The geometries of the alkoxy radicals investigated are presented in Fig. 1, where the relevant geometrical parameters have been indicated. In the following, R1, R2, R3 refer to the three alkoxy radicals investigated and TS1, TS2, TS3 designate the corresponding transition states for β-C–H scissions. It can be pointed out that reactant R1 has C1 symmetry, that is, the hydroxyl H-atom is twisted out of the OCO plane (HOCO = 54.3° at the MP2 level of theory), thus leading to the existence of two optical isomers. This result is obtained whatever the level of theory used and can be explained by orbital analysis. Out of plane position of hydrogen atom allows for a largest delocalisation of the two electrons in molecular orbital depicted Fig. 2, and simultaneously accounts for the (i) minimization of interaction between the 2p lone pairs of each oxygen, (ii) optimisation of the intramolecular hydrogen bonding forces involving the in-plane 2p lone pair on the open-shell oxygen atom, (iii) maximisation of the π overlap, involving the hydroxyl oxygen 2p orbital (out of plane) and a CH2 fragment orbital, that describes the unpaired electron π delocalisation across the OCO skeleton atoms. The reactant R2 geometry is consistent with the one predicted by Francisco[20] in his study of the CH3OCH2O˙ radical structure.
As aforementioned, we have taken a reaction path degeneracy α of two in the following rate constant calculations for each alkoxy decomposition, reactions (1), (2) and (3). Unlike the β-C–O decomposition (see Fig. 3), it is clear that H-elimination can occur by two distinct paths since two physically distinct β-C–H bonds can dissociate in each reactant (R1, R2, R3) to form a carbonyl group (see Fig. 1). Although these two H-atoms are differently situated and the two corresponding elimination paths are different, we have assumed, for computational purposes, that the loss of reactant proceeds with the same rate constant by each of the two pathways.
This can be rationalized by examining the structure of each transition state (see TS1, TS2, TS3 in Fig. 1) and keeping in mind that the two pathways differ only by the position of the leaving H-atom: situated above or under the HCOO plane. For reaction (1), the two corresponding transition states have exactly the same properties, while some differences appear for reactions (2) and (3) due to asymmetrical environment around the leaving H-atom. For instance, our CCSD(T)//MP2 computations give a difference in potential energy of 0.3 kcal mol−1 between these two similar transition states for reaction (3).
From an electronic structure point of view, the decomposition of the studied alkoxy radicals by β-C–H bond rupture involves transition state geometries (TS1, TS2, TS3) with more product-like character. For instance, in the transition state TS3 the C–O bond is shortened by about 0.13 Å (C–Oreactant = 1.36 Å, C–OTS3 = 1.23 Å) which corresponds to 90 percent of the final shortening (C–Oproducts = 1.21 Å). The breaking C–H bond in the transition state is significantly longer (by about 32 percent) than in the unperturbed radical. The structural parameters (vibrational frequencies and rotational constants) of reactants and transition states obtained at the MP2 level of theory for reactions (1)–(3) are presented in Table 1. It is worth noting that the transition state structures are well localized on the PES, with moments of inertia slightly lower than those of the corresponding radicals. This results in very small entropies of activation leading to low preexponential factors for these decomposition reactions (≈5 × 1012 s−1). It can be noticed that we have found, for reaction (2), another H-elimination channel involving at the beginning of this process a conformer of the reactant: R2′ (see Fig. 1). At the CCSD(T) level of theory, the calculated barrier is however about 2 kcal mol−1 larger, showing that this channel is not favoured. It is interesting to note that R3 conformer geometry reveals an intramolecular hydrogen bond. More explicitly, it forms a seven-membered ring structure making possible the isomerisation process (4). This possibility is examined below. The corresponding structure of the transition state (TS7) is given in Fig. 4 and structural parameters are reported in Table 1.
As can be seen from Table 2, the enthalpies of β-C–H scission (1), (2) and (3) at 0 K (ΔrH°(0 K)) are of the same order of magnitude for each level of theory. For instance, the heat of reactions (1)–(3) derived from the G2 method are −1.0, −1.4 and −1.2 kcal mol−1, respectively. The estimated enthalpy of each reaction indicates it to be thermoneutral at the two levels of theory G2 (about −1 kcal mol−1) or CCSD(T)//MP2 (between −2 and −3 kcal mol−1). Using isodesmic reactions for reaction (2), Good and Francisco[12] have estimated a ΔH value of 3.0 kcal mol−1, consistent with our findings. It is not surprising that B3LYP gives values significantly different with respect to the other methods since a larger basis set would be necessary to better describe the large electronic changes occurring between reactant and products.
Barrier heights (ΔH‡°(0 K)) for the β-C–H bond dissociations investigated in this work are summarized in Table 2. Comparison of these values to each other shows no significant difference at each level of theory for the three reactions. The studied β-C–H bond scissions involve a sizeable energy barrier (9–14 kcal mol−1). Hippler et al[11]. have computed activation barriers for such β-C–H bond dissociations for a set of twelve non-oxygenated alkoxy radicals and found values lying between 19 and 26 kcal mol−1. The fact that our computed barriers are about twice smaller than their values provides strong evidence for an effect of oxygen-substituents which increases the β-C–H dissociation rate.
Preceding the products, a shallow minimum has been spotted on the PES for each reaction corresponding to a weak-bound complex that is formed by just a few kcal mol−1 binding energy between H-atom and the corresponding co-product.
In order to establish the relative importance of the H-atom elimination and of the β-C–O dissociation, an alternative reaction pathway, we also present an estimation of the energetic features for the latter channel for each alkoxy radical, at B3LYP theoretical level: HOCH2O˙ + M → HO˙ + H2CO + MCH3OCH2O˙ + M → CH3O˙ + H2CO + MCH3OCH2OCH2O˙ + M → CH3OCH2O˙ + H2CO + MThe values of calculated barriers (20–30 kcal mol−1, Table 2) agree favourably with those obtained by Ferenac et al[10]. at the same level of theory for other similar oxygenated alkoxy radicals (CH3CH2OCH2O˙: 20 kcal mol−1, CH3COOCH2O˙: 22 kcal mol−1). Good and Francisco[12] have obtained an activation barrier of 23.1 kcal mol−1 at the G2 level of theory for reaction (12), which is in good agreement with our HF-DFT (B3LYP) value (21.4 kcal mol−1). The large activation energies obtained for the β-C–O bond scission (together with the Arrhenius factors found for this process at 298 K: A ≈ 1–2 × 1013 s−1) clearly show that this process is not favourable, as found by Good and Francisco[12] for the CH3OCH2O˙ radical. This is in line with experimental findings of Wallington et al[2]. and Geiger and Becker[5] who suggest that decomposition of CH3OCH2OCH2O˙ via reaction (13) is of minor importance.
There are no possible isomerisation reaction by 1,5 H-shift through a six-membered transition state for these three alkoxy radicals. However, Wallington et al[2]. suggest that an unknown reaction, accounting for about 9% of the fate of CH3OCH2OCH2O˙, should be an isomerisation reaction. Our CCSD(T)//MP2 calculations show that the 1,6-H-shift (reaction (4)) (ΔH‡°(0 K) = 10.7 kcal mol−1) is more favoured than the 1,4 H-shift (ΔH‡°(0 K) = 20.6 kcal mol−1). In view of our results (G2 and CCSD(T)//MP2 in Table 2), this 1,6-isomerisation channel through a 7-membered transition state may be energetically competitive with the β-C–H scission.
Concerning the CH3OCH2O˙ radical, the 1,2 and 1,4 H-shifts have not been taken into account in our kinetic treatment. Indeed, the corresponding B3LYP activation energies (ΔH‡°(0 K) = 27.6 and 24.4 kcal mol−1 respectively) are very large compared to the β-C–H one (ΔH‡°(0 K) = 13.3 kcal mol−1) (the B3LYP predicted Arrhenius A factors being similar at 298 K: Aβ-C–H = 6 × 1012 s−1, Aisom = 2–4 × 1012 s−1).
Structural parameters (vibrational frequencies and rotational constants) listed in Table 1 are employed in the RRKM calculations for H-elimination reactions (1), (2) and (3) and isomerisation reaction (4). There are no data available for the Lennard-Jones parameters, σ and ε/k, of the three studied alkoxy radicals. Values were evaluated from tabulated values of nearest species or were based on the recommendation of Gilbert and Smith[21] using the boiling point of similar species. These estimated values are also reported in Table 1. For the HOCH2O˙ radical, experiments[3] were performed in formaldehyde as buffer gas. Therefore the evaluated Lennard-Jones parameters of this species are reported in Table 1 along with those of N2 used as buffer gas for the two other alkoxy radicals.
When N2 is the buffer gas, the collisional efficiency βc is taken equal to 0.2 which is a reasonable value at 298 K. For reaction (1), the collisional efficiency of formaldehyde is unknown. Here, we have choosen to use βc = 0.2 as for N2. As discussed below, doubling this value has no significant effect on the k1 pressure dependence under atmospheric conditions. As proposed by Troe,[22] a temperature dependence for βc ∝ T−1 was used in calculations.
As above-mentioned, in first step of the calculation, all the energetic input data used in the statistical RRKM calculations were taken from the CCSD(T)//MP2 results. Then, taking into account that the ab initio values of the barrier height E0 are calculated with uncertainties of about 1–2 kcal mol−1, E0 was adjusted in order to fit at best the available kinetic experimental data. The best fit was obtained with E0 = 9.6, 10.7, 12.6 and 11.5 kcal mol−1 for reactions (1), (2), (3) and (4), respectively. These values are reported in Table 3, along with the values obtained from CCSD(T)//MP2 calculations. It can be seen that the adjustments (0.8–2.5 kcal mol−1) are within the margin of errors (both experimental and theoretical). Those values show that the β-C–H scission pathway can be predicted to occur with an activation energy of 10–13 kcal mol−1. It should be noted that in the particular case of the CH3OCH2OCH2O˙ radical, the 1,6-H-shift isomerisation reaction (4) was taken into account, using the data reported by Wallington et al.[2] In air conditions, they report that the β-C–H scission accounts for 7% of the total reaction, the 1,6-H-shift isomerisation for 9% and the reaction with oxygen for 84%. Consequently, at 296 K and 760 Torr, the experimental values are k3 ≈ 4 × 103 s−1 and k4 ≈ 5 × 103 s−1.
The falloff curves are shown in Fig. 5 for reactions (1)–(4). As can be anticipated for reactions having similar threshold energies, the increasing complexity of the radical from HOCH2O˙ to CH3OCH2OCH2O˙ results in a decrease of the broadening of the falloff.
In order to compare, the predicted pressure dependence of the β-C–H dissociation rate constants k1, k2 and k3, the values of the high-pressure limiting rate constant k∞ and the ratio Γ = k760/k∞ (k∞ ≡ kTST obtained using adjusted E0) are listed in Table 3. The same calculation was performed for upper tropospheric conditions (220 K and 150 Torr). It is interesting to point out that, for reactions (1) and (2) at 298 K and 760 Torr total pressure, the rate constant is still strongly pressure dependent (Γ = 0.05 and 0.35, respectively). This is in good agreement with measurements of Jenkin et al[1]. for reaction (2). The same situation is observed at 150 Torr and 220 K (upper tropospheric conditions). In contrast, reactions (3) and (4) are near the high pressure limit (Γ = 0.85 and 0.90, respectively). It appears from falloff curves (Fig. 5) that the extrapolated values at 760 Torr of the rate constants of reactions (1) and (2) are about one order of magnitude higher than that of reactions (3) and (4).
Uncertainties on E0 valuesFrom the calculated falloff of the rate constant k1 (HOCH2O˙), it is deduced that the experimental value[3] of the rate constant, obtained at 25 Torr, is in the low-pressure part of the falloff with k25/k∞ ≈ 10−3. Therefore errors on βc and ZLJ are directly propagated on the calculated value of the rate constant and thus, on the deduced value of E0. For example, doubling the formaldehyde collisional efficiency (βc = 0.4 rather than 0.2) leads to a new fitted E0 value of 10.1 kcal mol−1 (instead of 9.6 kcal mol−1). Similarly, by inspection of Fig. 5, it is clear that the k2 experimental value of Jenkin et al[1]. for CH3OCH2O˙ radical, obtained at 25 Torr, is in the low-pressure part of the falloff. This experimental value is obtained relatively to the rate constant of the reaction with O2, with [O2] in the range (1–6) × 1017 molecule cm−3. This results in large uncertainty on the experimental value of the dissociation rate constant of (1–6) × 103 s−1. Therefore, we consider that the calculated values of E0, obtained by fitting of our statistical calculations to experimental results, have an uncertainty of ±1 kcal mol−1.
It is worth noting that, for reaction (2), our RRKM calculated falloff fitted to the experimental value of rate constant at 25 Torr, has predicted the kinetic behaviour which is experimentally observed at 760 Torr[1] for this reaction. In fact, it is indicated in [ref. 1] that the rate constant is pressure dependent and that competition with the O2 reaction exists at 760 Torr, [O2] > 1018 molecule cm−3, which corresponds to rate constants >1 × 104 s−1. This is in good agreement with our calculated value of 5.2 × 104 s−1 at 760 Torr.
In addition to β-C–H bond scission (3) our calculations show that the 1,6-isomerisation (4) through a seven-membered transition state is a possible competing loss process for the CH3OCH2OCH2O˙ radical. For these two competitive unimolecular (reactions (3) and (4)), ab initio calculations predict dissociation barrier heights of the same order of magnitude (at the G2 and CCSD(T) levels of theory, Table 2). Similar values are returned by RRKM kinetic analysis of experimental results (in Table 3). Therefore, our ab initio calculations give theoretical support to the Wallington et al[2]. assumption suggesting that a competing isomerisation is likely to occur. However, further calculations are needed (for example CASPT2 computations) to confirm this. Actually, due to the existence of spin contamination (see computational details), the computed β-C–H activation energy for reaction (3) is less reliable than the isomerisation one. The predicted relatively low barrier height ≈11 kcal mol−1 for this isomerisation seems to arise from the presence of an oxygen atom next to the CH3 group. This is to link to the observation of Dibble et al[10]. who studied the 1,5 H-shift isomerisation reactions of CH3OCH2CH2O˙ and CH3CH2OCH2O˙ radicals. They observed that the activation barrier for isomerisation is lower by about 10 kcal mol−1 when the oxygen is adjacent to the methyl group from which abstraction occurs than when it is closer to the alkoxy radical center.
Estimation can be made of the competition between the β-C–H scission and the reaction with oxygen (the isomerisation process (4) has also been considered for CH3OCH2OCH2O˙ radical). Assuming kO2 ≈ 10−14 cm3 molecule−1 s−1 (see [ref. 4]), about 45% of HOCH2O˙, 50% of CH3OCH2O˙ and 7% of CH3OCH2OCH2O˙ dissociate, and 9% of CH3OCH2OCH2O˙ undergo isomerisation under conditions of the lower troposphere (760 Torr, 298 K and 20% O2). In the upper troposphere (150 Torr, 220 K and 20% O2), due to the barrier heights and the falloff effect of the unimolecular reactions, dissociations and isomerisation are negligible compared to reaction with oxygen.
Pressure and temperature dependencies of the β-C–H scission rate constant can be presented using the conventional Troe's equation:[23] in order to provide a quick and convenient way of calculating the pressure dependence of the rate constant values in the range (220 K–300 K). The three parameters k0, k∞ and Fc have been derived from the non-linear fitting of the above equation to the presented RRKM results at two temperatures. The resulting expressions of k0(T), k∞(T) and Fc(T) are given in Table 4, along with those corresponding to Fc = 0.6.