Theoretical Study of Radical-Molecule Reactions with Negative Activation Energies in Combustion: Hydroxyl Radical Addition to Alkenes.

Many of the radical-molecule reactions are nonelementary reactions with negative activation energies, which usually proceed through two steps. They exist extensively in the atmospheric chemistry and hydrocarbon fuel combustion, so they are extensively studied both theoretically and experimentally. At the same time, various models, such as a two transition state model, a steady-state model, an equilibrium-state model, and a direct elementary dynamics model are proposed to get the kinetic parameters for the overall reaction. In this paper, a conversion temperature T C1 is defined as the temperature at which the standard molar Gibbs free energy change of the formation of the reaction complex is equal to zero, and it is found that when T ≫ T C1, the direct elementary dynamics model with an inclusion of the tunneling correction of the second step reaction is applicable to calculate the overall reaction rate constants for this kind of reaction system. The reaction class of hydroxyl radical addition to alkenes is chosen as the objects of this study, five reactions are chosen as the representative for the reaction class, and their single-point energies are calculated using the method of CCSD(T)/CBS, and it is shown that the highest conversion temperature for the five reactions is 139.89 K, far below the usual initial low-temperature (550 K) oxidation chemistry of hydrocarbon fuels; therefore, the steady-state approximation method is applicable. All geometry optimizations are performed at the BH&HLYP/6-311+G(d,p) level, and the result shows that the geometric parameters in the reaction centers are conserved; hence, the isodesmic reaction method is applicable to this reaction class. To validate the accuracy of this scheme, a comparison of electronic energy difference at the BH&HLYP/6-311+G(d,p) level and the corrected electronic energy difference with the electronic energy difference at the CCSD(T)/CBS level is performed for the five representative reactions, and it is shown that the maximum absolute deviation of electronic energy difference can be reduced from 2.54 kcal·mol-1 before correction to 0.58 kcal·mol-1 after correction, indicating that the isodesmic reaction method is applicable for the accurate calculation of the kinetic parameters for large-size molecular systems with a negative activation energy reaction. The overall rate constants for 44 reactions of the reaction class of hydroxyl radical addition to alkenes are calculated using the transition-state theory in combination with the isodesmic correction scheme, and high-pressure limit rate rules for the reaction class are developed. In addition, the thermodynamic parameter is calculated and the results indicate that our dynamics model is applicable for our studied reaction class. A chemical kinetic modeling and sensitivity analysis using the calculated kinetic data is performed for the combustion of ethene, and the results indicate the studied reaction is important for the low-to-medium temperature combustion modeling of ethene.

Introduction

Although fuel combustion can bring convenience to us, it can also produce environmental pollution. How to achieve efficient and clean combustion is a major problem that we are facing now. Combustion reaction kinetics, including detailed mechanism construction, mechanism reduction, and combustion modeling, is an effective way to understand the nature of combustion. A reliable detailed reaction mechanism depends on not only its completeness of the reaction network but also the reliability of the thermodynamic and kinetic parameters. At present, the experiments and detailed chemical kinetic models for the high-temperature combustion of hydrocarbon fuels have been extensively studied,[1−4] and good progress has been made.[5] However, there are seldom theoretical or experimental reports on the rate constants for most of the reaction classes, more chemical species and more classes of elementary reactions are involved at low temperatures,[6,7] and the construction of detailed mechanisms for low-temperature combustion is more difficult than that for high-temperature combustion. In a lot of reviews,[1,4,8−11] the role of the reactions involved in low-temperature oxidation and ignition chemistry is highlighted.

Recently, there are a lot of studies on the radical–molecule reactions in the atmospheric and combustion chemistry, such as the addition and abstraction reactions of molecules (alkenes, <span class="Chemical">alcohols, and ethers) with radicals (oxygen and hydroxyl).[12−23] Wilke et al.[24] studied the reaction of C2H5 + O2 and found that the rearrangement of the ethylperoxy radical to form an epoxide passes through a two-step mechanism instead of the usual one-step mechanism. Many of the radical–molecule reactions feature negative activation energies, i.e., their high-pressure limit rate constants decrease with increasing temperature,[25] and it is widely accepted that these reactions occur through two steps: a reversible step that forms the reaction complexes and a subsequent irreversible reaction step that leads to the products.[14,26] In the theoretical study, to get the overall rate constants for these reactions, a variety of computational models have been proposed, such as the direct elementary dynamics model,[26] the equilibrium-state model,[16] the steady-state model,[17,18,26] and the two transition state model.[27,28] These models will be described and discussed in detail in the following computation method section, and the purpose of this study is to systematically identify the temperature range of these models and which model is the most applicable for radical–molecule reactions in combustion occurring at high temperatures (usually above 550 K[29]). In this study, the reaction class of OH addition to alkenes is taken as a test reaction class for radical–molecule reactions. This reaction class in the combustion of hydrocarbon fuels may contain large molecular systems, which provide a challenge in computational chemistry in their accurate electronic energy calculation. Another purpose of this study is to develop an effective method for the accurate calculation of reaction rates for large molecular systems.

Alkenes are of great importance in <span class="Chemical">alkane combustion, particularly in a rich flame where they play a vital role as intermediates in fuel oxidation and combustion and as a critical precursor of polycyclic aromatic hydrocarbons.[29,30] Thus, understanding the kinetics and thermodynamics of alkenes is crucial for combustion chemistry. However, compared to the extensive studies on alkanes, relatively less investigation is performed on the combustion of alkenes, especially at low temperatures. The typical low-temperature oxidation mechanisms of alkenes usually involved more species and elementary reactions, and there are scarce or no experimental data for these reactions. Therefore, theoretical research is more and more important in the study of low-temperature oxidation of alkenes, especially for the establishment of the rate rules for the reaction classes related to the low-temperature combustion mechanisms of alkenes.[31,32] Hydroxyl, as one of the most important radicals in combustion,[33] dominates the chain propagation, chain branching, and chain termination processes at variable temperature regions. Hence, the addition reactions of alkenes + OH are very important in the low-temperature combustion mechanisms, which lead to the acceleration of further decomposition, and are kinds of radical–molecule reactions with negative activation energy. Besides, alkenes eventually decompose into highly toxic aldehydes in the troposphere, and OH addition is the dominant pathway in alkene oxidation. Therefore, the kinetic study of the addition reaction of hydroxyl radicals with alkenes is of great significance in the evaluation of atmospheric pollution.

Moreover, radical addition to alkenes may lead to soot formation. They have been investigated in a large variety of experimental and theoretical studies, and an overview of the theoretical and experimental kinetic studies is presented in Table S1 in the Supporting Information. However, these studies are focused on small <span class="Chemical">alkenes and there are a few studies on the addition of large alkenes. Modeling of the combustion of hydrocarbons usually starts from an automatically generated mechanism that contains a core mechanism involving C0–C2, C0–C4, or C0–C6 chemistry and a mechanism involving large molecules, which are automatically generated by a computer software based on the rate rules for reaction classes. At present, there are no studies on the high-pressure limit rate rules for the addition reaction of OH + alkenes. Therefore, developing rate rules for the addition reaction of OH + alkenes is significant for the automatic mechanism construction for alkenes, which is another purpose of this study. However, to make the rate rules applicable for the addition reaction of OH with large alkenes, the representative set of reactions used for the development of the rate rule must include addition reaction of OH with large alkenes.

For the kinetic studies of large-molecule reaction systems, it is difficult to use high-precision ab initio methods such as the CCSD(T) method or QCISD(T) method. Truong and co-workers[34−41] proposed the reaction class transition-state theory (RC-TST), which is based on the traditional transition-state theory (TST). This theory points out that the electronic energy difference between the primary reaction and target reaction changes little with the ab initio level. In our previous studies,[42−44] an isodesmic reaction method (IRM) was used to interpret RC-TST theory and a correction scheme is used to get accurate electronic energy and rate constants for reaction classes. In this work, this correction scheme will be used to calculate rate constants of the addition reaction of OH with large alkenes for the construction of <span class="Disease">high-pressure limit rate rules.

Results and Discussion

In this study, a set of 44 reactions for the reaction class of hydroxyl radical addition to <span class="Chemical">alkene are studied, and the reactions are listed in Table S2 of the Supporting Information. In the isodesmic correction scheme, reaction R2 is chosen as the primary reaction and other reactions are chosen as target reactions.

Conversion Temperatures

First, the conversion temperatures are calculated for a representative set of five reactions including reactions R1, R2, R4, R6, and R8, where the thermodynamic calculation is carried out at the CCSD(T)/CBS level. The results are listed in Table .

Table 1

Reaction Enthalpy Changes, Reaction Entropy Changes, and the Conversion Temperatures of the Five Representative Reactions

reaction	reaction equation	ΔH1 (kcal·mol–1)	ΔS1 (cal·K−1·mol−1)	TC1 (K)
R1	C=C + OH• → C•COH	–1.94	–18.76	103.41
R2	CC=C + OH• → CC•COH	–2.53	–18.82	134.40
R4	CCC=C + OH• → CCC•COH	–2.65	–24.33	108.92
R6	C(C)C=C + OH• → CC•(C)COH	–3.07	–21.95	139.89
R8	cis-CC=CC + OH• → CC•C(C)OH	–3.03	–24.54	123.49

It can be seen from Table that the conversion temperatures of the five representative reactions are 103.41, 134.40, 108.92, 139.89, and 123.49 K, respectively. The highest conversion temperature for the five reactions is 139.89 K, which is far below the usual initial low-temperature (550 K[29]) oxidation chemistry of the hydrocarbon fuels. So the direct elementary dynamics model with an inclusion of the tunneling correction of the second step reaction is applicable for our studied reaction class.

For the overall radical–molecule reaction, it is a strong exothermic reaction and hence the concentration equilibrium constant decreases with temperature. In this study, a temperature Ti is defined as the inhibition temperature at which the concentration equilibrium constant KC for the overall reaction equals 0.01, and at T > Ti, the reaction can be considered as inhibited and not important in the combustion as the hydrocarbon radical R + O2 reactions are not important in high-temperature combustion modeling.[45−47] The calculated inhibition temperatures for the five representative reactions are given in Table . It can be seen from Table that at a higher temperature of 1000 K, the addition reaction of OH with <span class="Chemical">alkenes is not important. To further validate the results, a sensitivity analysis for an ignition delay time of ethene oxidation at a pressure of 10 atm and at ϕ = 1.0 in air at temperatures ranging from 780 to 1500 K using the AramcoMech 2.0 mechanism[48] has been carried out, and the results (see Section S3 in the Supporting Information) indicate that the reaction of the hydroxyl radical with ethene is important below 1000 K and this reaction is inhibited in ethene combustion above 1000 K.

Table 2

Concentration Equilibrium Constants for the Five Representative Reactions

	KC (T)
reaction equation	298.15 K	500 K	1000 K	1500 K	2000 K	2500 K	3000 K
C=C + OH• → C•COH	1.29 × 1010	8.87 × 102	5.06 × 10–3	1.08 × 10–4	1.77 × 10–5	6.41 × 10–6	3.42 × 10–6
CC=C + OH• → CC•COH	1.26 × 1010	8.58 × 102	4.95 × 10–3	1.07 × 10–4	1.77 × 10–5	6.47 × 10–6	3.47 × 10–6
CCC=C + OH• → CCC•COH	1.81 × 1010	1.21 × 103	6.77 × 10–3	1.45 × 10–4	2.39 × 10–5	8.69 × 10–6	4.66 × 10–6
C(C)C=C + OH• → CC•(C)COH	6.14 × 109	4.76 × 102	2.75 × 10–3	5.93 × 10–5	9.78 × 10–6	3.57 × 10–6	1.91 × 10–6
cis-CC=CC + OH• → CC•C(C)OH	4.00 × 1010	1.36 × 103	5.01 × 10–3	9.10 × 10–5	1.36 × 10–5	4.63 × 10–6	2.37 × 10–6

Conservation of the Reaction Center

In the reaction class of hydroxyl radical addition to <span class="Chemical">alkenes, it is well known that the OH addition can occur on either side of a double bond. In this study, it can be defined that each of the sites corresponds to a double-bonded C atom (Ca) on which OH adds, and its partner carbon in the double bond is Cb, as shown in Figure .

Figure 1

Radical addition reactions of R1R2C=CR3R4 + OH.

The labeling of atoms and bonds involved in the reaction centers is shown in Figure , where d1, d2, and d3 are the bond lengths and A1 and A2 are the bond angles. The optimized geometric parameters of the reaction centers at transition state for the reaction class are listed in Table S3 in the Supporting Information.

Figure 2

Labeling of atoms for the reaction center.

It can be seen from Table S3 that the geometries of the transition states of this reaction class are similar. The maximum average difference (Mad) of bond lengths d1, d2, and d3 are only 9.07 × 10–4, 0.13, and 7.34 × 10–3 nm, respectively. The Mad values of the bond angles A1 and A2 are only 4.33 and 7.53°, respectively. Therefore, the transition-state geometric structures of the addition reaction class are conserved and the isodesmic reaction method can be used to calculate the reaction kinetic parameters.

Relative Electronic Energies

In using the isodesmic correction scheme, reaction R2 is chosen as the primary reaction and the other reactions are chosen as target reactions. To validate the accuracy for electronic energy difference between the transition state of the reaction step 2 and the original reactants using the correction scheme mentioned above, the calculated results at the BH&HLYP/6-311+G(d,p) level and the corrected results using the correction scheme are compared with the CCSD(T)/CBS results in Table . It can be seen from Table that the maximum electronic energy difference at the BH&HLYP/6-311+G(d,p) level is 2.54 kcal·mol–1 and reduced to 0.58 kcal·mol–1 after correction, indicating that the electronic energy difference by a low-level ab initio method can be corrected by the correction scheme to obtain accurate values. Therefore, this correction scheme is applied for the calculation of the electronic energy difference for all target reactions, and the results are reported in Table S4 in the Supporting Information.

Table 3

Comparison of Relative Electronic Energies Using BH&HLYP and the Correction Scheme with the CCSD(T)/CBS Results (Unit, kcal·mol–1)

reactions	reaction equation	CCSD(T)/CBS	DFTa	Δ(DFT)b	IRMc	Δ(IRM)d
R1	C=C + OH• → C•COH	–0.20	1.47	–1.67
R2	CC=C + OH• → CC•COH	–1.29	0.63	–1.96
R4	CCC=C + OH• → CCC•COH	–1.47	1.07	–2.54	–0.89	–0.58
R6	C(C)C=C + OH• → CC•(C)COH	–2.04	0.09	–2.13	–1.88	–0.17
R8	cis-CC=CC+ OH• → CC•C(C)OH	–2.59	–0.25	–2.34	–2.22	–0.38

Calculated at the BH&HLYP/6-311+G(d,p) level of theory.

Difference between the CCSD(T)/CBS value and BH&HLYP value.

BH&HLYP results after validation by the isodesmic reaction method (IRM).

Difference between the CCSD(T)/CBS value and IRM value.

Rate Constants

In this study, the direct elementary dynamics model with an inclusion of the tunneling correction of the second step reaction is applied to calculate the approximate rate constants for our studied reaction class, where the Wigner method is employed to correct the quantum mechanical tunneling effect and accurate rate constants are obtained from the approximate rate constants using the isodesmic correction scheme.

For our studied reaction class of hydroxyl radical addition to <span class="Chemical">alkenes, only a few overall rate constants have been experimentally reported. The comparison of the rate constants between the calculated results in this paper and the experimental values is given in Table .

Table 4

Comparison of the Calculated Rate Constants with Experimental Data at 298.15 K for Alkene + OH Reactions

	k (cm3·mol–1·s–1)
alkenes	this work	selected literature data
CH2=CH2	5.18 × 10–12	8.52 × 10–12,[49] 8.49 × 10–12,[50] 8.2 × 10–12,[51] (7.85 ± 0.79) × 10–12 [52]
CH2=CHCH3	9.14 × 10–12	2.63 × 10–11,[50,53] 2.60 × 10–11,[53] (2.51 ± 0.25) × 10–11,[54] (2.46 ± 0.28) × 10–11 [55]
CH2=CHCH2CH3	4.93 × 10–12	3.13 × 10–11,[50] 2.95 × 10–11,[53] (3.53 ± 0.36) × 10–11,[54] (3.34 ± 0.25) × 10–11 [55]
CH2=C(CH3)2	1.43 × 10–11	5.14 × 10–11 [56]
cis-CH3CH=CHCH3	2.54 × 10–11	5.61 × 10–11 [56]
trans-CH3CH=CHCH3	8.66 × 10–12	6.37 × 10–11 [56]
CH2=CHCH2CH2CH3	5.08 × 10–12	3.13 × 10–11,[50] (3.97 ± 0.38) × 10–11 [55]
(CH3)2C=CHCH3	3.71 × 10–11	8.69 × 10–11 [56]
CH2=CH(CH2)3CH3	1.62 × 10–11	3.68 × 10–11 [50]

It can be seen from Table that our calculated results are lower than the experimental values, showing a systematic error, but the differences are within a factor of 10. Therefore, our kinetic model can provide reliable kinetic data for OH addition to alkenes and <span class="Disease">high-pressure limit rate constants from 298.15 to 3000 K for the 42 target reactions are calculated using the kinetic model; the calculated rate constants for this reaction class at the high-pressure limit and at temperature 298.15–3000 K are shown in Table S5 of the Supporting Information, and they are fitted to the modified Arrhenius equationwhere R is the ideal gas constant; T is the absolute temperature; and A, n, and E are the parameters to be fitted. The fitted (A, n, E) parameters are given in Table S6. The Wigner tunneling correction factors for these reactions are also given in Table S7 in the Supporting Information.

Negative Activation Energy Relationship

The variation trend of the overall rate constants with temperature for the five representative reactions is shown in Figure , from which it can be seen that in the temperature range of 298.15–600 K, the overall rate constants decrease with the increase of temperature, indicating that a negative activation energy relationship exists. In the temperature range of 600–3000 K, the overall rate constants grow with the increase of temperature. Therefore, it reveals that this reaction class only shows a negative temperature effect in the low-temperature section.

Figure 3

Plot of the temperature-dependent overall reaction constant k for five representative reactions.

High-Pressure Limit Rate Rules

For the expansion mechanism of large molecules, the reactions involved are usually classified into subclasses, and the mechanism is automatically generated by the rate rules.[45,57−59] In this paper, the rate rules are obtained by averaging the rate constants of the representative reactions. In the reaction class of hydroxyl radical addition to <span class="Chemical">alkenes, the reactions are divided into subclasses according to the atom type (a primary (p), a secondary (s), or a tertiary (t) atom) of the two carbons of the double bond in the alkenes. Due to symmetry, it is assumed that the first carbon of the double bond is the position of OH addition without the loss of generality. This division leads to nine subclasses of reactions including pp, ps, pt, sp, ss, st, tp, ts, and tt for each configuration of the alkenes. For the ss subclass, the configuration of the alkenes includes cis and trans configurations; therefore, this subclass is further divided into ss(trans) and ss(cis) subclasses. The rate rules for each subclass are derived by averaging the rate constants of a representative set of reactions for each subclass, and the results are given in Table , where the (A, n, E) parameters for the representative reactions are taken from Table S6 of the Supporting Information for these reactions. To measure the uncertainty of the rate rules, at 298.15 K, the ratio k/krule of the rate constant of a reaction in the subclass to the rate constant calculated from the rate rules is calculated, and the results are also listed in Table . It can be seen from Table that the values of k/krule for pp, sp, tp, ss(trans), ss(cis), pt, ps, ts, st, and tt are 1, 0.73–1.22, 0.01–2.98, 0.33–1.38, 0.65–1.70, 0.29–1.37, 0.77–1.43, 0.37–1.41, 0.63–1.22, and 0.40–1.93, respectively, indicating that these factors are below 3.00; hence, the developing rate rules are accurate.

Table 5

Rate Constants, Rate Rules, and Ratios of Rate Constants to the Rate Rules for Alkene + OH Reactions

	modified Arrhenius parameters			298.15 K
reactions	A (cm3·mol–1·s–1)	n	E (kcal·mol–1)	k	k/krule
pp rate rule	1.92 × 10–18	2.03	–7.97	5.18 × 10–12
R1	1.92 × 10–18	2.03	–7.97	5.18 × 10–12	1.0
sp rate rule	2.41 × 10–21	2.75	–11.6	1.64 × 10–12
R3	2.91 × 10–19	2.05	–10.02	2.00 × 10–12	1.22
R5	1.11 × 10–19	2.12	–10.18	1.21 × 10–12	0.73
R11	5.61 × 10–22	3.03	–11.31	1.72 × 10–12	1.05
tp rate rule	4.20 × 10–18	2.06	–10.44	3.66 × 10–11
R7	1.26 × 10–17	2.06	–10.45	1.09 × 10–10	2.98
R15	1.04 × 10–19	2.04	–8.18	3.33 × 10–13	0.01
R29	5.17 × 10–20	2.06	–10.27	4.29 × 10–13	0.01
ss(trans) rate rule	3.47 × 10–22	2.94	–17.07	6.25 × 10–12
R9	1.12 × 10–18	1.84	–13.36	8.66 × 10–12	1.38
R19	3.26 × 10–23	3.36	–13.6	2.06 × 10–12	0.33
R18	2.36 × 10–17	1.44	–11.24	8.04 × 10–12	1.29
ss(cis) rate rule	5.77 × 10–20	2.37	–14.62	1.49 × 10–11
R8	1.59 × 10–17	1.64	–12.16	2.54 × 10–11	1.7
R17	2.55 × 10–18	1.75	–12.79	9.67 × 10–12	0.65
R16	3.33 × 10–21	2.81	–14.26	9.73 × 10–12	0.65
pt rate rule	1.11 × 10–19	2.28	–13.42	1.05 × 10–11
R6	6.27 × 10–20	2.45	–13.15	1.43 × 10–11	1.37
R14	1.24 × 10–16	1.28	–10.79	1.40 × 10–11	1.34
R28	9.28 × 10–21	2.48	–13.58	3.03 × 10–12	0.29
ps rate rule	2.10 × 10–17	1.56	–9.31	6.38 × 10–12
R2	2.91 × 10–19	2.05	–10.02	9.14 × 10–12	1.43
R4	6.45 × 10–17	1.37	–8.5	4.93 × 10–12	0.77
R10	7.24 × 10–17	1.35	–8.66	5.08 × 10–12	0.8
ts rate rule	2.19 × 10–19	2.2	–15.1	2.62 × 10–11
R20	2.07 × 10–17	1.58	–13.35	3.71 × 10–11	1.41
R38	4.76 × 10–20	2.45	–15.78	9.80 × 10–12	0.37
R41	4.76 × 10–20	2.45	–15.78	3.18 × 10–11	1.21
st rate rule	7.97 × 10–21	2.68	–16.71	2.86 × 10–11
R21	1.16 × 10–21	2.98	–17.43	3.29 × 10–11	1.15
R39	5.43 × 10–20	2.41	–16.23	3.48 × 10–11	1.22
R40	3.88 × 10–20	2.41	–15.52	1.81 × 10–11	0.63
tt rate rule	6.20 × 10–20	2.4	–18.7	9.99 × 10–11
R42	3.54 × 10–20	2.39	–19.2	6.74 × 10–11	0.67
R43	3.06 × 10–20	2.39	–18.27	3.99 × 10–11	0.4
R44	1.29 × 10–19	2.39	–18.58	1.92 × 10–10	1.93

Conclusions

In this study, the kinetic models for radical–molecules in combustion chemistry, which exist extensively in the atmosphere and combustion chemistry, are studied with the addition reaction of OH to alkenes being taken as a case study. It is widely accepted that these reactions occur through two ste<span class="Chemical">ps: a reversible step that forms the reaction complexes and a subsequent irreversible reaction step that leads to the products and presents a negative activation energy relationship, and we conclude the following:

The direct elementary dynamic model, the equilibrium-state model, and the steady-state model are equivalent at high temperatures when tunneling correction is ignored, and the direct elementary dynamics model with an inclusion of the tunneling correction of the second step reaction is proposed in this study.

In this study, a temperature TC1 is defined as the conversion temperature at which ΔG1,mθ of the first step reactions equals zero and calculated at the CCSD(T)/CBS level. It is found that the highest conversion temperature for the studied reactions is only 139.89 K, which is far lower than the usual initial low-temperature (550 K) oxidation chemistry of <span class="Chemical">hydrocarbon fuels. Therefore, our dynamics model is applicable for radical–molecule reactions in combustion chemistry.

In addition, a temperature Ti is defined as the inhibition temperature at which the concentration equilibrium constant KC for the addition reaction of OH to alkenes equals 0.01, and it is found that the addition reaction of OH to <span class="Chemical">alkenes is inhibited above 1000 K and therefore is not important in the combustion, which is confirmed through the sensitivity analysis for ethene combustion modeling.

The isodesmic correction scheme is used to calculate the accurate electronic energy difference for the addition reaction of OH to alkene, and the calculated results are close to the available experimental results. Our studies indicate that the reaction class of <span class="Chemical">hydroxyl radical addition to alkenes presents a negative activation energy relationship below 600 K. In addition, the addition reactions of OH to alkenes are divided into ten subclasses and a high-pressure limit rate rule with an uncertainty factor less than 3.00 is developed for each subclass.

Computational Methods

Ab Initio Calculations

All of the electronic structure calculations are done by Gaussian09 software.[60] It is found that the BH&HLYP method is reliable to optimize geometries for the addition reactions of alkenes + OH,[61,62] which are very close to available experimental data.[14,63] In this study, the BH&HLYP method with the 6-311+G(d,p) basis set is employed to perform the frequency calculation and the geometry optimization for all involved species. The zero-point vibrational energy (ZPE) correction calculated at this level is scaled by a factor of 0.9540.[64] In addition, the transition states are confirmed by the intrinsic reaction coordinate (IRC).[65] In the isodesmic correction scheme, the approximate single-point energies are calculated by the low-level ab initio BH&HLYP/6-311+G(d,p) method and the accurate single-point energies are calculated by the high-level ab initio CCSD(T) method with cc-pVXZ (X = D, T, Q) basis sets that are extrapolated to the complete basis set (CBS) limit. The detailed extrapolation schemes are given below.[66−70]

The Hartree–Fock (HF) energy is assumed to approach its CBS limit by power lawswhere X = 2, 3, 4 for D, T, Q extrapolation with the basis set cc-pVXZ and B and a are constants.[66]

The correlation energy is assumed to approach its CBS limit through extrapolation formulas of the fromwhere X = 2, 3 for D, T extrapolation with the basis set cc-pVXZ and A is a constant.[69]

Then, the basis set limit for the total single-point energy is obtained bySignificant errors of thermodynamic parameters can result if the harmonic approximation is used to calculate the partition function for low-frequency modes that correspond to hindered internal rotation.[71] The general quantum mechanical problem of multidimensional internal rotation can be complicated and cumbersome to solve. However, simplifying the multidimensional problem to the product of one-dimensional rotors with simple cosine torsion potential will form a good approximation in many cases.[72,73] During the harmonic vibrational analysis, the internal rotation models are automatically recognized. If any modes are identified as internal rotation, hindered or free, the thermodynamic functions are corrected. Contributions of any very low frequency vibrational modes are listed separately so that their harmonic contributions can be subtracted from the totals. Thus, in this study, the specifying Freq = Hinderedrotor keyword is used in Gaussian09 to perform the low-frequency vibrations corresponding to the torsion of a single bond for the reactants, reaction complexes, transition states, and products, where if any modes are identified as internal rotation, hindered or free, the thermodynamic functions are corrected.[74]

In this study, the values of ⟨Ŝ2⟩ for all open-shell species calculated at the UBH&HLYP/6-311+G(d,p) level are very close to the exact doublet value of 0.75 for radicals; therefore, the spin contamination is not a problem for our studied reaction class. In addition, T1 diagnostic values for all stationary points along reaction paths are calculated to assess the suitability of the CCSD(T) method and whether the multireference effects can be ignored. In general, the multireference wave function is meaningful only when the T1 diagnostic value of closed-shell species is greater than 0.02.[75] In addition, several research studies[76−78] have shown that the T1 diagnostic value of the open-shell system of up to almost 0.045 is also acceptable. The T1 diagnostic values are similar for different molecules with the same type of reactions; thus, the T1 diagnostic values for some representative species in this work only are given in Table S8 of the Supporting Information. It can be seen that the values of these open-shell species are all lower than 0.045; therefore, in this study, the multireference effects for all species can be ignored and the single-reference (CCSD(T)/CBS) method is reasonable.

Kinetic Model for Radical–Molecule Reactions

As described in Section , it is widely accepted that the radical–molecule reactions proceed via a two-step complex mechanism as followswhere R, RC, and P are reactants, reaction complexes, and products, respectively.

A typical electronic energy profile for the radical–molecule reactions is shown in Figure , where the electronic energy of the transition state for the second step is lower than the electronic energy of the original reactants.

Figure 4

Electronic energy profiles along the reaction coordinates for the radical–molecule reactions.

In the two transition state model,[79] it is assumed that there are two transition states in the potential energy surface along the reaction coordinate corresponding to the two steps of the reactions: the outer transition state, which is corresponding to the first barrierless step and is determined variationally in the neighborhood of the reaction complex, and the inner transition state, which is corresponding to the second step of the reaction and is located in the neighborhood of the saddle point or at the energy barrier. Using the two transition state model, a general expression for the overall rate constant can be obtained[25]where Qreactants is the canonical partition function for the reactants including relative translation, kB is the Boltzmann constant, h is the Planck constant, T is the temperature, E is the total energy, and J is the total angular momentum. Neff†(E,J) is the effective flux through both transition stateswhere Nouter† and Ninner† are the E- and J-resolved transition state number of states at the outer and inner transition states, respectively.

The two transition state model has been successfully used in reactions of OH radicals with volatile organic compounds.[80,81] It is shown that the outer transition state provides the dominant bottleneck at low temperatures and the inner transition state provides the dominant bottleneck at higher temperatures and at intermediate temperatures, and that the bottleneck in the flux through both transition states will contribute to determining the overall rate constant. It is concluded that two transition state model is required at low temperatures, such as atmospheric temperatures.[82] However, in the two transition state model, the outer transition state is determined in a variational way; hence, its location depends on the temperature, which makes the application of the two transition state model complicated.[83]

If there are canonical rate constants for the two steps of the radical–molecule reactions, the reaction schemes can be expressed aswhere k1 and k–1 are the forward and reverse rate constants for the first step and k2 is the rate constant for the second step.

In the equilibrium-state model, it is assumed that the reaction complex is in equilibrium with separated reactantsThe overall rate constants, which are controlled by the second reaction step, can be obtained[84]According to the relationship between the equilibrium constant and Gibbs free energy changeand traditional transition-state theory for k2where ΔG2‡ is the Gibbs free energy difference between the transition states and the reaction complex and κ2 is the tunneling correction factor for the second reaction step. Finally, the overall rate constant in equilibrium-state model can be expressed asHowever, the equilibrium-state model can only be applied when the condition of equilibrium-state approximation satisfies, i.e., both the forward reaction and the backward reaction of the first step are fast relative to the second reaction step.

In the direct elementary dynamics model,[26,85] it is assumed that the reaction complex can be ignored and the overall reaction can be treated as an elementary reactionwhere κ is the tunneling correction factor for the overall reaction, which is a problem or meaningless because the electronic energy of the transition state is lower than the electronic energy of the reactants for radical–molecule reaction, as shown in Figure , and in this case, the tunneling correction can be ignored.

It is evident that the overall rate constant expressions for the equilibrium-state model and the direct elementary dynamics model are equivalent except for the tunneling factors, and hence, the equilibrium-state model and direct elementary dynamics model are equivalent when ignoring the tunneling correction, which may cause significant discrepancies between the calculated rate constants and the experimental results.[26] Therefore, the equilibrium-state model is more accurate than the direct elementary dynamics model and it is used in this study and the direct elementary dynamics model is not considered.

According to eq , the net rate of the reaction complex RC satisfies the following equationInitially, [R] ≠ 0, [RC] = 0, and the net rate will decrease with time t. If (k–1 + k2) ≫ k1, i.e., the consuming process of the reaction complex is much faster than the forming process of the reaction complex; then, in a very short time, the net rate will become zero, i.e., the reaction complex RC reaches a steady state

The steady-state model leads to a rate constant for the overall reaction, which can be written as[86]The steady-state model has been applied to predict the temperature-dependent rate coefficients for reactions involving the van der Waals complex at high temperatures.[87−89]

The first step of the radical–molecule reactions is a barrierless reaction, and the rate constants are usually calculated with the variational transition-state theory (VTST), in which the transition is determined variationally along the free energy profile. The activation entropy ΔS1 is negative for the first step of the radical–molecule reactions because three translational plus three rotational are converted into six vibrational degrees of freedom. The Gibbs free energy of the reaction complex is less than that of the reactant along the reaction coordinates for the radical–molecule reactions at low temperatures. However, as temperatures increases, the Gibbs free energy of the reaction complex is higher than that of the reactants.[44] The Gibbs free energy profiles for the radical–molecule reaction can be shown as follows.

The equilibrium-state model is a method based on the hypothesis that both the forward reaction and the backward reaction of the first step are fast relative to the second reaction step, i.e., k1, k–1 ≫ k2. As shown in Figure , for the radical–molecule reaction, k–1 is less than k2 at low temperatures and hence the equilibrium-state model is not applicable. With the temperature increasing, the Gibbs free energy of the reaction complex increases, making k–1 increase and k–1 ≫ k2 and hence the equilibrium-state model is applicable at high temperatures.

Figure 5

Gibbs free energy profiles along the reaction coordinates for the radical–molecule reactions at (a) low temperatures and (b) high temperatures.

In the steady-state model,[12] it is assumed that the concentration of the reaction complex as an active intermediate remains constant during the reaction and the application condition of the the steady-state model is that the consuming process of the reaction complex is much faster than the forming process of the reaction complex.[86] For radical–molecule reactions, when (k–1 + k2) ≫ k1, the steady-state approximation model is applicable. However, as shown in Figure , at low temperatures, k1 > k–1 and k1 > k2; hence, the steady-state model is not applicable at low temperatures. With the temperature increasing, the Gibbs free energy of the reaction complex increase, making k–1 increase and k1 decrease, and hence, the steady-state model is applicable at high temperatures. Furthermore, as discussed in the papers of Shojaie et al.[90] and Shannon et al.,[13][13]k–1 is considered to be much larger than k2 at high temperatures. In this case, eq for the steady-state model can be reduced aswhich is equivalent to the equation of the equilibrium-state model. In eq , Keq can be expressed aswhere QRC and QR are partition functions corresponding to the reaction complex and the original reactants, respectively. Under the high-pressure limit, in a unimolecular process, an equilibrium distribution of reactants is maintained, and the TST approach can be applied to calculate k2 (the second step) aswhere κ2 is the Wigner tunneling coefficient of the second step and QTS is the transition-state partition function. Thus, the overall rate constant iswhere ΔE‡ is the difference of the electronic energy between the transition state of the reaction step 2 and original reactants. Comparing eq with eq , it can be seen that eq is equivalent to the rate constant of the direct elementary dynamics model with an inclusion of the tunneling correction of the second step reaction and this equation will be used throughout this study. Finally, the problem is to determine the temperature at which the equilibrium-state model and the steady-state model satisfy their approximate conditions and therefore eq can be used to calculate the overall rate constant for radical–molecule reactions.

According to the thermodynamic theory, the standard molar Gibbs free energy change ΔG1,mθ for the formation reaction of the reaction complex can be expressed aswhere ΔH1,mθ is the standard enthalpy change, ΔS1,mθ is the standard entropy change, and K1θ is the standard equilibrium constant between the original reactants and the reaction complex. The standard equilibrium constant K1θ can be written aswhere Pθ is the standard atmospheric pressure, R is the ideal gas constant, and Kc,1 is the concentration equilibrium constant between the original reactants and the reaction complex. The expression for the concentration equilibrium constant Kc,1 can also be written asThe enthalpy change and the entropy change are negative in the formation of the reaction complex for radical–molecule reactions, and therefore, from eq , the entropy term is not dominant if T is low and hence the standard molar Gibbs free energy ΔG1,mθ is less than zero at low temperatures, which leads to a k1 value that is larger than k–1. With the increase of temperature, ΔG1,mθ is increasing to zero and the corresponding temperature is defined as the conversion temperature TC1 in this studyWhen the temperature is much higher than TC1, ΔG1,mθ is much greater than zero, which leads to that k–1 are larger than k1 and k1 and k–1 are larger than k2 at higher temperatures, and hence, the steady-state model and the equilibrium-state model are applicable for the radical–molecule reactions when the temperature is much higher than TC1.

Isodesmic Reaction Correction Scheme

In our previous studies,[42−44] an isodesmic reaction method was proposed to interpret RC-TST theory[35−37,91−93] and a correction scheme based on the isodesmic reaction method was proposed to get accurate rate constants for reaction classes at a relatively low ab initio level. A small reaction in the reaction class is usually chosen as a primary reaction RP, and it is assumed that its accurate electronic energy difference obtained from the high-level ab initio method is ΔEP‡′ and its approximate electronic energy difference obtained from the low-level ab initio method is ΔEP‡ and their difference iswhich can be used as the correction to calibrate the electronic energy difference ΔET‡ calculated at the low-level ab initio method for any target reactionwhere ΔET‡′ is the corrected electronic energy difference. Using the above isodesmic correction scheme, accurate electronic energy difference between the transition state of the reaction step 2 and original reactants for our studied radical–molecule reaction class. The isodesmic correction scheme only involves the low-level ab initio calculation for the target reactions, and hence, it is applicable to large-molecule systems. Moreover, the accurate rate constant kT′ for the target reaction can be calculated by correcting the approximate rate constant kT calculated at a low ab initio level using the following expression