4.3.1). 100 kJ mol⁻¹ ≙ 1.04 eV; 0 kJ ≙ −5 eV, details in S2; Table T3 (ESI†).

3.3. The influence of radical electron affinity e_x

As explained above, we developed our method with reactions of OH˙ radicals, where ample data is available. The predictions were very satisfactory. However, when we extended our database with controls for HOO˙, H˙, and CH₃˙ (Tables T2–T5, ESI†), some minor systematic, radical-specific deviations from prediction became evident.

Eqn (8) contains 2 radical-specific properties: pK_a(X–H) and E_SOMO,X˙. The error, therefore, is introduced in one of the two values. As we used the experimentally determined value of pK_a(X–H) for HOO˙, H˙ and HO˙, we assume the error to relate to our calculation of E_SOMO,X˙ (S2, ESI†). We then decided to derive experimental values for E_SOMO,X˙ (via fitting) and obtained an ‘electron affinity’, e_x. If we compare the two values, Fig. 8, they are identical for the OH˙ radical: e_OH˙ = 8.0 (±0.5) ≈ E_SOMO,OH˙ = 8.0 (±0.2) eV. For best intercomparison between radicals, E_SOMO,X˙ has to be replaced with e_x in eqn (8):

Fig. 8 Comparison of the electron affinity, e_x, with E_SOMO,X˙. The dashed line denotes interaction between frontier orbitals allowing oxidative outer-sphere electron transfer. Data for O₂˙⁻ (both abstraction and addition reactions) and HOO˙s indicated by squared symbols were taken from ref. 21. p < 0.05 (2SD).

4. Discussion

With our method for calculus, we can estimate rate constants of one-electron oxidation processes with a statistical reproducibility of R² = 0.98 and with a confidence (±1 SD, i.e. 68%) of 30–40%. In other words, the activation energy is predicted with an accuracy of around 1 kJ mol⁻¹. We want to compare the result to errors from experimental data: rate data for fast kinetics with rate constants >10⁶ M⁻¹ s⁻¹ are often reported with precisions better than 10%. However, results of different labs routinely yield a distinctly higher scatter (20–40%). One may assume, therefore, that the accuracy of data often cannot keep up with their reported precision. Our calculations have accuracies comparable to the accuracy of preliminary experiments.

Our method uses a minimal set of parameters to predict kinetic rate constants, though similar (but substantially larger) sets recur in both machine learning and ab initio methods. Because of the minimization, parameters may embed more than 1 influence. However, for our examples and controls, the parametrization appears sufficient. Rather than quantifying explicit contributions to Δ(ΔG^‡) via ‘expensive’ ab initio methods, our parametrization is computationally ‘cheap’. Calculations were carried out on a standard desktop computer and for the ±40% prediction accuracy they took 5 seconds per molecule. Such a low-cost approach is very handy for selecting possibly relevant reactions in a complex reaction scheme. It enables, for example, large-scale screening of aquatic reactivity in environmental chemistry^36,37 that can single out possibly relevant reaction pathways. This is the application for which the current method was developed. Then, in a second step, more accurate and precise data can be acquired either by experimental work, more sophisticated LFER-based methods,^22,24,38,39 or even extensive ab initio calculation.

We initially assumed that the influence of ET and PT would be largest, whereas other influences were assumed to be distinctly smaller. Statistical evaluation of data supports this notion (Fig. S10, ESI†). Thus, our parametrization mainly relies on the assumption of a linear free-energy relationship between Gibbs energy of reaction and the Gibbs activation-energy of the same reaction. The former is given by the sum of the electron transfer and, if applicable, a proton transfer. We calculated the energies of electron and proton transfer via the energies of frontier orbitals and pK_a values, respectively. We combined the terms via a thermodynamic cycle and Klopman's equation.²⁵

The exact values for activation energies were then refined by estimation of relative (de-)stabilization of transition states by local delocalization and polarization. This process is reiterated over all possible reactive sites of a molecule. The measured rate constant then is equal to the sum of all rate constants for individual reactive sites. Our initial hypothesis of a closely similar rate determining step for the one-electron oxidations by HOO˙, H˙, CH₃˙ and HO˙ appears to be correct; our method is applicable for multiple radicals (Fig. 6–8 and Fig. S12, ESI†).

4.1. Statistical factor

With symmetric molecules, inclusion of the statistical factor q via Σ(σ) + Σ(π) (see Fig. 3A) led to better correlations between k_r and ΔG between and within H-donor types (Fig. S8C compared to Fig. S8B, ESI†). Similar findings were reported for unsaturated organics.^40,41 The inclusion of q via c_HOMO (Fig. S8D, ESI†) marginally improved the correlations (Fig. S10, ESI†), whereas it slightly converged the offsets of the regressions for π-type X–H compared to σ-type X–H bonds (Fig. S8D compared to Fig. S8C, ESI†). In contrast, the statistical factor q (as described via c_HOMO,R⁻, within a factor 5) did not improve the LFER for ET (N = 36, Fig. S2 and Table T1, ESI†). Note that we can also interpret the influence of c_HOMO in terms of ΔG_rvia Klopman's equation (eqn (S1), ESI†).²⁵

If q was included in the calculation model, the slopes of the LFERs (i.e. the substituent constants, σ_LFER) for OH˙ converged to ≈2.3 p(k_r)/eV (Fig. 6B and Fig. S2, S8D (ESI†)). The slopes do not differ between π and σ H-atom types. Thus, q improves the prediction accuracy of k_r. The overall reaction rate constant is a combination of the accessibility of each individual hydrogen atom that can be abstracted (i.e. q), and depends on the reaction dynamics.^42–44 Depending on the dynamics, symmetry arguments (Table T2, ESI†) may reduce the computational effort needed.

4.2. Driving forces ΔG_r

4.2.1. Electron transfer versus hydrogen abstraction. For electron transfer by OH˙ (eqn (2) and (3)), the σ_LFER (Hammett substituent constant) value, ±2SD, is 2.2 ± 0.2 p(k_r)/eV, similar to that of HOO˙/O₂˙⁻ (σ_LFER = 2.1–2.5 p(k_r)/eV) including ET.²¹ Using a thermodynamic cycle (eqn (6)), we found that slopes of the LFERs (σ_LFER) for OH˙ involving different types of H-donors all converged to ∼2.3p(k_r)/eV (0.024p(k_r)/(kJ mol⁻¹)). Thus, it appeared to us that the values for ET and HA do not significantly differ. For both, it implies doubling of the rate constant with every −13 kJ mol⁻¹ decrease of ΔG_r.

HA by OH˙ is usually faster than ET.^45,46 Hence, the observed pathway is usually HA (similar for addition, Fig. S14, ESI†). Our calculations give clear hints on the expected reaction mechanism. For example, the inclusion of ET only in the calculus may predict a very low rate constant as compared to the case of PCET/HAT. In that case, the contribution from proton-transfer is significant, and ET is not the prominent mechanism. In other words, the influence of pK_a on the ΔG_r values (eqn (6)) is non-negligible.

If experimental data are available, and if the calculated rate constant is significantly lower than the observed one, a misconception about the mechanism is to be expected. An obvious case is the reaction of the hydrogen atom with propene (outlier in Fig. 7). There, the observed reaction with H˙ is probably addition,⁴⁷ not an oxidation (Tables T2–T5, ESI†). Compared to OH˙ and OOH˙, which react as electron acceptors with most organic chemicals (Fig. S3B, ESI,†²¹), H˙ and CH₃˙ might not (eqn (4)). Provided a correct assumption of mechanism, the applicability of the method is retained and eqn (8) can predict k_r for different radicals (Fig. 6–8).

The influence of pK_a on the reaction rate (eqn (6)) can be interpreted as a Coulombic interaction between the hydrogen atom of R–H and X˙, e.g. the oxygen atom of OH˙. A larger value for Q_H (lower pK_a for R–H) implies greater electrostatic stabilization and higher k_r. Provided that the solvent is inert, a higher pK_a of R–H implies a proportionally lower contribution of proton transfer to ΔG_r. (eqn (6)). Within the model of LFER, HA is expected to be faster than ET if the reaction is more favorable thermodynamically, i.e. the products including H₂O (HA) are more stable than those including OH⁻ (ET).

4.2.2. Comparison with other methods. Machine learning and ab initio methods require separate models and simulations for individual chemicals. For example, charge-transfer interactions are described explicitly in ab initio methods, which consider charge transfer as a function of time and distance. Charge-based descriptors also recur in many QSPR models.^2,48 However, proton transfer can also be characterized by static (cheap) descriptors; the use of pK_a (Section 2.2.2) takes less computational effort.

LFERs for ET and HA exist for OH˙, OOH˙/O₂˙⁻, H˙ and CH₃˙, with similar σ_LFER values (Table T5, ESI†). Thus, the individual LFERs are not phenomenological,¹⁸ and the potential energy curves of the reactants can be described by a uniform method, e.g. a harmonic oscillator (depicted in Fig. 1) or Morse potential. In contrast, the relevance of E_HOMO and MoRSE potential^2,49,50 parameters and the QSPR model algorithm architecture differs between radical species. As MoRSE derives from molecular vibrational structure,^51,52 specification of potential curves^24,53 might facilitate extrapolation to a broader set of radicals.

The optima in k_r for OH˙, OOH˙, H˙ and CH₃˙ generally correspond to cases when |E_SOMO − E_HOMO| → 0 (Fig. 8); the reactivity sequence (Tables T2–T4, ESI†) is in line with previous studies.^54–57E_SOMO relates to electron accepting/donating ability, electrophilicity (S5, ESI†)⁵⁴ and ‘electron affinity’, e_x (Fig. 8 and Fig. S11, S12, ESI†).⁵⁴ We did not consider specific terms for orbital shapes of radicals (eqn (S1) and (S2), ESI†) in the current study. Machine learning methods use associated terms (e.g. radial density function descriptors^58,59). Hence, a direct description of orbital–orbital overlap and orbital symmetry factors may also characterize electron affinity and improve ΔG_r predictions.

4.3. Transition state stabilization: Δ(ΔG^‡_TS)

4.3.1. Proton-coupled electron transfer versus hydrogen atom transfer. The LFERs for HA were developed using a thermochemical cycle (Fig. 3) based on the assumption that the relative proton affinities of the reactants are not influenced by ET, and vice versa. In the case of a concerted mechanism, e.g. proton-coupled electron transfer (PCET),^60,61 ‘low’ ΔG^‡ might be involved. Though at elevated temperature, abstraction of a ring and benzylic hydrogen dominate over addition, indicating larger ΔG^‡ values.⁶² In such cases, sequential mechanisms might be involved. Indeed, Fig. 6B and 7 illustrate the involvement of Δ(ΔG^‡_TS) and the presence of mechanistically distinct pathways. In analogy, Pompe et al. predicted k_r based on differentiation between C–H bond types.^63,64

For H-donors with neighboring π-electrons (π in Fig. 5), μ_π positively correlates with ΔG^‡, eqn (8). This is because charge redistribution in the TS is involved. Abstraction of ‘π-type hydrogens’ normally occurs via proton transfer between the R–H and X˙ (σ-orbitals), but the electron transfers from the adjacent sp²-π orbital (only atomic orbitals of the same symmetry can combine⁶⁵). This often involves relatively high ΔG^‡. In analogy, captodative stabilization can influence ΔG^‡.

The rate is faster in H₂O if the reaction involves movement of only 1 electron, rather than 3 electrons, or when the proton and electron are transferred between the same sets of orbitals⁶⁶ (e.g. PCET reactions). This fits with the data for e.g. ammonia and 2-methylalanine (Table T2). These are polar and zwitterionic compounds, respectively, but OH˙ abstracts both the electron and proton via the σ R–H bond (‘σ-type’ H-atom) since there are no neighboring π-electrons. Hence their ‘high’ reactivity as compared to molecules containing polar and zwitterionic ‘π-type’ H atoms (Fig. S8E, ESI†). The principles also apply to reactions involving pairs of electrons: in water, 2-electron reactions involving singlet oxygen (¹O₂) are faster than 4-electron reactions.⁴⁹

4.3.2. Connecting mechanisms to thermodynamics. Reactivity depends on the relative probability and location of the electron transferred during the reaction, i.e. the reaction coordinate. For example, while the proton is abstracted from the σ-orbital, the σ-electron might delocalize into a sp²-π system. In this case, the delocalization might be interpreted as an electronic coupling⁶⁷ or a third (intermediate) vibrational state.¹⁸ Only before or after the delocalization, rearrangement of solvent/intermediates can occur (Franck–Condon approximation).

Theoretically, ΔG^‡_TS is composed of entropic (ΔS^‡_TS) and enthalpic (ΔH^‡_TS) terms. We found offsets of Δ(ΔG_r) ≈ 100 kJ mol⁻¹ (Δ(ΔG^‡_TS) ≈ 13 kJ mol⁻¹) between LFERs (Fig. S8E, S11A and S12A, ESI,† involving HA by HO˙, H˙ and CH₃˙, resp.) for non-polar (μ_π 0–1 D) and zwitterionic compounds (μ_π = 10–15 D). The difference between non-polar (μ_π = 0–1 D) and polar (μ_π = 5 ± 2 D) compounds is Δ(ΔG_r) ≈ 30 kJ mol⁻¹ (Δ(ΔG_TS) ≈ 4 kJ mol⁻¹, involving HA by HO˙). Solvent (H₂O) rearrangement enthalpies depend on the polarity of the molecule and its effective solvation, but values in the same range have been reported^68–70 (note also that 1 hydrogen bond in H₂O is ∼18 kJ mol⁻¹).

For similar mechanisms such as sequential HAT, ΔS^‡_TS may be a constant factor, in which case ΔG^‡_TS is only determined by ΔH^‡_TS.⁷¹ Thus, μ_π is likely to characterize ΔH^‡_TS. We infer that σ and π could characterize ΔS^‡_TS, e.g. by distinguishing between concerted and stepwise mechanisms. Analogously, we might discern between outer- and inner-sphere ET via Marcus Theory⁷² (S2, ESI†). We may further investigate the components of ΔG^‡_TS by looking at ‘exotic’ radicals with lower Δ(ΔG_r) (Fig. 1) e.g. ‘nucleophillic’ tert-butyl or HS˙. This may help extending eqn (8) to addition reactions.^73,74

4.4. Outlook

Empirical prediction methods often use many parameters and ambiguous algorithms, which does not facilitate interpretation and can be error-prone. Therefore, one should not accept beforehand that complex methods are needed. Alternatively, TS and frontier molecular orbital theory are closer to chemical intuition than a purely mathematical fitting process. Indicators, or proxies, for the free energy of activation ΔG^‡ are more easily retrievable than e.g. with ‘ab initio’ methods. The approach presented here reflects textbook knowledge of basic chemistry, implying that the parameters used have a physical basis. Despite the uncertainties involved, mathematical fitting yields surprisingly accurate results and may easily compete with machine learning methods with little computational effort. For the examples and controls shown here, it might even compete with computationally expensive quantum chemical (‘ab initio’) methods. The method presented here can effectively screen a large number of reactions, for multiple radicals.

5. Summary

In this study we used a thermodynamic/quantum-mechanical framework and kinetic data to parametrize the Eyring equation to predict the rate constants of one-electron oxidation reactions. Our method assumes a linear-free-energy relationship between activation and reaction energies. Via a thermodynamic cycle we combined frontier orbital interaction, |E_SOMO − E_HOMO| and charge (proton) transfer to adequately describe hydrogen abstraction reactions. On the basis of Hammett substituent constants, we inferred that all one-electron oxidation reactions entail similar rate limiting steps. Thus, offsets between different reaction pathways were approximately constant, which we attribute to specific transition state effects. These effects can be described by local polarity (dipole moment, μ) and atom type orbital delocalizability indices. Our method, parametrized to OH˙, successfully predicted rate constants for H˙ and CH₃˙ and it is within expectation that rate constants for OOH˙ and O₂˙⁻ can also be evaluated. Predicted and experimental values for k_r show appreciable correlations.

Conflicts of interest

The authors declare that they have no conflicts of interest.

Acknowledgements

Funding by the Swiss National Science Foundation (SNSF) is gratefully acknowledged (grant no. 175493). This work is part of the research programme TTW financing the Contaminants of Emerging Concern in the Water Cycle (CERCEC) project number 15759, which is financed by the Dutch Research Council (NWO). Discussions with Minju Lee (Glasgow University) and Jingwen Chen (Dalian University) were greatly appreciated.

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Footnote

† Electronic supplementary information (ESI) available. See DOI: 10.1039/d0cp03750h

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