Benchmark ab initio characterization of the abstraction and substitution pathways of the OH + CH₄/C₂H₆ reactions

Abstract

We report a comprehensive ab initio investigation of the OH + CH₄/C₂H₆ reactions using a high-level composite approach based on CCSD(T)-F12b/aug-cc-pVTZ geometries and CCSD(T)-F12b/aug-cc-pVnZ n = 5/Q energies augmented with additive corrections of post-CCSD(T) correlation, core correlation, scalar relativity, spin–orbit coupling, and zero-point energy. Besides the hydrogen-abstraction (HA) channel leading to H₂O + CH₃/C₂H₅ (ΔH₀ = −14.37/−18.19 kcal mol⁻¹), we reveal, for the first time, hydrogen-substitution (HS) and methyl-substitution (MS) pathways resulting in H + CH₃OH/C₂H₅OH (ΔH₀ = 13.19/7.12 kcal mol⁻¹) and CH₃ + CH₃OH (ΔH₀ = −2.20 kcal mol⁻¹) products, respectively. The adiabatic barrier heights for HA, MS, and HS in OH + CH₄/C₂H₆ are 4.78/2.18, 39.60, 43.53/41.73(52.48) kcal mol⁻¹, respectively, where substitution proceeds with Walden-inversion or (front-side-attack retention). In the entrance channels van der Waals wells with depths of 0.5–0.8 kcal mol⁻¹ are found and in the exit channels the HOH⋯C₂H₅, HOH⋯CH₃, H₃C⋯CH₃OH, and H⋯C₂H₅OH complexes are characterized with D_e values of 2.4, 1.7, 0.7, and 0.3 kcal mol⁻¹, respectively.

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I. Introduction

Moving beyond the reactions of atoms with alkanes,^1–16 the reactions of the OH radical with methane (CH₄) and ethane (C₂H₆) have been thoroughly investigated by theory and experiment to uncover the fundamental rules of polyatomic chemical reactivity.^16–39 The main product channel of the OH + CH₄/C₂H₆ reactions results in H₂O + CH₃/C₂H₅via hydrogen-abstraction (HA). In the early years the kinetics of the OH + CH₄/C₂H₆ reactions were investigated using experimental techniques as well as transition-state theory.^{17,18,28,30,31} The stationary points along the HA pathway were characterized by various ab initio and density functional methods reporting pre- and post-reaction complexes separated by a transition state for each reaction.^{19,22,23,27,28,33,35,38} In the case of the OH + CH₄ system, the CH₄⋯OH and H₂O⋯CH₃ complexes were also studied by different experimental techniques using stimulated Raman, infrared and electronic excitation,^24,36,37 photoelectron–photofragment coincidence,³⁴ and infrared spectroscopy in helium nanodroplets.^26,32 Furthermore, detailed reactive scattering experiments were performed for the different isotope-variants of the OH + CH₄ → H₂O + CH₃ reaction by Liu and co-workers.²⁹ For the OH + C₂H₆ system, the nascent H₂O vibrational distributions were measured by infrared chemiluminescence.²¹ In order to simulate some of the above-mentioned experiments the development of a full-dimensional potential energy surface (PES) was necessary. Following the early work of Espinosa-García and co-workers,²² Li and Guo³⁸ developed such a PES for the OH + CH₄ reaction and investigated its dynamics using quasi-classical trajectory and reduced-dimensional quantum methods.³⁹

In this study we focus on the high-level ab initio characterization of the PES of the OH + CH₄/C₂H₆ reactions providing benchmark structures and energies for the stationary points, thereby complementing our recent work on atom (F, Cl, Br, I) + CH₄/C₂H₆ systems.^13–15 Several aspects of the present study move beyond the previous work: (1) unlike most of the early work except ref. 38, we use the explicitly correlated CCSD(T)-F12 method⁴⁰ to obtain structures, frequencies, and energies; (2) basis sets as large as aug-cc-pV5Z (OH + CH₄) and aug-cc-pVQZ (OH + C₂H₆) are employed; (3) post-CCSD(T) correlation effects are considered up to the CCSDT(Q) level of theory; (4) the correlation energy contribution of the core electrons is taken into account; (5) scalar relativistic corrections are determined; and (6) spin–orbit effects are computed. Furthermore, besides the above-mentioned quantitative advances, we investigate several new alternative reaction pathways for the OH + CH₄/C₂H₆ systems, namely hydrogen-substitution and methyl-substitution forming H + CH₃OH/C₂H₅OH and CH₃ + CH₃OH products, respectively. Knowing the energetics of the above channels is essential to develop global PESs for the title reactions and to advance our knowledge on complex reaction mechanisms of multi-channel reactions.

In Section II we describe the computational details of the composite approach used to determine the best technically feasible stationary-point properties. The results are presented and discussed in Section III. The paper ends with summary and conclusions in Section IV.

II. Computational details

Stationary-point search is performed on the basis of previous studies^14,38 and chemical intuition using the second-order Møller–Plesset perturbation theory⁴¹ (MP2) with the aug-cc-pVDZ basis set.⁴² The obtained minima and saddle points are further optimized with the explicitly-correlated coupled-cluster singles, doubles, and perturbative triples (CCSD(T)-F12b) method⁴⁰ using the aug-cc-pVDZ and aug-cc-pVTZ basis sets.⁴² Harmonic vibrational frequencies are computed for each stationary point using the MP2/aug-cc-pVDZ, CCSD(T)-F12b/aug-cc-pVDZ, and CCSD(T)-F12b/aug-cc-pVTZ (except for some of the OH + C₂H₆ stationary points) levels of theory.

In order to achieve sub-chemical accuracy we perform the following singe-point energy computations at the best geometries obtained at the CCSD(T)-F12b/aug-cc-pVTZ level:

(1) CCSD(T)-F12b computations are carried out using the aug-cc-pVQZ and, in the case of the OH + CH₄ system, the aug-cc-pV5Z basis sets.

(2) CCSD(T),⁴³ CCSDT,⁴⁴ and CCSDT(Q)⁴⁵ methods are used with the cc-pVDZ and, in the case of the OH + CH₄ system, the aug-cc-pVDZ basis sets.

(3) All-electron (AE) and frozen-core (FC) computations are performed at the CCSD(T)-F12b/cc-pCVTZ-F12 level of theory.^40,46 In the usual FC treatment only the valence electrons are correlated, whereas the AE computations correlate the 1s² electrons of the C and O atoms as well.

(4) Second-order Douglas–Kroll⁴⁷ (DK) relativistic energies are computed at the AE-CCSD(T)-F12b/cc-pCVTZ-F12 level of theory.

(5) Spin–orbit (SO) corrections are determined using the Breit–Pauli Hamiltonian in the interacting-states approach⁴⁸ using the Davidson-corrected⁴⁹ multi-reference configuration interaction⁵⁰ (MRCI+Q) method with the aug-cc-pVDZ and aug-cc-pVTZ basis sets. The MRCI computations utilize an active space of 15/21 electrons in 8/11 spatial orbitals keeping all the 4/6 core electrons frozen for OH + CH₄/C₂H₆. Two doubly-degenerate electronic states are determined, non-SO₁ and non-SO₂, resulting in a 4 × 4 SO matrix, whose eigenvalues correspond to the two-fold ground (SO₁) and excited (SO₂) spin–orbit states.

The benchmark classical relative energies are calculated by the following composite energy expression:

CCSD(T)-F12b/aug-cc-pVnZ + δ[CCSDT] + δ[CCSDT(Q)] + Δ_core + Δ_rel + Δ_SO, (1)

where n = 5 and 4(Q) for the OH + CH₄ and OH + C₂H₆ systems, respectively, and

δ[CCSDT] = CCSDT/(aug-)cc-pVDZ − CCSD(T)/(aug-)cc-pVDZ, (2)

δ[CCSDT(Q)] = CCSDT(Q)/(aug-)cc-pVDZ − CCSDT/(aug-)cc-pVDZ, (3)

Δ_core = AE-CCSD(T)-F12b/cc-pCVTZ-F12 − FC-CCSD(T)-F12b/cc-pCVTZ-F12, (4)

Δ_rel = DK-AE-CCSD(T)-F12b/cc-pCVTZ-F12 − AE-CCSD(T)-F12b/cc-pCVTZ-F12 (5)

Δ_SO = SO₁(MRCI+Q/aug-cc-pVTZ) − non-SO₁(MRCI+Q/aug-cc-pVTZ). (6)

The benchmark adiabatic relative energies are obtained as

CCSD(T)-F12b/aug-cc-pVnZ + δ[CCSDT] + δ[CCSDT(Q)] + Δ_core + Δ_rel + Δ_SO + Δ_ZPE (7)

including the terms defined in eqn (1)–(6) and the zero-point energy correction (Δ_ZPE) determined at the CCSD(T)-F12b/aug-cc-pVTZ level, except for some of the OH + C₂H₆ stationary points, where the frequency computations are performed without symmetry at the CCSD(T)-F12b/aug-cc-pVDZ level.

For open-shell systems the MP2 method is used in a restricted formalism (RMP2),⁵¹ unless otherwise noted, whereas all the CCSD(T)-F12b computations utilize the unrestricted UCCSD(T)-F12b⁵² method based on restricted open-shell Hartree–Fock (ROHF) orbitals. For the determination of the post-CCSD(T) correlation effects we use unrestricted Hartree–Fock (UHF) reference and the unrestricted UCCSD(T), UCCSDT, and UCCSDT(Q) methods. Note that, for the sake of simplicity, we usually omit the reference function as well as the R (restricted) and U (unrestricted) abbreviations in the notations.

All the MP2, CCSD(T)-F12b, MRCI, and SO computations are carried out with the MOLPRO program package,⁵³ whereas the CCSD(T), CCSDT, and CCSDT(Q) energies are obtained with MRCC⁵⁴ interfaced to MOLPRO.

III. Results and discussion

The topology of the PESs with the benchmark classical and adiabatic energies of the OH + CH₄ and OH + C₂H₆ reactions are shown in Fig. 1 and 2, respectively, and the corresponding stationary-point geometries showing the most important structural parameters at different levels of theory are given in Fig. 3 and 4. The HA pathways are exothermic with benchmark classical(adiabatic) ΔE(ΔH₀) values of −13.07(−14.37)/−16.70(−18.19) kcal mol⁻¹ for the OH(²Π_3/2) + CH₄/C₂H₆ → H₂O + CH₃/C₂H₅ reactions. The HA barrier is significantly higher for OH + CH₄, i.e., 6.30(4.78) kcal mol⁻¹ than in the case of OH + C₂H₆, 3.69(2.18) kcal mol⁻¹. The structures of the hydrogen-abstraction transition-states (HA TSs) are reactant like in both cases as the reactive C–H bonds are only stretched by 0.118/0.087 Å relative to the corresponding bond lengths in CH₄/C₂H₆, whereas the forming O–H bonds are stretched by 0.363/0.428 Å relative to the H₂O product as shown in Fig. 3 and 4. Thus, the exothermic HA processes feature early barriers in accord with the Hammond postulate⁵⁵ and the more exothermic OH + C₂H₆ reaction has lower and more reactant-like barrier as the above-discussed data show. In the product wells of both systems there are post-reaction minima (HA PostMINs), where H₂O binds to the CH₃/C₂H₅ units with a single hydrogen-bond. The D_e(D₀) values of the HA PostMIN complexes are 1.73(0.65)/2.36(1.35) kcal mol⁻¹, in accord with the ratio of the intermolecular C⋯H distances of 2.383/2.312 Å. The greater stability of the latter can be explained by the fact that the C₂H₅ unit has larger dipole moment than the CH₃ fragment.

Fig. 1 Schematic potential energy surface of the OH(²Π_3/2) + CH₄ reaction showing the benchmark classical (adiabatic) relative energies of the stationary points along the different reaction pathways.

Fig. 2 Schematic potential energy surface of the OH(²Π_3/2) + C₂H₆ reaction showing the benchmark classical (adiabatic) relative energies of the stationary points along the different reaction pathways.

Fig. 3 Stationary-point structures of the OH + CH₄ system showing the most important distances (Å) and angles (degree) obtained at the MP2/aug-cc-pVDZ, CCSD(T)-F12b/aug-cc-pVDZ, and CCSD(T)-F12b/aug-cc-pVTZ levels of theory.

Fig. 4 Stationary-point structures of the OH + C₂H₆ system showing the most important distances (Å) and angles (degree) obtained at the MP2/aug-cc-pVDZ (see footnote k of Table 2), CCSD(T)-F12b/aug-cc-pVDZ, and CCSD(T)-F12b/aug-cc-pVTZ levels of theory.

For the OH + CH₄ reaction we have found an endothermic, ΔE(ΔH₀) = 14.33(13.19) kcal mol⁻¹, hydrogen-substitution (HS) pathway resulting in H + CH₃OH products via a Walden-inversion TS (HS W TS) with a classical(adiabatic) barrier height of 44.30(43.53) kcal mol⁻¹ as shown in Fig. 1. The HS W TS has C_s symmetry with nearly collinear O–C–H arrangement, where the O–C and C–H distances are 1.717 and 1.415 Å, respectively, stretched by 0.297 and 0.327 Å relative the corresponding bonds in CH₃OH and CH₄ (Fig. 3); thus, HS has a central or slightly-late barrier. The HS pathway of the OH + CH₄ reaction is revealed for the first time in the present study, but for atom + methane reactions HS is not unprecedented.^4,56–60 For the H and O(³P) + CH₄ reactions HS was investigated both experimentally^4,56 and theoretically.^4,57 For Cl + CH₄ we found a Walden-inversion barrier height of 42.08(38.84) kcal mol⁻¹,⁵⁸ which is similar to that of OH + CH₄. Besides Walden inversion, for the O(³P) and Cl + CH₄ reactions front-side attack TSs with C_2v and C_4v symmetry, respectively, were reported.^4,60 In the case of OH + CH₄ we do not find front-side attack TS; however, reaction dynamics simulations may reveal retention trajectories as we found that the retention pathways of the Cl + CH₄ → H + CH₃Cl reaction avoid the high-energy C_4v TS.⁶⁰

In the case of the OH + C₂H₆ reaction HS pathways leading to H + C₂H₅OH also exist as shown in Fig. 2. The endothermicity of the HS channel, ΔE(ΔH₀) = 9.10(7.12) kcal mol⁻¹, is less than that of the OH + CH₄ reaction, ΔE(ΔH₀) = 14.33(13.19) kcal mol⁻¹, whereas the Walden-inversion classical(adiabatic) barrier heights of the two systems are similar, i.e., 43.32(41.73) kcal mol⁻¹ for OH + C₂H₆ and 44.30(43.53) kcal mol⁻¹ for OH + CH₄. Furthermore, these HS barrier heights are also very similar to that of the Cl + C₂H₆ system, 41.60(37.66) kcal mol⁻¹.¹⁴ For the OH + C₂H₆ reaction we have found a front-side (FS) attack retention pathway via a higher barrier with classical(adiabatic) height of 54.13(52.48) kcal mol⁻¹.¹⁴ At the HS W TS the forming O–C and breaking C–H bonds are nearly collinear (168.3°) and the H₂C–C unit is almost planar, whereas at the HS FS TS the O–C–H angle is 58.2° along the reaction coordinate and the H₂C–CH₃ unit is ethane-like retaining its reactant configuration. Similar HS FS TS was found for the X + C₂H₆ [X = F, Cl, Br, I] reactions, comparing the barrier heights again the X = Cl case has similar, though slightly larger, value of 59.14(54.97) kcal mol⁻¹. For the OH + C₂H₆ reaction we have found weakly-bound H⋯C₂H₅OH complexes in the exit channels of the HS inversion and retention pathways with small D_e values of 0.26 and 0.32 kcal mol⁻¹, respectively. However, these shallow HS product-channel wells may not support a bound vibrational state, because the ZPE-corrected energies of the HS W PostMIN and HS FS PostMIN complexes are slightly above the vibrationally ground-state product asymptote by 0.05 and 0.04 kcal mol⁻¹ (Fig. 2).

Unlike for OH + CH₄, for the OH + C₂H₆ reaction a third product channel is possible resulting in CH₃OH + CH₃via methyl substitution (MS). As Fig. 2 shows, MS is slightly exothermic ΔE(ΔH₀) = −0.94(−2.20) kcal mol⁻¹ and goes over a classical(adiabatic) Walden-inversion barrier of 39.89(39.60) kcal mol⁻¹ height. Thus, similarly to the halogen + C₂H₆ reactions,¹⁴ MS is both thermodynamically and kinetically favored over HS. At the central MS W TS the O–C–C atoms are nearly collinear and the forming O–C and breaking C–C bonds are stretched by 0.407 and 0.413 Å, respectively, relative to their corresponding equilibrium values in the product and reactant (Fig. 4). For the halogen + C₂H₆ systems we reported a higher-energy front-side attack MS pathway as well,¹⁴ however, for OH + C₂H₆ MS FS TS is not found, though its existence cannot be ruled out. Nevertheless, we have found a H₃C⋯CH₃OH complex in the product channel of MS with D_e(D₀) values of 0.69(0.38) kcal mol⁻¹. The stability trend, D_e = {2.4, 0.7, 0.3} kcal mol⁻¹, of the {HA, MS, HS} PostMIN complexes, {HOH⋯C₂H₅, H₃C⋯CH₃OH, H⋯C₂H₅OH}, reflects the facts that HA PostMIN is stabilized by dipole–dipole interaction and CH₃ is more polarizable than the H atom.

The above-discussed stationary-point properties correspond to our new benchmark values obtained by the composite ab initio approach described in Section II. Now let us discuss the accuracy of the computed results. The most important structural parameters of the stationary points obtained by the MP2/aug-cc-pVDZ, CCSD(T)-F12b/aug-cc-pVDZ, and CCSD(T)-F12b/aug-cc-pVTZ levels of theory are shown in Fig. 3 and 4. As seen, the MP2 and CCSD(T)-F12b results may differ by about 0.01 Å, whereas the CCSD(T)-F12b distances with the aug-cc-pVDZ and aug-cc-pVTZ basis sets usually agree within 0.001 Å, except for large intermolecular distances where an order of magnitude larger uncertainties are found due to the flatness of the potential along the dissociation coordinate.

The convergence of the relative energies are shown in Tables 1 and 2 for the OH + CH₄ and OH + C₂H₆ systems, respectively, and the corresponding auxiliary corrections are detailed in Tables 3 and 4. The MP2 and CCSD(T)-F12b relative energies differ by 1–6 kcal mol⁻¹, for example, MP2 overestimates the HA barrier heights by 2.4–2.8 kcal mol⁻¹ and underestimates the endoergicity of the HS channels by 5.8–6.0 kcal mol⁻¹ as seen in Tables 1 and 2. Thus, it is clear that the use of the CCSD(T)-F12b method is needed to achieve chemical accuracy (uncertainty less than 1 kcal mol⁻¹). For the OH + CH₄ system we have CCSD(T)-F12b energies up to the large aug-cc-pV5Z basis and on the basis of Table 1 we find that the average absolute deviations of the aug-cc-pVDZ, aug-cc-pVTZ, and aug-cc-pVQZ relative energies from the aug-cc-pV5Z data decrease as 0.25, 0.06, 0.04 kcal mol⁻¹, respectively, showing the excellent basis convergence of the explicitly-correlated CCSD(T)-F12b method. Thus, we expect that even the CCSD(T)-F12b/aug-cc-pVQZ result is basis converged within 0.1 kcal mol⁻¹. This finding for OH + CH₄ is useful for the larger OH + C₂H₆ system, where we do not perform the CCSD(T)-F12b/aug-cc-pV5Z computations. For OH + C₂H₆, the aug-cc-pVDZ and aug-cc-pVTZ CCSD(T)-F12b relative energies are converged with average absolute deviations of 0.27 and 0.04 kcal mol⁻¹, respectively, with respect to the CCSD(T)-F12b/aug-cc-pVQZ results.

Table 1 Energies (kcal mol⁻¹) of the stationary points and product channels of the OH + CH₄ reaction relative to the reactants at different levels of theory

Stationary points	MP2	CCSD(T)-F12b				∑Δ^f	Classical^g	Δ ZPE	Adiabatic^i
	aVDZ^a	aVDZ^b	aVTZ^c	aVQZ^d	aV5Z^e
a MP2/aug-cc-pVDZ relative energies obtained at MP2/aug-cc-pVDZ geometries. b CCSD(T)-F12b/aug-cc-pVDZ relative energies obtained at CCSD(T)-F12b/aug-cc-pVDZ geometries. c CCSD(T)-F12b/aug-cc-pVTZ relative energies obtained at CCSD(T)-F12b/aug-cc-pVTZ geometries. d CCSD(T)-F12b/aug-cc-pVQZ relative energies obtained at CCSD(T)-F12b/aug-cc-pVTZ geometries. e CCSD(T)-F12b/aug-cc-pV5Z relative energies obtained at CCSD(T)-F12b/aug-cc-pVTZ geometries. f ∑Δ = δ[CCSDT/aVDZ] + δ[CCSDT(Q)/aVDZ] + Δcore + Δrel + ΔSO(aVTZ). g Benchmark classical relative energy obtained as CCSD(T)-F12b/aV5Z + ∑Δ. h ZPE correction obtained at CCSD(T)-F12b/aug-cc-pVTZ. i Benchmark adiabatic relative energy obtained as classical + ΔZPE.
HA TS	8.50	6.08	6.27	6.30	6.38	−0.08	6.30	−1.52	4.78
HA PostMIN	−18.22	−15.15	−14.99	−15.00	−14.96	+0.16	−14.80	−0.22	−15.02
HS W TS	41.07	44.52	44.40	44.38	44.42	−0.12	44.30	−0.77	43.53
CH3OH + H	7.62	13.39	14.00	13.95	13.98	+0.35	14.33	−1.14	13.19
CH3 + H2O	−16.28	−13.19	−13.15	−13.25	−13.25	+0.18	−13.07	−1.30	−14.37

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Table 2 Energies (kcal mol⁻¹) of the stationary points and product channels of the OH + C₂H₆ reaction relative to the reactants at different levels of theory

Stationary points	MP2	CCSD(T)-F12b			∑Δ^e	Classical^f	Δ ZPE	Adiabatic^h
	aVDZ^a	aVDZ^b	aVTZ^c	aVQZ^d
a MP2/aug-cc-pVDZ relative energies obtained at MP2/aug-cc-pVDZ geometries. b CCSD(T)-F12b/aug-cc-pVDZ relative energies obtained at CCSD(T)-F12b/aug-cc-pVDZ geometries. c CCSD(T)-F12b/aug-cc-pVTZ relative energies obtained at CCSD(T)-F12b/aug-cc-pVTZ geometries. d CCSD(T)-F12b/aug-cc-pVQZ relative energies obtained at CCSD(T)-F12b/aug-cc-pVTZ geometries. e ∑Δ = δ[CCSDT] + δ[CCSDT(Q)] + Δcore + Δrel + ΔSO(aVTZ). f Benchmark classical relative energy obtained as CCSD(T)-F12b/aVQZ + ∑Δ. g ZPE correction obtained at CCSD(T)-F12b/aug-cc-pVTZ (for exemptions see j). h Benchmark adiabatic relative energy obtained as classical + ΔZPE. i Obtained at CCSD(T)-F12b/aug-cc-pVDZ geometries. j ZPE correction obtained at CCSD(T)-F12b/aug-cc-pVDZ. k Obtained with UMP2.
HA TS	6.21	3.45	3.65	3.70	−0.01	3.69	−1.51	2.18
HA PostMIN	−21.78	−19.28	−19.07	−19.09	+0.03	−19.06	−0.48	−19.54
HS W TS	39.89	43.57	43.49^i	43.51^i	−0.19	43.32	−1.59^j	41.73
HS W PostMIN	2.11	7.96	8.62	8.63	+0.21	8.84	−1.67	7.17
HS FS TS	55.68^k	54.33	54.47	54.53	−0.40	54.13	−1.65^j	52.48
HS FS PostMIN	2.00^k	7.90	8.56	8.57	+0.21	8.78	−1.62^j	7.16
MS W TS	44.80^k	40.29	40.21^i	40.21^i	−0.32	39.89	−0.29^j	39.60
MS W PostMIN	−3.14^k	−2.02	−1.85	−1.89	+0.26	−1.63	−0.95^j	−2.58
C2H5OH + H	2.29	8.29	8.91	8.89	+0.21	9.10	−1.98	7.12
CH3OH + CH3	−2.24	−1.16	−1.13	−1.21	+0.27	−0.94	−1.26	−2.20
C2H5 + H2O	−18.96	−16.64	−16.61	−16.75	+0.05	−16.70	−1.49	−18.19

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Table 3 Auxiliary corrections (kcal mol⁻¹) for the relative energies of the stationary points and product channels of the OH + CH₄ reaction

Stationary points	δ[CCSDT]^a		δ[CCSDT(Q)]^b		Δ core	Δ rel	Δ SO		Δ ZPE
	VDZ	aVDZ	VDZ	aVDZ			aVDZ	aVTZ	MP2/DZ	CC/DZ	CC/TZ
a CCSDT – CCSD(T) obtained with the cc-pVDZ (VDZ) and aug-cc-pVDZ (aVDZ) basis sets at CCSD(T)-F12b/aug-cc-pVTZ geometries. b CCSDT(Q) – CCSDT obtained with the cc-pVDZ (VDZ) and aug-cc-pVDZ (aVDZ) basis sets at CCSD(T)-F12b/aug-cc-pVTZ geometries. c Core-correlation correction obtained as the difference between all-electron and frozen-core CCSD(T)-F12b/cc-pCVTZ-F12 relative energies at CCSD(T)-F12b/aug-cc-pVTZ geometries. d Scalar relativistic effect obtained as the difference between Douglas–Kroll and non-relativistic all-electron CCSD(T)-F12b/cc-pCVTZ-F12 relative energies at CCSD(T)-F12b/aug-cc-pVTZ geometries. e Spin–orbit (SO) corrections obtained as the difference between the SO and non-SO ground-state MRCI+Q/aug-cc-pVDZ (aVDZ) or MRCI+Q/aug-cc-pVTZ (aVTZ) relative energies at CCSD(T)-F12b/aug-cc-pVTZ geometries. f ZPE corrections obtained at MP2/aug-cc-pVDZ (MP2/DZ), CCSD(T)-F12b/aug-cc-pVDZ (CC/DZ), and CCSD(T)-F12b/aug-cc-pVTZ (CC/TZ).
HA TS	−0.11	−0.12	−0.15	−0.23	0.06	0.02	0.18	0.19	−1.05	−1.67	−1.52
HA PostMIN	0.00	0.04	−0.13	−0.15	−0.04	0.12	0.18	0.19	−0.06	−0.01	−0.22
HS W TS	−0.26	−0.18	−0.35	−0.43	0.25	0.05	0.17	0.19	−0.75	−0.84	−0.77
CH3OH + H	0.09	0.18	−0.20	−0.19	0.02	0.15	0.18	0.19	−1.24	−1.12	−1.14
CH3 + H2O	0.01	0.05	−0.13	−0.14	−0.04	0.12	0.18	0.19	−1.26	−1.26	−1.30

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Table 4 Auxiliary corrections (kcal mol⁻¹) for the relative energies of the stationary points and product channels of the OH + C₂H₆ reaction

Stationary points	δ[CCSDT]^a	δ[CCSDT(Q)]^b	Δ core	Δ rel	Δ SO		Δ ZPE
					aVDZ	aVTZ	MP2/DZ	CC/DZ	CC/TZ
a CCSDT – CCSD(T) obtained with the cc-pVDZ basis set at CCSD(T)-F12b/aug-cc-pVTZ geometries. b CCSDT(Q) – CCSDT obtained with the cc-pVDZ basis set at CCSD(T)-F12b/aug-cc-pVTZ geometries. c Core-correlation correction obtained as the difference between all-electron and frozen-core CCSD(T)-F12b/cc-pCVTZ-F12 relative energies at CCSD(T)-F12b/aug-cc-pVTZ geometries. d Scalar relativistic effect obtained as the difference between Douglas–Kroll and non-relativistic all-electron CCSD(T)-F12b/cc-pCVTZ-F12 relative energies at CCSD(T)-F12b/aug-cc-pVTZ geometries. e Spin–orbit (SO) corrections obtained as the difference between the SO and non-SO ground-state MRCI+Q/aug-cc-pVDZ (aVDZ) or MRCI+Q/aug-cc-pVTZ (aVTZ) relative energies at CCSD(T)-F12b/aug-cc-pVTZ geometries. f ZPE corrections obtained at MP2/aug-cc-pVDZ (MP2/DZ), CCSD(T)-F12b/aug-cc-pVDZ (CC/DZ), and CCSD(T)-F12b/aug-cc-pVTZ (CC/TZ). g Obtained at CCSD(T)-F12b/aug-cc-pVDZ geometries. h Obtained with UMP2.
HA TS	−0.11	−0.14	0.03	0.02	0.18	0.19	−2.08	−1.74	−1.51
HA PostMIN	−0.04	−0.13	−0.11	0.12	0.18	0.19	−0.08	−0.31	−0.48
HS W TS	−0.27^g	−0.40^g	0.24^g	0.05^g	0.18^g	0.19^g	−1.55	−1.59
HS W PostMIN	0.10	−0.20	−0.02	0.14	0.18	0.19	−1.82	−1.69	−1.67
HS FS TS	−0.38	−0.50	0.26	0.03	0.18	0.19	−1.88^h	−1.65
HS FS PostMIN	0.10	−0.20	−0.02	0.14	0.18	0.19	−1.80^h	−1.62
MS W TS	−0.48^g	−0.43^g	0.39^g	0.01^g	0.18^g	0.19^g	1.19^h	−0.29
MS W PostMIN	−0.02	−0.12	0.09	0.12	0.18	0.19	−0.84^h	−0.95
C2H5OH + H	0.10	−0.20	−0.02	0.14	0.18	0.19	−2.08	−1.92	−1.98
CH3OH + CH3	−0.02	−0.11	0.09	0.12	0.18	0.19	−1.15	−1.21	−1.26
C2H5 + H2O	−0.03	−0.12	−0.11	0.12	0.18	0.19	−1.51	−1.47	−1.49

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The large-basis CCSD(T)-F12b computations provide a very good estimate of the complete-basis-set limit of the CCSD(T) relative energies. If one aims to approach the “exact” energies additional small corrections, such as post-CCSD(T) and core electron correlation as well as scalar and SO relativistic effects should be considered. Furthermore, to get experimentally observable quantities the ZPE corrections have to be taken into account. These so-called auxiliary corrections are given in Tables 3 and 4 for the OH + CH₄ and OH + C₂H₆ systems, respectively. For OH + CH₄ we have determined the δ[CCSDT] and δ[CCSDT(Q)] corrections with the cc-pVDZ and aug-cc-pVDZ basis sets. As Table 3 shows even the small cc-pVDZ basis provides good estimates for these post-CCSD(T) correlation effects suggesting that it is sufficient to perform only the CCSDT(Q)/cc-pVDZ computations for the larger OH + C₂H₆ system. In the case of the products and HA PostMIN of the OH + CH₄ reaction the δ[CCSDT] and δ[CCSDT(Q)] absolute corrections are between 0–0.2 kcal mol⁻¹ and partially cancel each other. However, for the HA TS and HS W TS the aug-cc-pVDZ(cc-pVDZ) corrections add up to −0.35(−0.26) and −0.61(−0.61) kcal mol⁻¹, respectively, showing the good performance of the smaller basis set. For the OH + C₂H₆ system most of the post-CCSD(T) corrections have the same negative sign, except the HS products and PostMINs (Table 4). The largest cumulative post-CCSD(T) corrections are found for the TSs, namely −0.25, −0.67, −0.88, and −0.91 kcal mol⁻¹, for HA TS, HS W TS, HS FS TS, and MS W TS, respectively, which are clearly not negligible if sub-chemical accuracy is desired.

For both reactions the core correlation corrections are in the range from −0.11 to +0.39 kcal mol⁻¹, whereas the scalar relativistic effects on the relative energies are always small positive values between 0.01 and 0.15 kcal mol⁻¹. The largest core corrections are obtained for the HS TSs (0.24–0.26 kcal mol⁻¹) and for the MS W TS (0.39 kcal mol⁻¹) as seen in Tables 3 and 4.

SO interaction almost fully quenches at all the stationary points, while lowers the reactant asymptote, thereby increasing all the relative energies by 0.19 kcal mol⁻¹. As seen in Tables 3 and 4 we have performed SO computations with the aug-cc-pVDZ and aug-cc-pVTZ basis sets, which result in almost the same corrections of 0.18 (once 0.17) and 0.19 kcal mol⁻¹, respectively. These values are in good agreement with the experimental data of 0.20 kcal mol⁻¹ deduced from the measured SO splitting (ε = 0.40 kcal mol⁻¹) of the OH radical as ε/2. To get deeper insight, we have computed SO and non-SO potential energy curves along the intermolecular coordinates of the CH₄⋯OH and C₂H₆⋯OH systems while the OH approaches the CH₄ and C₂H₆ molecules from different directions as shown in Fig. 5 and 6. The ground electronic state of the OH radical is the 2 × 2-fold degenerate ²Π, which splits to a 2-fold SO ground (²Π_3/2) and a 2-fold SO excited state (²Π_1/2). As OH approaches CH₄ the two doublet non-SO state remains quasi-degenerate and the SO ground and excited states are below and above the non-SO states by ε/2 as shown in Fig. 5. There is a van der Waals well in the entrance channel whose depth is 0.5 and 0.7 kcal mol⁻¹ with HCH₃⋯OH and H₃CH⋯OH C_3v orientations, respectively. The well depths and positions are not affected by the SO interaction, because the wells are at C⋯O distances of 3–4 Å, whereas the SO correction is almost constant until the C⋯O distance decreases to about 2 Å, where the difference between the SO and non-SO ground state energies rapidly drops to zero (see insets in Fig. 5). This fast quenching of SO interaction occurs at high relative energies above 100 kcal mol⁻¹, where the quasi-degeneracy of the non-SO states starts lifting. Interestingly, in the case of the halogen (X) + CH₄ reactions, the HCH₃⋯X minimum is the deeper with depth of 0.6(0.9) kcal mol⁻¹ and the H₃CH⋯X well is the shallower, 0.3(0.3) kcal mol⁻¹, with(without) SO correction.^13,58 In the case of the H₃CCH₃⋯OH (C_3v) and H₃CH₂CH⋯OH (C_s) arrangements in the entrance well of OH + C₂H₆ reaction, the conclusions are qualitatively the same as for OH + CH₄, i.e., the former minimum is 0.6 kcal mol⁻¹ deep, whereas the latter is deeper, 0.7 kcal mol⁻¹, and these are not affected by the SO interactions. If OH approaches C₂H₆ perpendicularly to the C–C bond, the well is the deepest, 0.8 kcal mol⁻¹, and slightly affected by the SO interaction as the quenching and the departure of the two non-SO states occur in the 2.5–3.5 Å range of the CC⋯O distance as seen in Fig. 6. For the X + C₂H₆ systems, the H₃CCH₃⋯X (C_3v) minimum was found to be the deepest, slightly below the perpendicular well, and the H₃CH₂CH⋯X (C_s) arrangement gave the shallowest minimum.¹⁴ These entrance-channel wells may play an important role in the dynamics of the title reactions at low collision energies, especially for OH + C₂H₆, which has a low barrier for HA.

Fig. 5 Potential energy curves along the C₃ axis of the CH₄⋯OH system obtained at the MRCI+Q/aug-cc-pVDZ level of theory while the structures of the CH₄ and OH units are kept frozen at their equilibrium geometries. SO₁ and SO₂ denote the spin–orbit ground and excited states, whereas non-SO₁ and non-SO₂ are the non-relativistic ground and excited electronic states, respectively. The insets show the distance dependence of the spin–orbit corrections obtained as difference between the SO₁ and non-SO₁ energies.

Fig. 6 Potential energy curves along the C₃ (left), CH (middle), and CC-perpendicular (right) axes of the C₂H₆⋯OH system obtained at the MRCI+Q/aug-cc-pVDZ level of theory while the structures of the C₂H₆ and OH units are kept frozen at their equilibrium geometries. SO₁ and SO₂ denote the spin–orbit ground and excited states, whereas non-SO₁ and non-SO₂ are the non-relativistic ground and excited electronic states, respectively. The insets show the distance dependence of the spin–orbit corrections obtained as difference between the SO₁ and non-SO₁ energies.

The ZPE effects on the relative energies are always negative and in the most cases the absolute corrections are in the range of 1–2 kcal mol⁻¹ as shown in Tables 1 and 2. The mean absolute deviation of the MP2 ZPE corrections from the CCSD(T)-F12b results is about 0.2 kcal mol⁻¹, whereas the CCSD(T)-F12b/aug-cc-pVTZ data are converged within 0.1 kcal mol⁻¹ as Tables 3 and 4 show. These ZPE effects are clearly not negligible if we aim to compute chemically accurate, measurable adiabatic relative energies.

For the reaction enthalpies comparison between the present computed adiabatic relative energies and the “experimental” data deduced from 0 K enthalpies of formation taken from the Active Thermochemical Tables (ATcT)⁶¹ is possible. Note that ATcT collects the best measured and computed thermochemical data, thereby providing the best available predictions for enthalpies of formation and their uncertainties of several chemical species. We call these ATcT data as “experimental” even if they rely on both theory and experiment. As Table 5 shows the agreement between the present ab initio 0 K reaction enthalpies and experiment is excellent; the mean absolute deviation between theory and experiment is only 0.07 kcal mol⁻¹ and, in most cases, the theoretical predictions are within the error bars of the experiment. This comparison demonstrates that one needs to consider the auxiliary corrections (post-CCSD(T), core, scalar relativistic, SO) to achieve this outstanding accuracy, because the cumulative effect of these corrections averaged for the five reaction channels is 0.21 kcal mol⁻¹, which is significantly larger than the above-mentioned mean deviation between theory and experiment. Furthermore, the excellent agreement for these measurable quantities confirms the accuracy of the present theoretical predictions for the experimentally hardly accessible properties, such as barrier heights.

Table 5 Comparison between the best available experimental and our computed benchmark 0 K reaction enthalpies (kcal mol⁻¹) for the OH + CH₄/C₂H₆ reactions

Reaction	Theory^a	Experiment^b
a Benchmark theoretical reaction enthalpies (this work). b Data obtained from the latest version (1.122 g) of the Active Thermochemical Tables (ATcT).^61 Uncertainties are derived from the uncertainties of each 0 K enthalpy of formation given in ATcT using the Gaussian error-propagation law.
OH + CH4 → CH3OH + H	13.19	13.22 ± 0.04
OH + CH4 → CH3 + H2O	−14.37	−14.28 ± 0.02
OH + C2H6 → C2H5OH + H	7.12	7.13 ± 0.06
OH + C2H6 → CH3OH + CH3	−2.20	−2.16 ± 0.05
OH + C2H6 → C2H5 + H2O	−18.19	−18.35 ± 0.07

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IV. Summary and conclusions

The reactions of the OH radical with methane and ethane have become benchmark systems to understand the dynamics and mechanisms of polyatomic reactions. Despite many previous experimental and theoretical studies^16–39 focusing on the exothermic (ΔH₀ = −14.37/−18.19 kcal mol⁻¹) hydrogen-abstraction reaction resulting in H₂O + CH₃/C₂H₅via low adiabatic barriers of 4.78/2.18 kcal mol⁻¹, other product channels and their energetic requirements were unknown until the present study. Here, we show that hydrogen-substitution leading to H + CH₃OH/C₂H₅OH is endothermic with ΔH₀ = 13.19/7.12 kcal mol⁻¹ and can proceed via a Walden-inversion barrier with adiabatic height of 43.53/41.73 kcal mol⁻¹ or for the latter we have also found a front-side attack pathways via an adiabatic barrier of 52.48 kcal mol⁻¹. For OH + C₂H₆ a methyl-substitution channel forming CH₃ + CH₃OH products is also possible, which is exothermic, ΔH₀ = −2.20 kcal mol⁻¹, but has a large adiabatic barrier of 39.60 kcal mol⁻¹. For the product channel several complexes have been revealed and characterized, showing a stability order of HOH⋯C₂H₅, HOH⋯CH₃, H₃C⋯CH₃OH, and H⋯C₂H₅OH with D_e values of 2.4, 1.7, 0.7, and 0.3 kcal mol⁻¹, respectively. For the first time, we have performed SO computations for the entrance channel, thereby revealing van der Waals wells with depths of 0.5–0.8 kcal mol⁻¹ depending on the relative orientation of the reactants. Unlike for the halogen + CH₄/C₂H₆ systems,^13,14 the well is the deepest for perpendicular C–C⋯OH approach (0.8 kcal mol⁻¹), followed by the H₃CH/H₃CH₂CH⋯OH (0.7 kcal mol⁻¹) and HCH₃/H₃CCH₃⋯OH (0.5–0.6 kcal mol⁻¹) arrangements. These pre- and post-reaction wells may play significant roles in the dynamics of the hydrogen-abstraction processes, especially at low collision energies, by steering the reactants in the entrance channels and affecting product rotation in the exit channels, respectively.

The stationary-point properties have been computed using an accurate composite ab initio approach which goes beyond the widely-used standard quantum chemistry. The complete-basis-set limit of CCSD(T) is approached well within 0.1 kcal mol⁻¹ with explicitly-correlated CCSD(T)-F12b/aug-cc-pVnZ computations with n = 5 and 4(Q) for OH + CH₄ and OH + C₂H₆, respectively. Post-CCSD(T) correlation (−0.9 to −0.1), core correlation (−0.1 to +0.4), scalar relativistic (+0.0 to +0.2), SO (+0.2), and ZPE (−2 to −1) effects are determined resulting in typical values for the title reactions as indicated in parentheses in kcal mol⁻¹. We conclude that the present theoretical predictions provide sub-chemically accurate relative energies with an estimated uncertainty of around 0.1 kcal mol⁻¹, which is about an order of magnitude better than that of the standard quantum chemistry studies. The outstanding accuracy of the present results can be confirmed by the comparison of the computed reaction enthalpies with the corresponding experimental data⁶¹ showing only 0.07 kcal mol⁻¹ mean absolute deviation.

The new insights into the mechanisms and alternative reaction pathways of the title reactions are essential to develop global PESs for the OH + CH₄/C₂H₆ systems, allowing dynamical investigations over a large collision energy range. Furthermore, the relative energies of the stationary points may guide future experimental investigations showing the thermodynamical and kinetical controls of the different processes. Finally, the present composite approach may be utilized in several similar ab initio investigations if high accuracy is desired.

Conflicts of interest

There are no conflicts of interest to declare.

Acknowledgements

We thank the National Research, Development and Innovation Office-NKFIH, K-125317, the Ministry of Human Capacities, Hungary grant 20391-3/2018/FEKUSTRAT, and the Momentum (Lendület) Program of the Hungarian Academy of Sciences for financial support.

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