Polar Effects in Hydrogen Atom Transfer Reactions from a Proton-Coupled Electron Transfer (PCET) Perspective: Abstractions from Toluenes

Abstract

Rate constants for hydrogen atom transfer (HAT) reactions of substituted toluenes with tert-butyl, tert-butoxy, and tert-butylperoxyl radicals are re-analyzed here using the free energies of related proton transfer (PT) and electron transfer (ET) reactions, calculated from an extensive set of compiled or estimated pK_a and E° values. The Eyring activation energies ΔG^‡_HAT do not correlate with the relatively constant ΔG°_HAT, but do correlate close-to-linearly with ΔG°_PT and ΔG°_ET. The slopes of correlations are similar for the three radicals except that the ^tBu• barriers shift in the opposite direction from the oxyl radical barriers—a clear example of the qualitative “polar effect” in HAT reactions. When cast quantitatively in free energy terms (ΔG^‡_HAT vs. ΔG°_PT/ET), this effect is very small, only 5–10% of the typical Bell-Evans-Polanyi (BEP) effect of changing ΔG°_HAT. This analysis also highlights connections between polar effects and the concepts of “asynchronous” or “imbalanced” HAT reactions, in which the PT and ET components of ΔG°_HAT contribute differently to the barrier. Finally, these observations are discussed in light of the traditional explanations of polar effects and the potential for a rubric that could predict the extent to which contra-thermodynamic selectivity may be achieved in HAT reactions.

Graphical Abstract

INTRODUCTION

Hydrogen atom transfer (HAT) has been one of the fundamental reactions of organic chemistry for over a century (eq 1). It is common in biological transformations, atmospheric chemistry, combustion, catalysis, and synthetic chemistry. In catalysis and synthesis, achieving the needed selectivity is often difficult. The rates of HAT reactions are primarily governed by the strengths of the bonds being made and broken, so reagents that make stronger bonds are more reactive, and weaker bonds are easier to break.

$$\text{X}-\text{H}+{\text{Y}}^{\cdot }\to {\text{X}}^{\cdot }+\text{H}-\text{Y}$$ (1)

“Polar effects” have emerged as a way to achieve selectivities that differ from the normal radical selectivity based on bond strengths.¹ Electrophilic radicals such as alkoxyl or fluoroalkyl radicals (RO^•, R_F^•) show a kinetic preference for electron-rich X–H bonds, while nucleophilic radicals such as simple alkyl radicals (R•) react faster with electron-poor X–H bonds.^2,3 For example, F₃C^• abstracts H from HCl ~70 times slower than H₃C^•, despite having an equilibrium constant ~30 times greater.³ A dramatic effect is also seen in HAT reactions between acylated amino acids and electrophilic radicals such as Cl^• or HO^•, which preferentially abstract from distal positions instead of from the more thermodynamically favorable α-carbon.^4–6 However, the effect is often more subtle, as in the abstractions from substituted toluenes analyzed here.

While polar effects are valuable, they remain largely qualitative and incompletely understood. They have been attributed to the electronegativities of the X and Y groups involved, and rationalized by dipole-dipole (electrostatic) stabilization of transition states.³ Frontier-orbital and valence bond (VB) models have provided important insights, emphasizing interaction of the X• SOMO orbital with the H–Y HOMO or LUMO.^7,8,9 Some of the semiempirical models developed to predict HAT barrier heights have related perspectives.^9–13

However, the dipole or frontier molecular orbital pictures are not easily applied to all HAT reactions. In the widely studied H-abstractions by compound I of cytochrome P450s, the H^• being removed is ‘separated’ into an H⁺ that binds to the Fe=O oxygen and an e⁻ that transfers to a porphyrin/thiolate radical cation (eq 2).¹⁴ Polar effects are more complex to rationalize in such reactions since the SOMO is distant from the H being transferred. More broadly, HAT reactions are a subset of concerted proton-coupled electron transfer (PCET) reactions in which the proton and electron transfer “together,” and this “together” may mean into the same orbital, onto the same atom, or onto the same protein.¹⁵ A different approach to polar effects is needed to encompass this continuum.

(2)

Analyses of HAT reactions have recently been expanded to include ‘imbalances’ in the ET or PT components of a concerted e^–/H⁺ (H^•) transfer.^16–24 Computational and experimental studies alike have found that the relationship between ΔG°_PT and ΔG°_ET can affect barrier heights in ways not captured by ΔG°_HAT. These effects have sometimes been attributed to a so-called “asynchronous” or “imbalanced” transition state, wherein the PT and ET reaction coordinates have proceeded to different extents.^16–24 Srnec in particular has developed theoretical models for the effects of asynchronicity and discussed the possible connection with polar effects.^17,25 Others have incorporated these effects into empirical models through statistical analysis and machine learning.^{9,16,21,26,27}

From one perspective, the proposal of asynchrony or imbalance in a PCET reaction harkens back to the classic physical-organic discussions of ‘asynchrony’ that were often illustrated as off-diagonal paths in More O’Ferrall-Jencks diagrams such as in Figure 1.^28,29 In his Principle of Nonperfect Synchronization (PNS), Bernasconi related this imbalance in PT reactions to changes in their intrinsic barriers.^30,31 The PNS has also been invoked in HAT reactions to account for an increased intrinsic barrier in abstractions from allylic and benzylic C–H bonds compared to saturated substrates.³² However, the electron and the proton should be treated quantum-mechanically, not as classical particles with well-defined positions. Current theory often describes HAT as a simultaneous e^–/H⁺ double-tunneling event,^33,34 as discussed below.

Figure 1.

Generic potential energy surface for an exoergic PCET from XH to Y• and its projection onto a schematic More O’Ferrall-Jencks diagram. The HAT reaction pathway is drawn over the saddle point between the barriers for initial PT (left) or ET (right).

The goal of this paper is to connect the extensive but qualitative literature of polar effects with the emerging discussions of imbalanced HAT based on the relevant reduction potentials (E°) and acidities (pK_as). Specifically, we re-analyze the reported rate constants for HAT from substituted toluenes to the nucleophilic tert-butyl radical and to the electrophilic tert-butoxyl and tert-butylperoxyl radicals, using E° and pK_a values that we have assembled. As shown below, this is almost an ideal system to explore non-BEP effects because the toluenes have very similar BDFEs but vary widely in their pK_as and E°s.

We examine these experimental data using simple linear free energy relationships (LFERs), exploring how the HAT rates and barriers depend on ΔG°_ET, ΔG°_PT, and ΔG°_HAT. Through the ΔG°_ET/ΔG°_PT lens, we draw an explicit, experimental, and quantitative connection between the classical polar effects and emerging discussions of imbalance in HAT reactions. Finally, we discuss the results within the context of (i) the traditional explanations of polar effects, (ii) the asynchronicity parameter η developed by Srnec and coworkers,¹⁷ and (iii) quantum-mechanical treatments of HAT reactions. We hope that the analysis presented here will move the field towards a rubric that could predict the extent to which contra-thermodynamic selectivity may be achieved in HAT reactions.

LINEAR FREE ENERGY RELATIONSHIPS, briefly

Relationships between rates and driving forces are a foundation of physical organic chemistry and have been applied to HAT reactions for close to a century.^35–42 These can be written in terms of rate constants and equilibrium constants in the Brønsted catalysis ‘law’ (eq 3) or as LFERs between ΔG^‡ and ΔG° (eq 4). The latter are often referred to as BEP relationships, for Bell-Evans-Polanyi or Brønsted-Evans-Polanyi. Equations (3) and (4) are conceptually the same and the same series of reactions will give the same value of α in either treatment.

$$\text{log}(k)=\alpha \text{log}\left(K_{eq}\right)+b$$ (3)

$$\Delta G^{‡}=\alpha \Delta G^{°}+c$$ (4)

The proportionality constant, the Brønsted α, captures the sensitivity of the barrier height to changes in driving force. α is almost always between 0 and 1 and is often close to ½ (Marcus theory pins α to ½ for isoergic reactions, and it is frequently taken as ½ in its electrochemical incarnation as the symmetry factor β in the Butler-Volmer equation). α is expected from the Hammond Postulate to be closer to 1 for highly endoergic reactions where the transition state is close to the products, and smaller for highly exergonic reactions. This has suggested a rough connection between α and the position of the transition state along the reaction coordinate.^43,44

Traditional applications of BEP relationships to HAT reactions have used enthalpies such as bond dissociation enthalpies (BDEs), correlating Arrhenius activation energies E_a with reaction ΔH°. However, free energies are at the core of the original Evans and Polanyi papers,^38–40 the Brønsted law, LFERs, and Marcus-type analyses, so BEP relationships should use free energies. For HAT reactions, bond dissociation free energies (BDFEs) are appropriate for the typical solution reactions of organic, inorganic, and biological chemistry (despite the dominance of BDEs in organic chemistry textbooks). BDFEs are discussed and tabulated in a recent review.⁴⁵ The free energy of an HAT reaction is then the difference between the BDFEs of the reactant and product.

RESULTS AND DISCUSSION

I. Compilation of data

A. Kinetic data

The many studies of HAT from substituted toluenes to tert-butyl,^46–49 tert-butoxy,^50–57 and tert-butylperoxyl^58–60 radicals differ in the solvent and temperature in which the reactions were performed, the techniques used, and whether the measured rates are relative or absolute. The kinetic data for the reactions of ^tBuO^• and ^tBuOO^• used here are from a 1994 compilation by Héberger that evaluated, normalized, and averaged reported rate constants for these reactions.⁶¹ The compiled rate constants were measured in various solvents, ranging in polarity from carbon tetrachloride to acetonitrile, and were normalized to 40 °C for ^tBuO^• and 20 °C for ^tBuOO^•. The relatively small differences in temperature should not affect the general conclusions. For the ^tBu^• HAT reactions, we have instead analyzed data from a single study.⁴⁸ This is because the trends discussed in this work hold for each individual study (see SI Section S2), but the reported rate constants shift from one report to another, introducing needless scatter (for our purposes). The kinetic data used here are listed in Table 1.

Table 1.

Kinetic and thermochemical data for HAT from substituted toluenes^a

		$\mathit{\text{log}}\left(\raisebox{1ex}{$\mathit{K}$}\!\left/ \!\raisebox{-1ex}{${\mathit{M}}^{-1}{\mathit{S}}^{-1}$}\right.\right)\mathit{\text{for}}\mathit{Z}{\mathit{C}}_6{\mathit{H}}_4\mathit{M}\mathit{e}+{\mathit{Y}}^{•}$ b			Thermochemical parameters g
	σ c	tBu• d	tBuO• e	tBuOO• f	BDFEh	E°(R•/−)i	E°(RH•+/0)i	pKa(RH)j	pKa(RH•+)j
p-OMe	−0.27		5.96	−1.08	81.8	−2.19	1.43	58.1	−2.7
p-t-Bu	−0.20	1.25			84.5	−2.09k	1.54	58.5	−2.6
p-Me	−0.17	1.15	5.73	−1.30	84.2	−2.02	1.63	57.1	−4.3
m,m-Me2	−0.14	1.17	5.59		84.6	−2.03k	1.61	57.5	−3.8
m-Me	−0.07	1.18	5.66	−1.46	84.7	−1.90	1.70	55.4	−5.2
p-OPh	−0.03		5.73	−1.18	85.4	−1.92k	1.68k	56.2	−4.3
p-Ph	−0.01		5.84			−1.90k	1.70k
H	0.00	1.16	5.51	−1.22	84.9	−1.83	1.87	54.4	−7.9
p-F	0.06	1.15			83.9	−1.90	1.76k	54.9	−6.7
m-OMe	0.12			−1.57	85.7	−1.76k	1.52	53.8	−1.4
p-Cl	0.23	1.45	5.49	−1.52	84.9	−1.80	1.86	53.9	−7.7
p-Br	0.23		5.36		86.0	−1.65k	1.83	52.2	−6.5
m-F	0.34	1.26			84.3	−1.54k	1.99k	49.0	−10.3
m-Cl	0.37	1.32	5.37	−1.73	85.1	−1.51k	2.01k	49.1	−10.1
p-CO2Me	0.45			−1.60	84.6l	−1.42k	1.99	47.3	−10.1
p-C(O)Me	0.50			−2.10		−1.11	2.24
m-CN	0.56		5.33	−1.96	83.9	−1.51	2.21	48.3	−14.3
m,p-Cl2	0.60	1.57			83.5	−1.27k	2.20k	44.0	−14.5
p-CN	0.66	1.67	5.18	−1.85	83.2	−1.17	2.24	42.1	−15.3
m-NO2	0.71		4.97	−1.98	84.8	−1.16k	2.30	43.0	−15.2
p-NO2	0.78		4.91	−1.84	84.0	−1.08k	2.33	41.2	−16.2
Range m	1.05	0.52	0.93	0.92	4.2	1.10	0.90	17.3	14.8

^aAn empty cell indicates a value not known.

^bBimolecular rate constants in M⁻¹ s⁻¹.

^cHammett σ_m and σ_p substituent constants taken from refs^62,63.

^dRate constants at 48 °C in neat substituted toluene from ref ⁴⁸.

^eRate constants at 40 °C in various systems and solvents, compiled in ref⁶¹.

^fRate constants at 20 °C in various systems and solvents, compiled in ref⁶¹.

^gFor the sources of the BDFEs, E_½s, and pK_as, see Section 2 of the text and Section S3 of the SI.

^hBDFEs are ± 2.0 kcal mol⁻¹.

ⁱPotentials are ± 0.10 V vs Fc+/0.

^jpK_as are ± 2.0 units.

^kPotentials extrapolated from Hammett correlation (see SI Section S3 for details).

^lBDFE given is that of p-CO₂Et-C₆H₄Me.

^mEach range given is that of the column above.

B. Thermochemical data

The BDFEs, reduction potentials, and pK_as needed for the analysis were obtained from multiple sources, as described here and in the Supporting Information (SI Section S3). The values are in Table 1.

The reversible potentials for the 1e^– reduction of about half of the substituted benzyl radicals (E°(R^•/–)) have been measured in acetonitrile (MeCN) using photomodulation voltammetry.⁶⁴ The values correlate well with the corresponding para- and meta-substituent Hammett constants, allowing estimation of the unmeasured E°(R^•/–) values (see Figure S5). The 1e⁻ oxidation potentials of the toluenes, E°(RH^•+/0), were calculated from the established linear correlation^65,66 with the measured gas-phase vertical ionization potentials obtained from NIST⁶⁷ or by extrapolation using the Hammett constants.

The bond dissociation free energies (BDFEs) were estimated from the measured gas-phase bond dissociation energies (BDEs)⁶⁸ using previously described thermochemical schemes involving an entropic correction to the BDE and the free energies to solvate RH, R^•, and H^•.⁴⁵ When multiple BDEs have been reported, the recommended or most recent values were used.

Finally, the pK_as of the toluenes and their radical cations, pK_a(RH) and pK_a(RH^•+), were calculated from the redox potentials and BDFEs using eq 5 (and its analogue with pK_a(RH^•+) and E°(RH^•+/0); all terms in kcal mol⁻¹).⁴⁵ Similar values and trends were obtained upon converting from gas-phase acidities.⁶⁹ These extrapolations are discussed in greater detail in Section S3 of the SI.

$$\mathit{\text{BDFE}}=1.37p{K}_a(XH)+23.06E^{°}\left(X^{•/-}\right)+C_G$$ (5)

The thermochemical parameters of interest were also estimated for the abstracting radicals in MeCN (Table 2). The solution-phase BDFEs were calculated from the gas-phase BDEs as described above. The E°(Y^•/–) values have been measured in MeCN for tert-butyl⁷⁰ and tert-butoxy radicals.⁷¹ The pK_a(YH) values of each were calculated from values in other solvents using established correlations (see SI Section S3 for details). Each E°(XH^•+/0) value was estimated from the gas-phase vertical ionization potentials as described above. The E°(^tBuOO^•/–), as well as the pK_a(YH^•+) of each abstractor, were determined from the above thermochemical parameters using eq 5. It is worth noting that even generous error bars do not detract from the trends discussed below.

Table 2.

BDFE, E°, and pK_a values for ^tBu^•,^tBuO^•, and tBuOO^{•/– a}

	BDFE	E°(Y•/-)	E°(YH•+/0)	pKa(YH)	pKa(YH•+)
tBu•	90.9	−2.94	3.37	78	−29
tBuO•	101.5	−0.70	2.81	45	−12
tBuOO•	79.4	−0.87	2.80	34	−28

^aFor the sources of the BDFEs, E_½s, and pK_as, see Section 2 of the text and Section S3 of the SI. BDFEs are ± 2.0 kcal mol⁻¹, potentials are ± 0.10 V vs Fc^+/0, and pK_as are ± 2 units.

II. Correlations kinetic and thermodynamic data

A. Comparing k_HAT for ZC₆H₄Me + ^tBu^•, ^tBuO^•, and ^tBuOO^•

The rate of HAT from the toluenes to ^tBu^• is about 4 orders of magnitude slower than to ^tBuO^• and 2 orders of magnitude faster than to ^tBuOO^•. This is the result of two primary effects. First, there are large differences in driving forces due to the different strengths of the Y–H bonds formed: the ^tBuO–H BDFE is ~11 kcal mol⁻¹ higher than the Me₃C–H bond, which is another ~11 kcal mol⁻¹ stronger than the ^tBuOO–H bond. However, the HAT rates of ^tBu^• do not sit directly in between those of ^tBuO^• and ^tBuOO^• but nearly 2 orders of magnitude slower. This can be explained by a higher intrinsic barrier for the HAT reactions of C-centered radicals as compared to those of O-centered radicals.^72,73 We note that the BEP driving force effect and the intrinsic slowness of C-centered radicals are large effects, orders of magnitude (~10⁶ and ~10², respectively). These contrast with the smaller magnitude of the polar effects discussed below.

B. Correlating HAT rate constants with the toluene BDFEs, pK_as, and E°s

With consistent sets of HAT rate constants and thermochemical values in hand, we have analyzed rate/driving force LFERs for changes in the toluene substituents. Rate constants were converted to free energy barriers (ΔG^‡_HAT) using the Eyring equation. The ΔG°_HAT for each reaction is the difference in the BDFEs of the R–H and Y–H bonds involved (Scheme 1, eq 6).

Scheme 1.

Reactions and energies for initial HAT, PT, and ET

The plot of ΔG^‡_HAT vs. ΔG°_HAT shows essentially no correlation within the dataset for each radical (Figure 2A). This is likely due to the very small range of toluene BDFEs, only 4.2 kcal mol⁻¹, which is on the order of the uncertainties themselves (±2 kcal mol⁻¹). One way to view the constancy of the BDFEs is that the substituents cause offsetting changes in the pK_a and E°. For example, para-nitrotoluene is 13.2 orders of magnitude more acidic than toluene, but the nitrobenzyl anion is 12.5 orders of magnitude (750 mV) less reducing. These shifts mostly cancel, leaving the BDFE of nitrobenzene only 0.9 kcal mol⁻¹ weaker than that of toluene.

Figure 2.

Correlation of the HAT barrier height relative to the parent toluene with (A) the overall driving force, (B) the driving force for initial PT, and (C) the driving force for initial ET. Black points are the reactions of tert-butyl radical; blue, tert-butoxy; and red, tert-butyl peroxyl, and m indicates the slope of each linear fit. Note that in B and C the horizontal axes for ΔG° values are 40 times larger than the vertical axis for ΔΔG^‡_HAT. R² values of fits: (B) 0.70 (^tBu•), 0.88 (^tBuO•), 0.77 (^tBuOO•); (C) 0.65 (^tBu•), 0.90 (^tBuO•), 0.73 (^tBuOO•). See Figure S7 for plots with data points labeled with their corresponding toluenes.

The similarity of the toluene BDFEs and insensitivity of the ΔG^‡_HAT values to these BDFEs is key to this study. It means that the comparison made here highlights the factors other than the BDFEs that affect the HAT barriers (i.e., non-BEP effects).

Plots of the HAT barriers ΔG^‡_HAT vs. the free energies for initial PT or ET—ΔG°_PT and ΔG°_ET—are shown in Figures 2B and 2C. The reactions do not proceed through initial PT or ET (which are +50 to +120 kcal mol⁻¹ uphill); they are all concerted HAT. Nevertheless, the ΔG°_PT and ΔG°_ET are valuable because they are the energies of the opposite corners of the More O’Ferrall-Jencks diagram (the green hills in Figure 1). In contrast to the roughly constant ZC₆H₄CH₂–H BDFEs, the pK_as and E°s vary substantially, by 1.1 V in E° and by 17 units in pK_a (both are a range of ~25 kcal mol⁻¹ in ΔG°_PT and ΔG°_ET).

Two immediate observations should be made from the plots of ΔΔG^‡ vs. ΔG°_PT and ΔG°_ET in Figures 2B and 2C. First, the ΔG^‡ are not very sensitive to ΔG°_PT or ΔG°_ET (or (ΔG°_PT – ΔG°_ET) in a Figure below). The x-axes in these plots cover a 40-fold wider range than the y-axes (15-fold in Figure 2A). Second, the ΔG^‡ nonetheless correlate linearly with ΔG°_PT or ΔG°_ET. Moreover, the correlations of ^tBuO^• and ^tBuOO^• are opposite in sign of the ^tBu^• correlations. These observations and their implications will be discussed in the following sections.

The scatter in the data is in part due to the assumptions made to obtain the pK_as and E°s. The rate data being from a host of different systems, temperatures, solvents, etc. likely also contributes to the scatter (see SI Section S2). Still, without the removal of any outliers, it is striking that these trends hold quite well. Plots of ΔΔG^‡ vs. pK_a, E_½ and Hammett parameters are similar to the ΔG°_PT and ΔG°_ET plots shown here (SI Section S1). The shallow and opposite slopes of these lines are discussed in the next section.

C. Variation of HAT rate constants with substituents: polar effects and ΔG°_PT / ΔG°_ET correlations.

As emphasized in the original reports, HAT to the electrophilic ^tBuO^• and ^tBuOO^• radicals are faster with more electron-rich toluenes, while HAT to the nucleophilic ^tBu^• radical is faster when electron withdrawing groups are present. For example, ^tBu^• reacts ~3 times faster with the electron-poor para-cyanotoluene than with unsubstituted toluene, while ^tBuO^• and ^tBuOO^• react ~3 times slower. This is a clear polar effect. Since the range of k_HAT for each radical approaches a factor of ten, changing the character of the radical can significantly affect abstraction selectivity.

Here we analyze the same rate data from a ΔG°_ET/ΔG°_PT perspective. The nucleophilic ^tBu^• reacts faster with substrates that are better acids and poorer electron donors (e.g., para-cyano-toluene). The electrophilic oxyl radicals are the opposite, preferring substrates with a lower E° (more easily oxidized) and higher pK_a (less acidic). Thus, qualitatively, there are good analogies between the traditional polar effects and the ΔG°_ET/ΔG°_PT perspective.

The ΔG°_ET/ΔG°_PT analysis is more quantitative than the traditional analyses of polar effects because the E° and pK_a are (in principle) measurable thermochemical parameters. The slopes of the fit lines in Figure 2 are all unitless (kcal mol⁻¹/kcal mol⁻¹) and can therefore be compared. As noted above, the Brønsted plot (Figure 2A, ΔΔG^‡_HAT vs. ΔG°_HAT) shows that the variation in the barrier is not due to differences in BDFEs but to differences in ΔG°_ET and/or ΔG°_PT. While the other plots are also ΔΔG^‡_HAT vs. ΔG°, they are not true Brønsted plots or BEP correlations because the free energy changes (ΔG°_PT, ΔG°_ET) are not for the full HAT reaction.

As noted above, the first lesson from these plots is that the magnitude of the polar effects is very small from an LFER perspective. The rate constants vary by about one order of magnitude for all three radicals, even though the K_as vary by 17 orders of magnitude (and similarly, E°s by 1.1 V). The unitless slopes in Figure 2B–C are all between 0.03 and 0.06. Compared to a typical BEP relationship with α ≈ ½, the barriers respond far less sensitively to changes in pKa and E°, about an order of magnitude less sensitively.

This conclusion is substantiated by the global electrophilicity index (ω), which quantifies the nucleo- or electrophilicity of a radical based on its electron affinity and ionization potential.^74,75 The ω values computed for the benzyl radicals formed in the present reactions are all very similar, ranging from “moderately nucleophilic” to “weakly nucleophilic.” Furthermore, local electrophilicity indices, which describe the electrophilicity condensed at the radical center, describe the benzyl radicals as even more similar to one another (see Figure S9),⁷⁵ despite their large differences in pK_a and E°. This parallels the ΔG°_ET/ΔG°_PT analysis, that large swings in pK_a and E° only induce weak to moderate polar effects.

A closer look at the slopes of the ΔΔG^‡_HAT vs. ΔG°_PT or ΔG°_ET plots in Figure 2 reveals several deeper implications. First, the slopes of the ^tBuOO^• and ^tBuO^• data are remarkably similar. This is most evident in Figure 2C, where the blue and red lines almost overlay. This is surprising because the ^tBuO^• reactions are highly exoergic (–17 kcal mol⁻¹ on average) while the ^tBuOO^• reactions are endoergic (+5 kcal mol⁻¹). Still, their barriers exhibit the exact same sensitivity to changes in reactant pK_a/E°. The Hammond postulate would suggest that the uphill ^tBuOO^• reactions have much “later” transition states and should therefore be more sensitive to changes in substituents. Viewed another way, this is a violation of the reactivity-selectivity principle: ^tBuO^•, which reacts 10⁷ times faster, shows virtually the same selectivity pattern as ^tBuOO^•. Thus, the polar effect on the barrier height is insensitive to the traditional measure of the position of the transition state along the reaction coordinate.

The slopes are also roughly the same between the oxyl radical reactions and those of ^tBu^•, except opposite in sign. This is mirrored in the Hammett ρ values for the reactions as well (Figure S2). The similarity is rough due to the large uncertainty for the ^tBu^• slope. Still, the magnitude of the polar effect here appears independent of whether the incipient bond is C–H or O–H. This is surprising given the marked differences between carbon- and oxygen-centered radicals in both their intrinsic HAT rate and ability to hydrogenbond.^72,73 This similarity between carbon and oxygen radicals has not been previously noted, to our knowledge, and we hope that its origin can be examined in future experimental and computational studies.

A final observation is that the constancy of ΔG°_HAT in these reactions necessitates that the substituent effects are the same in the forward and reverse directions; k_f and k_r must vary by the same factor upon changing Z because their ratio does not change (Figure 3, eqs 9, 10). This, too, is a violation of the reactivity-selectivity principle. For instance, ^tBuO^• + RH reactions are very downhill and the reverse ^tBuOH + R^• are very uphill which would normally imply very different selectivities. Viewed another way, the changes in reactivity of the ZC₆H₄CH₃ substrate are identical to the changes in reactivity of the ZC₆H₄CH₂^• radical. Again, this perhaps counterintuitive result should be general to any system where substituents change the barrier but not the driving force.^76,77

Figure 3.

Simple free energy surface for the radical reactions in equation 9, illustrating that variation of the aryl substituent Z changes the barrier height but not the driving force. Eq 10 shows the implication of the essentially constant ΔG°_HAT across the series.

In summary, the plots in Figure 2 reveal that the HAT rates correlate well with ΔG°_ET and ΔG°_PT, and that this is not attributable to changes in the overall ΔG°_HAT. Moreover, ^tBu^• and oxyl radicals respond oppositely to these changes in ΔG°_ET/ΔG°_PT, the former preferring acidic toluenes and the latter preferring electron-rich toluenes. When compared through unitless slopes, these polar effects are miniscule in comparison to BEP effects. Interestingly, the two oxyl radicals respond quantitatively the same to changes in ΔG°_ET/ΔG°_PT, despite the reactions of ^tBuOO^• being endoergic and those of ^tBuO^• being highly exoergic—an apparent violation of the reactivity-selectivity principle. Finally, the magnitude of the polar effect appears to be the same between the oxyl and ^tBu^• radicals, which is surprising given the differences between the intrinsic HAT reactivity of C–H vs. O–H bonds.

D. Other models: “Imbalance,” the Srnec “asynchronicity” parameter, and quantum models.

The historical picture of polar effects in HAT reactions starts from transition state theory, with a classical hydrogen atom crossing over a transition state. From valence bond (VB) theory, the transition state can be stabilized by mixing with excited state configurations.^78,79 For HAT reactions, a “polar” transition state is one that is stabilized by an excited state charge transfer configuration, namely, by X^– + HY^•+ or by XH⁺ + Y^–. In the More O’Ferrall-Jencks diagram in Figure 1, the off-diagonal corners represent these excited state configurations. This is analogous to considering charged resonance structures of the transition state (namely, [X⁻ ··H··Y⁺]^‡ or [X⁺··H··Y^–]^‡). Conceptually, these approaches all can give HAT transition states in which the PT component is more (or less) developed than the ET component; [X^∂– ···H··Y^∂+]^‡ or [X^∂+··H···Y^∂–]^‡). This is the backdrop for the following sub-sections on “asynchrony” or “imbalance” in HAT reactions and quantitative models which describe it. We finish by discussing these classical pictures in view of a quantum-mechanical treatment of HAT reactions.

1). Imbalance.

As described above, a number of recent studies have discussed the possibility of an “imbalanced” or “asynchronous” transition state, wherein PT and ET are unevenly developed.^16–24 This would correspond to a classical off-diagonal path in a More O’Ferrall-Jencks diagram (Figure 1). Indicators of such a pathway are (i) a difference between Brønsted α values for changes that predominantly affect the pK_a or the E° of a reactant,¹⁸ or (ii) rates that correlate better with the pK_a or E° of the changing reactant than with its BDFE.^16,19,24,80

The present reactions, viewed through a ΔG°_ET/ΔG°_PT lens, highlight links between this imbalance and polar effects. ^tBu^• rates increase as PT becomes more favorable but decrease as ET becomes more favorable. This demonstrates the greater sensitivity of ^tBu^• barriers to changes along the PT coordinate. The opposite is true of ^tBuO^• and ^tBuOO^•, whose barriers show a greater sensitivity to the ET coordinate. In other words, the HAT reactivities of all three radicals suggest an imbalanced transition state: the nucleophilic ^tBu^• in favor of PT and the electrophilic oxyl radicals in favor of ET.

2). The Srnec “asynchronicity” parameter η.

Srnec and coworkers have developed a mathematical model to predict the effects of thermodynamic “asynchronicity” on the barrier height for an HAT reaction, using the asynchronicity parameter η.^17,25 η is essentially ΔG°_PT – ΔG°_ET (eq 11; F is Faraday’s constant), so we will use η and (ΔG°_PT – ΔG°_ET) interchangeably in this discussion. (ΔG°_PT – ΔG°_ET) is the difference in energy between the tops of the green hills in Figure 1.

For an electrophilic radical and/or hydridic X–H bond, η > 0 (ΔG°_PT > ΔG°_ET), and the opposite is true for a nucleophilic radical and/or acidic X–H bond. Interestingly, Srnec predicted and computationally showed that barriers will be lowest when |η| or |ΔG°_PT – ΔG°_ET| is largest. In other words, the rates will be fastest when the polarity of the H atom donor and acceptor are most different. Their quantitative relationship is in eq 12.^17,25

$$\eta F=\frac{1}{\sqrt{2}}\left(\Delta {G^{°}}_{PT}-\Delta {G^{°}}_{ET}\right)$$ (11)

⁸¹

$$\Delta G^{‡}\approx \frac{\lambda }{4}+\frac{\Delta G^{°}}{2}=\frac{{\lambda }_0}{4}-\frac{F\left|\eta \right|}{4}+\frac{\Delta G^{°}}{2}$$ (12)

⁸²

The dataset of rate constants and thermochemical values developed here provide a detailed experimental test of eq 12 across a series of reactions. For each of the three radicals in this study, the HAT rate constants correlate linearly with |η| (Figure 4A). For the electron-rich tert-butyl radical, |η| is largest and rates are fastest with electron-poor toluenes. For the electrophilic oxygen-centered radicals, |η| is largest and rates are fastest with electron-rich toluenes. This is perhaps illustrated most simply in the plot of ΔG^‡_HAT versus ΔG°_PT – ΔG°_ET (Figure 4A): the highest barrier is when ΔG°_PT and ΔG°_ET are most similar (smallest η). For the ^tBu^• reactions, the data tentatively suggest a peak or plateau of the barriers around η = 0. These results agree with the predictions of the Srnec model. The Srnec model has also been shown by Barman et al. and by Schneider et al., to provide improved correlations with experimental data for C–H bond activation by metal-oxo complexes.^16,21

Figure 4.

(A) Correlation of the relative HAT barriers with ΔG°_PT – ΔG°_ET (bottom axis), or equivalently, the Srnec η (top axis). Slopes are given for ΔΔG^‡ vs. ΔG°_PT – ΔG°_ET. (B) The relative HAT barriers predicted from eq 12. The dashed line indicates perfect agreement between the predicted and experimental relative barriers. Black points are the reactions of tert-butyl radical; blue, tert-butoxy; and red, tert-butyl peroxyl. R² values: (A) 0.68 (^tBu^•), 0.91 (^tBuO^•), 0.78 (^tBuOO^•); (B) 0.34 (^tBu^•), 0.94 (^tBuO^•), 0.79 (^tBuOO^•). See Figure S8 for plots with data points labeled with their corresponding toluenes.

Despite the good correlation of the barrier height with |η| or |ΔG°_PT – ΔG°_ET|, the magnitude of the experimental effect is much smaller than that predicted by eq 12. The range of η for each of the oxygen-centered radical reactions is 1.34 V (31 kcal mol⁻¹). According to eq 12, this should give a span of almost 8 kcal mol⁻¹ in ΔG^‡ and a 10⁶ span in rate constant (Figures 4B and S6). However, within each radical series, the rate constants are all within a single order of magnitude. Therefore, η or (ΔG°_PT – ΔG°_ET) are informative predictors of polar effects in the present system, but their effects are small. Reference 21 similarly concluded that adding η to their analysis provided only a small improvement. The conclusion here of a small effect of η for the toluene reactions parallels the small influence of changes in pK_a and E° discussed above.

The radical HAT reactions that exhibit more pronounced kinetic polar effects seem to have much larger thermodynamic “asynchronicities,” i.e., larger |ΔG°_PT – ΔG°_ET|). For instance, the classic contra-thermodynamic faster HAT from HCl to CH₃^• vs. CF₃^• (an effect of 10^2–3 in rate constant) can be traced to the ~10²⁷ lower K_a of methane (in DMF^83,84). A similar effect has been reported for the (pseudo-) self-exchange reactions of CH₃/CF₃^• + CH₄.³ In each case, the effect of |ΔG°_PT − ΔG°_ET| on the barrier seems to be 5–10% that of changing the overall HAT driving force.

The relationship between |ΔG°_PT – ΔG°_ET| and the height and position of the barrier is illustrated in Figure 5. Such “3D” More O’Ferrall-Jencks diagrams have been used in non-PCET contexts for some time.^{28,29,85–88} Figure 5A depicts the pathway for an HAT reaction in which ET and PT are equally unfavorable and the corresponding “synchronous” transition state. Figure 5B then shows the shift of the saddle point when ΔG°_PT >> ΔG°_ET: lower and closer to the ET corner of the More O’Ferrall-Jencks diagram. Taken together with the observation that the present reactions violate the reactivity-selectivity principle, we might then say that polar effects are insensitive to the position of the transition state along the HAT reaction coordinate, but sensitive to its position along the perpendicular coordinate (the off-diagonal). However, we emphasize that the “ET” and “PT” reaction coordinates in Figure 5 are not the positions of the quantum mechanical e^– and H⁺, as discussed in section II.C.4 below.

Figure 5.

Pictorial schematic of the relationship between the energies of the off-diagonal species (ΔG°_ET and ΔG°_PT) and the position and height of the HAT transition state, when ΔG°_PT = ΔG°_ET (A) and when ΔG°_PT >> ΔG°_ET (B). Red arrows highlight the changes from (A) to (B), and gray curves in (B) represent the corresponding black curves in (A).

3). Comparison of ΔG°_PT/ΔG°_ET with Hammett treatments.

Polar effects have historically been probed and understood through Hammett correlations (eq 13), as discussed above. This treatment relates changes in rate constants to standard substituent constants σ typically determined from quite different reactions. The use of standard constants allows comparisons of Hammett ρ values across a wide range of transformations. For the toluene HAT reactions discussed here, the ρ values are −0.8 for ^tBuO^• and ^tBuOO^• and +0.5 for ^tBu^•. The negative ρ values for the oxyl radical reactions are interpreted as buildup of positive charge at the HAT transition state, in contrast to the negative charge buildup for the ^tBu^• radical reactions (ρ > 0). These values are one of the traditional indications of a significant polar effect, that the oxyl radicals are electrophilic while ^tBu^• is nucleophilic.

$$\text{log}\left(\frac{k}{k_0}\right)=\sigma \rho$$ (13)

As emphasized by a reviewer, the ρ values for the ^tBuO^•, ^tBuOO^•, and ^tBu^• HAT reactions have large magnitudes for reactions that do not have charged intermediates. This leads to the intuition that these are quite polar reactions given that only a neutral atom is being transferred. In contrast, the BEP plots above indicate that the barriers are not very sensitive to the ΔG°_PT and ΔG°_ET. The origin of this difference in intuitions is the extremely large ρ values for ΔG°_PT and ΔG°_ET, for instance, ρ = 16.5 for the change in pK_a (a 10^17.3 change in K_a over for a change in σ of 1.05 (Table 1)).

In our view, the ΔG^‡/ΔG° correlations (BEP or Brønsted) are valuable because they compare two things that relate to the set of reactions in question, the ΔG° between minima on the free energy surface and the transition state energies. In other words, the ΔG^‡/ΔG° correlations describe relationships between features on the free energy surface of a single reaction. The Hammett analysis here is complementary; while it fundamentally compares k_HAT with benzoic acid pK_a values, it contextualizes the HAT reaction in the canon of organic reactions.

4). Quantum mechanical treatments of PCET.

The discussion above has been purely classical, but current PCET theories^33,34 treat the proton and the electron as quantum particles. The PCET step is described in a Marcus-like model, where the [classical] heavy atoms and solvent rearrange to allow simultaneous electron and proton tunneling to move the system from the reactant to the product vibronic free energy surface. In this model, the surface hopping must involve synchronous ET and PT (though recent results suggest a 24 fs lag in one system⁸⁹).

The quantum nature of the proton and electron make the use of More O’Ferrall-Jencks diagrams conceptually complicated. The off-diagonal axes cannot represent the physical motion of the particles. In Marcus theory, the barrier to ET is in large part the reorganization of the solvent to enable the hopping of the electron. Such charge separation and solvent involvement does not appear to be involved in HAT, at least from hydrocarbons. Ingold et al. showed that rate constants for H-abstraction from cyclohexane by cumyloxyl radical were completely independent of solvent from cyclohexane to acetic acid (±10%), a large range of static dielectric constants.^90,91 More generally, there is typically a close correspondence of hydrocarbon radical chemistry in gas-phase and in solution. In those cases, invoking asynchrony that involves substantial buildup of charge seems unlikely—although polar effects have been explained by invoking dipolar interactions at the transition state.³

Even though the e^– and H⁺ do not have classical positions, versions of More O’Ferrall-Jencks diagrams for PCET can be envisioned where the off-diagonal corners represent collective motions of heavy atoms that facilitate PT or ET. In the Marcus theory treatment of electron transfer, the reaction coordinate is usually referred to as a solvent coordinate, describing the collective motion of solvent molecules to allow the movement of charge.⁹² The Kiefer-Hynes theory of proton transfer similarly emphasizes the solvent reorganization due to charge movement.⁹³ HAT reactions, however, likely have less charge transfer because formally a neutral atom is being transferred. Therefore, for HAT the important heavy atom motions probably emphasize inner-sphere reorganizations within the donor and acceptor molecules.⁹² In the reactions examined here, this could be the rehybridization of the benzylic carbon or collective changes in the C–C bond lengths with changes in unpaired spin density. If such collective nuclear motions responded differently to changes in PT vs. ET thermochemistry, this could be a source of “imbalance” despite the simultaneous double tunneling of the e^– and H⁺.

CONCLUSIONS AND IMPLICATIONS

H-atom transfer (HAT) rate constants from substituted toluenes to tert-butyl, tert-butoxy, and tert-butylperoxyl radicals have long been known to show polar effects. The electrophilic oxyl radicals abstract H• faster from electron-rich toluenes, while the tert-butyl radical reacts faster with electron-poor toluenes. This report analyzes the relative rate constants in terms of the free energies of the highly unfavorable initial electron- or proton-transfer steps from the same reactants (ΔG°_PT, ΔG°_ET). While the primary determinant of such rate constants are the relevant bond dissociation free energies (BDFEs) and free energies of reaction (ΔG°_HAT), these do not vary significantly for substituted toluenes. Therefore, this system provides a good test of other effects on the HAT barriers.

The most striking observation is that changes in PT or ET energetics of ~17 kcal mol⁻¹ shift ΔG^‡_HAT by little more than 1 kcal mol⁻¹. Polar effects here are thus an order of magnitude smaller than the typical BEP effect of changing the overall ΔG°_HAT of HAT reactions. Strongly polar radicals are needed to achieve mildly contra-thermodynamic selectivity.

Nevertheless, the variations of ΔG^‡_HAT in each series correlate linearly with the changes in ΔG°_PT or ΔG°_ET (at essentially constant ΔG°_HAT). This connects polar effects and ‘imbalanced’ or ‘asynchronous’ HAT reactions. Moreover, the reaction barriers within each set of reactions correlate inversely with |ΔG°_PT – ΔG°_ET|, with the largest barriers for the smallest ‘asynchrony.’ This is as predicted by Srnec and coworkers,¹⁷ though the magnitude of the effect is significantly smaller than their first-order analysis.

For all three radicals, the plots of ΔG^‡_HAT vs. ΔG°_PT or ΔG°_ET have the roughly the same magnitude of slope. The same behavior is found for the highly exoergic reactions of ^tBuO^• and the endoergic reactions of ^tBuOO^•. This is a violation of the reactivity-selectivity principle: the traditional metric of the position of the transition state has no bearing on polar effects in these reactions. These may be general properties of reaction sets where the barrier varies but the driving force does not.

These results suggest that polar effects in an HAT reaction may be predicted from the relative thermodynamics of single ET and PT steps, though the origin(s) of these correlations are not fully resolved. The pK_a and E° parameters are complimentary to traditional predictors such as Hammett parameters and electrophilicity indices, with the advantage that these thermodynamic parameters have an independent meaning. Furthermore, the ΔG°_ET/ΔG°_PT lens brings VB theory, multidimensional Marcus theory, and Bernasconi’s PNS to bear on polar effects in a concise and tractable way. Further studies are needed to explore these effects across reaction sets where the ΔG°_HAT or ΔG°_PCET are not constant.

Polar effects are one of a number of smaller effects that modulate HAT barriers—small compared to changing the ΔG°_HAT. Radical reactivity can also be modulated with steric differences, with the differing reactivities of C–H vs. O/N–H, for allylic vs. saturated bonds,³² and by introduction of hydrogen-bonding additives.⁹⁴ Though these are small effects, they can have a significant bearing on reaction selectivity. After all, the difference between 70:30 vs. 30:70 selectivity is just 1 kcal mol⁻¹ in barrier height.

Data Availability Statement

The data underlying this study are available in the published article and its Supporting Information.