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Ab Initio Study of Hydrogen Migration in 1-Alkylperoxy Radicals Alexander C. Davis* and Joseph S. Francisco* Department of Chemistry and Department of Earth and Atmospheric Science Purdue UniVersity West Lafayette, Indiana 47907-1393, United States ReceiVed: May 10, 2010; ReVised Manuscript ReceiVed: September 14, 2010
Alkylperoxy and hydroperoxyalkyl radicals are key reactive intermediates in hydrocarbon oxidation mechanisms. An understanding of the interconversion of these two species via a hydrogen migration reaction is of fundamental importance to the prediction of chain branching reactions and end product composition. An extensive ab initio investigation of the hydrogen migration reaction in 1-ethyl, 1-propyl, 1-butyl, 1-pentyl, and 1-hexylperoxy radicals is conducted to assess the validity of using cycloalkanes to model the ring strain of their transition states as well as the effect of both location of the migrating hydrogen and directionality of the remaining alkyl chain in the transition state of the reaction involving a secondary hydrogen. The G2 and CBS-Q composite methods are used to determine the activation energy and enthalpy of reaction relative to the alkylperoxy radical. Both methods show good agreement with five experimentally determined reaction enthalpies, having root mean squared deviations of 0.7 and 1.3 kcal mol-1 for the CBS-Q and G2 methods, respectively. The effect of hydrogen abstraction site and transition state geometry, particularly axial and equatorial geometries of the remaining alkyl chain, on the activation energy, Arrhenius A-factor, tunneling, and rate coefficient are discussed. Differences between terminal adjacent and nonterminal adjacent secondary sites result in small but consistent differences in barrier height. Failure of key assumptions within the cycloalkane based estimation method leads to the break down in the accuracy for both small and large transition states. For large transition states, the breakdown of these assumptions also results in the failure of the current cycloalkane method as a conceptual model. Of great interest is the observed alteration in the preferred H-migration from the 1,5 to the 1,6 H-migration within the temperature region where these reactions are particularly important to the combustion mechanism. I. Introduction With the growing interest in improving the efficiency of combustion systems and the movement toward alternative fuels, a detailed understanding of the combustion mechanism is needed. Under low temperature and moderate pressure conditions, similar to those present in the troposphere, the reaction between alkyl radicals (R•) and O2 leads to the formation of relatively stable alkylperoxy radicals (RO2•) (rxn 1, Figure 1).1 At more moderate temperatures (>500 K), such as those found in combustion systems, RO2• becomes less stable and can react unimolecularly to form bimolecular products, the primary of which are olefins and HO2 with OH and cyclic ethers as minor products.1-3 As the temperature increases further, reaction 1 (Figure 1) begins to favor the reactants resulting in an inverse temperature dependence for the formation of the end products. However, because neither the forward nor reverse reaction are chain termination steps, and the R• continues to react with O2 to reform RO2•.1,3 At temperatures above 900 K, unimolecular reactions involving the alkyl radical begin to dominate the oxidation mechanism.4,5 It is in the more moderate temperature range (∼500 to 900 K) that RO2• and the hydroperoxy alkyl radicals (•QOOH) lead to significant chain branching in the hydrocarbon oxidation mechanism. •QOOH are formed through unimolecular hydrogen abstraction reactions, commonly referred to as 1,n H-migrations, where n is the location along RO2• where the hydrogens are abstracted (rxn 3 in Figure 1). In their n-heptane and iso-octane oxidation modeling studies, Curran * To whom correspondence should be addressed. E-mail: davisac@ purdue.edu (A. C. D.),
[email protected] (J. S. F.).
Figure 1. Some of the important reactions that are present in the hydrocarbon oxidization mechanism. The reaction in red is the focus of this study.
and co-workers identify 25 different classes of reactions, 11 of which contain either RO2• or •QOOH.4,5 As a result, H-migration reactions play a central role in the overall oxidation mechanism of alkyl radicals (Figure 1).1,4,5 To better characterize the combustion mechanism under moderate to high temperatures, the ethyl radical + O2 is often used as a prototype for all R• + O2 reactions.1,2,6,7 One of the primary advantages of this model system is that the resulting ethylperoxy radical is the smallest alkylperoxy radical that can form an olefin, allowing for the determination of the dominant pathway. At the same time, ethylperoxy is unable to undergo the larger, more competitive, 1,n H-migration (n g 5) and is instead limited to the 1,3 and 1,4 H-migration reactions, which contain more strained transition states. However, as Taatjes points out, it is for this very reason that the ethyl radical may
10.1021/jp1042393 2010 American Chemical Society Published on Web 09/30/2010
Hydrogen Migration in 1-Alkylperoxy Radicals not be the best prototype for these reactions.1 Because the 1,5 and larger H-migration reactions have less ring strain associated with the formation of their transition states, other RO2• will contain multiple competing reaction pathways that are unavailable to ethylperoxy radicals. The majority of experimental and theoretical work on RO2• has focused on determining the mechanism for olefin and HO2 formation. Prior to the work of Walker and co-workers, the 1,4 H-migration was believed to be responsible for these products (rxn 7 in Figure 1).8-10 Several key characteristics of the overall mechanism were reported by Slagle et al., and later confirmed by Kaiser, that contradicted the perceived importance of this pathway.11-13 First, the rate of olefin formation decreases with increasing temperature, indicating that the barrier for olefin formation lies below the entrance channel of reaction 1 (Figure 1). Second, there is a decreasing pressure dependence with increasing temperature, which lead Slagle et al. to propose that the reaction occurred via a coupled mechanism with two alternative reaction paths that utilize a common reactive intermediate.11 On the basis of these results and their own prior work, Walker and co-workers proposed a novel pathway for the formation of an olefin and HO2 from RO2• involving a concerted elimination mechanism (rxn 2, Figure 1).9,10 High level computational studies by Schaefer and co-workers, using ethyl radical and oxygen as a model system, determined that the concerted elimination path has a barrier height approximately 3 kcal mol-1 below the reactant, R + O2, energy and that the corresponding 1,4 H-migration path lies 2.3-9.9 kcal mol-1 above.6,8 Miller and co-workers also provided an in-depth quantitative analysis of the concerted elimination pathway over the low, middle and high temperature regimes.2,14 There is now little debate over the validity of the 1,4 H-migration versus concerted elimination for the formation of the observed olefin and HO2. However, the 1,4 and other H-migration reactions are still important in the overall combustion mechanism of hydrocarbons. In an early n-hexane combustion study, Jones and Fenske reported the formation of significant quantities of cyclic ethers in addition to olefins between 580 and 930 K.15 These results highlight the importance of not only the concerted elimination reaction, but also the H-migration reactions that can lead to the formation of cyclic ethers via reaction 9 (Figure 1). Much of the empirical data on the larger RO2• H-migration reactions comes from the research group of Baldwin and Walker.9,10,16-23 However, because these studies were conducted prior to the elucidation of the concerted elimination pathway, their reported 1,4 H-migration reaction rates are invalid for this process. More recently, propyl and larger alkyl radical + O2 reactions have been investigated by Kaiser, Wagner et al., and Taatjes and co-workers.1,7,12,13,24-27 Unfortunately, due to the reactivity of the radical species involved in the hydrocarbon oxidation mechanism, experimentally determined values for elementary reaction must be inferred from the relative concentrations of the end products. Because cyclic ethers are formed in a separate reaction that occurs after the H-migration reaction, using their relative concentrations to determine the rates of H-migration reactions relies on the assumption that the barrier height for reaction 9 (Figure 1) is significantly lower than reaction 3 (Figure 1). Taatjes and co-workers have partially overcome this problem by combining experimental photolysis and computational methods to study ethylperoxy and propylperoxy radicals.1,7,26,27 In addition to highlighting the importance of the concerted elimination reaction, they also found that propylperoxy can
J. Phys. Chem. A, Vol. 114, No. 43, 2010 11493 undergo a 1,5 H-migration reaction with a barrier height that is significantly below the R• + O2 entrance channel.1,7 Other studies have investigated H-migration reactions for a broader set of alkylperoxy radicals using computational methods alone. Chan et al. studied the H-migration reactions involving a primary and radical adjacent secondary abstraction site in the ethylperoxy through pentylperoxy radicals.28 Pfaendtner et al. looked at 1,5 and 1,6 reactions involving a primary, terminal adjacent secondary, and tertiary abstraction site in methylated and unmethylated alkyl-2-peroxy radicals.29 Recently, Sharma et al. reported on a series of 1,3 through 1,7 H-migrations involving primary, terminal adjacent secondary, and tertiary abstraction sites in alkylperoxy and hydroperoxyalkylperoxy radicals.30 Following their formation, •QOOH can undergo several different reactions (rxns 7-10, Figure 1) depending on the location of the abstracted hydrogen. Sharma et al., Huynh et al. and Asatryan and Bozzelli looked at not only the RO2• H-migration, but also H-migrations in •O2QOOH.30-32 Cullis et al., while studying the oxidation of n-hexane and focusing on the detection of cyclic ethers, observed the formation of oxiranes, oxetanes, and tetrahydrofurans. Baldwin et al. investigated, through the addition of HO2 to ethene and propene, the reactions that 1-hydroperoxy-2-alkyl radicals, the product of a 1,4 H-migration, would undergo.10 They observed the formation of ethylene oxide and 1,2 propylene oxide (rxn 9, Figure 1). Wijaya et al. expanded on these effort using computational methods to study on the formation of cyclic ethers starting from • QOOH.33 They found that the activation energy for the formation of cyclic ethers from •QOOH are between 21.9 and 13.0 kcal mol-1, making them lower than the entrance channel for R• + O2, except 1,3 propylene oxide, which is slightly higher. One of the goals of studying the alkyl radical oxidation mechanism is the development of computer simulations that can predict end product concentration. With the large number of reactions present in the combustion and decomposition pathways of alkyl radicals, the determination of every possible reactant, product, and transition state through either experimental or computational means is impractical. As a result many of the current computer models rely on a group additivity value (GAV) scheme, originally described by Benson, to aid in the determination of unknown parameters.4,5,34 This method uses the thermodynamic properties of the moieties present within a molecule or transition state to estimate the overall thermodynamic properties of a molecule or reaction. On the basis of this concept, Benson and Fish suggested that the activation energy, Ea, for RO2• H-migration reaction can be estimated through the summation of the reaction barrier for a bimolecular hydrogen abstraction between an alkane and a RO2•, Eabs, and the ring strain of an (n + 1)-cycloalkane, Estrain, where n is the sum of carbon and oxygen atoms in the ring portion of the H-migration transition state.34,35
Ea ) Eabs + Estrain
(1)
When it was originally developed, the cycloalkane model system allowed for the estimation of parameters that had not yet been experimentally determined. Another advantage of this model is that it allows for easy visualization of the transition state geometries. However, soon after it was proposed Baker et al. clearly demonstrated that the cycloalkane model was flawed, particularly in the relative heights of the 1,5 and 1,6 H-migration reactions.17 By using the sum of the bimolecular abstraction and cycloalkane ring strain energies to predict the activation energy of the H-migration reactions, two significant underlying
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TABLE 1: Comparison of Experimental ∆H°rxn at 298 K with Values Determined Using the Two Computational Methods Used in This Study (kcal mol-1) ∆H°rxn experimental57
∆H°rxn CBS-Q
∆H°rxn G2
-35.2 ( 2.3
-35.3
-37.8
-22.3 ( 2.3
-21.8
-21.5
CH3CH2OO• T CH3C•HOOH
13.2 ( 2.3
11.9
12.6
CH3CH2OO• T •CH2CH2OOH
17.8 ( 2.5
18.2
18.3
-20.2 ( 0.4
-20.7
-19.2
0.7
1.3
reactions •
•
CH3C H2 + O2 T CH3CH2OO •
CH3CH2OO T CH2CH2 +
HO•2
CH3CHdCH2 + CH3C•H2 T CH3(CH2)3C•H2 rms deviation from experimental
assumptions are made: first that the migrating hydrogen traverses the same path along the potential energy surface as a hydrogen in a bimolecular abstraction reaction; and second that the Estrain in the transition states is approximately the same as the modeled cycloalkane system. Another implication of using cycloalkanes as models for the estimation of kinetic parameters is that for H-migrations with nonplanar transition states, there should exist a difference in barrier height between axial and equatorial arrangement, based on 1,3 diaxial interactions. In a recent study, Huynh et al., while investigating the oxidation of propane, reported on a difference in both Arrhenius pre-exponential factors (A-factor) and activation energy for the isomerization of CH3CH(OO•)CH3 leading to C•H2CH(OOH)CH3 based on the directionality of the methyl group relative to the ring.31 Because the kinetic parameters are often expressed on a per hydrogen basis, there may also exist a difference in both activation energy and A-factor between the each hydrogen on a secondary abstraction site; however, to date no study has reported on this phenomena. A more recent estimation method has been developed by Truong and co-workers called reaction class transition-state theory (RC-TST).36,37 Instead of relying on the ring strain of a cycloalkane and barrier height for a bimolecular H-abstraction, RC-TST uses reactions, within the same class (e.g., 1,4 H-migrations), involving the smallest system to predict the kinetic and thermodynamic parameters for novel reactions. This system, therefore, relies on the assumption that the simplest form of a reaction can accurately predict the reactions of larger, more computationally demanding, molecules.36,37 To assess the validity of each of these estimation methods for H-migration reactions, the present work investigates all possible hydrogen migration pathways for the 1-ethyl, 1-propyl, 1-butyl, 1-pentyl, and 1-hexylperoxy radicals. This work also attempts to determine the impact of methyl and peroxy substituents, on the R- and β- carbon, on radical site stability and the effect that they, as well as axial and equatorial geometries, have on activation energies, A-factor, tunneling coefficients, and reaction rates. II. Computational Methodology Hydrogen migration reactions are often described in one of two methods: Either as an a,b H-migration, where a and b are the locations of the radical on the reactant and product side, respectively; or the method described by Hardwidge, which uses an Ncd format, where N is the number of atoms in the ring of the transition state, including the hydrogen, and c and d are the types of radical sites (p ) primary, s ) secondary, and t ) tertiary) for the product and reactant, respectively.38 Following these methods, a 1,5 H-migration in a pentylperoxy radical would be described as a 6sp transition. This paper will utilize both to highlight different aspects of the isomerization reactions.
However, the secondary sites using Hardwidge’s method will be sublabeled as either sR, or sβ, depending on the location of the secondary radical site relative to the end of the alkyl chain. The symbol sR will be used for secondary sites that are adjacent to the terminal carbon, and sβ for those that are two or more carbon groups away from a terminus. Also, the notation of axial or equatorial geometries for the remaining chain will be included through an A or E superscript, respectively, when deemed relevant. Thus the above pentylperoxy radical 1,5 H-migration reaction with an equatorial geometry will be indicated as 6sβEp. All calculations are performed using the Gaussian 03 suite of programs.39 Geometries and frequencies are determined using the CBS-Q and G2 composite methods.40-42 These composite methods are selected for their high accuracies, having reported standard deviations of 1.1 and 1.0 kcal mol-1, respectively.40 However, because these reported values include reactions that are dissimilar to those investigated here, five reaction enthalpies (∆H°rxn), with radical species similar to those investigated, are used to assess the reliability of these composite methods. The resulting values are compared to experimentally determined literature values (Table 1). Due to the computational cost of both methods, the largest system investigated is the hexylperoxy radical which can undergo the 1,3 through 1,8 H-migrations. The geometries for the transition states are selected based on the Benson model, which suggests that both the shape and ring strain energy of the transition state can be predicted by an (n + 1)-cycloalkane, where n is the number of carbon and oxygen atoms in the ring structure.34,35 For systems with more than one ring conformation (i.e., cyclohexane through cyclononane), the lowest energy conformation was used: the chair conformation of cyclohexane and the lowest energy conformations of cycloheptane through cyclononane, based on the findings of Anconi et al., Franco et al., and Wilberg.43-45 In each case, the carbon that serves as a model for the migrating hydrogen is selected based on an attempt to reduce any gauche interactions within the transition state. The peroxy group then replaced the next two carbons in the ring. The resulting transition state structures are consistent with those reported by Sharma et al.30 Transition states are confirmed by the presence of a single negative frequency and IRC calculations at MP2/6-31G(d). Because both the CBS-Q and G2 methods determine low level HF/6-31G(d) frequencies, additional geometry and frequency calculations are carried out using MP2/6-31G(d) for the reactants, products, and transition state species. The MP2/631G(d) geometry optimization is selected for its similarity to both the MP2(full)/6-31G(d) and the MP2/6-31G(d‡‡), where ‡‡ denotes the use of polarization functions taken from 6-311G** basis set, optimizations used by G2 and CBS-Q, respectively.41 Tunneling correction factors, κ(T), are calculated using the MP2/ 6-31G(d) frequencies and both the CBS-Q and G2 energies for temperatures from 200 to 2500 K, using the Wigner, and Skodje
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and Truhlar (S&T) methods.46-48 The Wigner method is the simpler and more commonly used method:
k(T) ) 1 -
2
(2)
where νs is the imaginary frequency of the transition state, T is the temperature, and h and KB are Planck’s and Boltzmann’s constants, respectively. However, because tunneling is highly dependent on both the barrier height and width, more rigorous methods are needed, particularly at lower temperatures.47 S&T represents an improvement over the Wigner method:46,47,49
k(T) )
βπ/R β e[(β-R)(V)] for β < R sin(βπ/R) R-β
(3a) and
k(T) )
β [e[(β-R)(V)] - 1] for β > R β-R
(3b)
where
R)
2π hVs
β)
1 kBT
and
V ) Ea - ∆Hrxn for ∆Hrxn > 0 V ) E2 for ∆Hrxn e 0 Ea and ∆Hrxn are the activation energy and enthalpy of reaction, respectively. The A-factor is calculated using the MP2/6-31G(d) frequencies and rotational constants:
A(T) )
∏ i)1
( )
1 hVs 24 KBT
n
Qvib )
kBT Q‡ × h Qr
(4)
Where Q‡ and Qr are the total partition functions for the transition state and reactant, respectively. These functions can be further broken down into four components: rotational (Qrot), vibrational (Qvib), electronic (Qelec), and translational (Qtrans). The product of these four terms results in the total partition function (5):
Q ) QrotQvibQelecQtrans
(5)
For a unimolecular reaction the reactant Qelec and Qtrans are roughly equivalent to their transition state counterparts, thus they do not need to be determined due to the ratio of Q‡ and Qr in eq 4. Qvib and Qrot are calculated using eq 6 and 7 respectively.
Qrot ) π1/2 ×
1 1-e
hVi/
(6) kBT
( ) ( ) ( ) kBT hA
1/2
×
kBT hB
1/2
×
kBT hC
1/2
(7)
Where n is the number of vibrational modes, νi are the harmonic frequencies (s-1); and A, B, and C are the rotational constants for the molecule. Frequencies and vibrational modes are listed in Table S1 in the Supporting Information. III. Results and Discussion A. Calibration of Methods for Alkylperoxy Radical Systems. The computational methods used in this study were selected based on their reliability. Curtiss et al. and Ochterski et al. evaluated the accuracy of the G2 and CBS-Q methods against 125 experimental values, resulting in mean absolute deviations of 1.0 and 1.1 kcal mol-1 for the G2 and CBS-Q, respectively.40-42 Although these deviations are comparable to experimental uncertainties, both studies include reactions that are unrelated to those presented here. To better assess the reliability of these methods for their use with •QOOH and RO2•, five reaction enthalpies, whose enthalpy of formation values have been experimentally determined, are used as test cases (Table 1).43 Both methods demonstrate good agreement with the empirical values, having root mean squared (rms) deviations of 0.7 and 1.3 kcal mol-1 for CBS-Q and G2, respectively. The large deviation observed in the first calibration reaction (Table 1) for G2 is most likely due to a known 2.4 kcal mol-1 error in the enthalpy of formation for O2, specifically.50 This particular error inherent in the G2 composite method is not observed in the enthalpy of formation values of CH3• or H2O2 in the same study and is therefore not expected to significantly impact the other values reported in this study, with the exception of the reaction enthalpies for the R• + O2 in Table 2. The remaining reaction enthalpy values either fall within or are very close to the experimental uncertainty for each of the test reactions (Table 1). In fact, if the first test reaction is removed from consideration, the rms deviations become 0.8 and 0.7 kcal mol-1 for CBS-Q and G2, respectively. B. Alkylperoxy and Hydroperoxyalkyl Radicals. The first step in the formation of n-alkylperoxy radicals involves the reaction between an n-alkyl radical and oxygen (rxn 1 in Figure 1). The exothermic reaction produces a vibrationally excited RO2•; Table 2 lists the enthalpy of reaction values determined using both the CBS-Q and G2 methods. The CBS-Q method yields values in reasonable agreement with the experimentally determined values for the formation of methylperoxy. The predicted CBS-Q enthalpy of formation for ethylperoxy radicals shows excellent agreement with both the experimental and computational values of Schaefer and co-workers, and the HL2 values of DeSain et al.6-8 However, as the alkyl chain lengthens, the binding energy continues to increase by between 0.6 and 0.8 kcal mol-1 for each additional carbon, except the hexylperoxy radical, which has an enthalpy of reaction that is 0.5 kcal mol-1 less than pentylperoxy. This alkyl group size dependence of the CBS-Q data set is also observed in the activation energies and leads to significant differences between the values for the two methods. The G2 reaction enthalpies are generally ∼3 kcal mol-1 greater than the experimental values; these large deviations in the G2 reaction enthalpy are likely the result of the known 2.4 kcal mol-1 error in the G2 methods determination of the enthalpy of formation for O2, as described in the previous
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TABLE 2: Enthalpy of Reaction Values for the •R + O2 T ROO• Reaction at 298 K (kcal mol-1)
•R + O2 T ROO• •CH3 + O2 T CH3OO• •C2H5 + O2 T C2H5OO• •C3H7 + O2 T C3H7OO• •C4H9 + O2 T C4H9OO• •C5H11 + O2 T C5H11OO• •C6H13 + O2 T C6H13OO• a
experimental -30.3 ( 1.2
computational
-35.2 ( 2.357
CBS-Q
G2
-32.1
-34.8
32.5,7 a 34.2,6 34.9,7 b 35.98
-34.9
-37.8
33.27 a
-35.4
-37.9
7a
-36.1
-37.9
-36.7
-38.0
-35.8
-38.0
57
33.2
DeSain’s HL1 values. b DeSain’s HL2 values.
section.50 Unlike the CBS-Q values, the G2 reaction enthalpies, for ethylperoxy through hexylperoxy, they are all within 0.2 kcal mol-1 of each other, ranging between -37.8 and -38.0 kcal mol-1. This trend in the G2 data set suggests that the use of ethyl radical + O2 as a prototype for larger systems may be valid for the 1,3 and 1,4 H-migrations and for the concerted elimination reactions. Prior to collisional stabilization, the RO2• radicals, larger than ethylperoxy, should contain roughly 35 kcal mol-1 of internal energy for CBS-Q and G2 data sets, if the size dependence of the CBS-Q and 2.4 kcal mol-1 known error associated with O2 for G2 are taken into account. This value is important when considering the viability of subsequent reactions. Often the thermodynamic parameters for reactions that proceed from RO2• are reported in relation to the entrance channel of reaction 1 (Figure 1), however, due to the known error in the G2 method for O2 and hence R• + O2 reaction enthalpy, the values presented here will be in reference to RO2•. Following reaction 1 (Figure 1), the vibrationally excited RO2• can undergo the concerted elimination of HO2 to produce an
olefin (rxn 2 in Figure 1), isomerize to form a •QOOH (rxn 3 in Figure 1), or they can release energy through collisional stabilization (rxn 4 in Figure 1). For both the G2 and CBS-Q data sets, a clear pattern in the reaction enthalpies is observed between RO2• and the location of the radical site along the • QOOH (Figures 2 and S1). For both data sets, the RO2• form is the most stable. Moving the radical site location to the C1 carbon results in a reaction enthalpy of 12.9 and 13.1 kcal mol-1 for the CBS-Q and G2 hydroperoxymethyl radical, respectively. Elongation of the alkyl chain reduces the endothermicity of this process to between 11.4 and 11.9 kcal mol-1 for CBS-Q and between 12.2 and 12.3 kcal mol-1 for the G2 data sets. This ∼1 kcal mol-1 decrease is significantly less than the 2-4 kcal mol-1 difference in activation energy between an Npp and Nsp H-migration observed in the CBS-Q and G2 data sets, as well as in previous studies.30 Although the enthalpies of reaction values for reaction 1 (Figure 1) differ between G2 and CBS-Q by up to 3 kcal mol-1 (Table 2) the reaction enthalpies for the
Figure 2. A plot of the G2 enthalpy of reaction for the H-migrations for each alkylperoxy radical (at 0 K), and the labeling scheme used for the discussion of abstraction sites.
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TABLE 3: Thermochemical Data for Transition States of a Hydrogen Atom Migration Across Alkylperoxy Radical (kcal mol-1) at 0 Ka migration type alkylperoxy radical
1,n
Nab
methyl peroxy ethyl peroxy propyl peroxy butyl peroxy pentyl peroxy hexyl peroxy ethyl peroxy propyl peroxyE butyl peroxyE pentyl peroxyE hexyl peroxyE propyl peroxyA butyl peroxyA pentyl peroxyA hexyl peroxyA propyl peroxy butyl peroxyE pentyl peroxyE hexyl peroxyE butyl peroxyA pentyl peroxyA hexyl peroxyA butyl peroxy pentyl peroxyE hexyl peroxyE pentyl peroxyA hexyl peroxyA pentyl peroxy hexyl peroxyE hexyl peroxyA hexyl peroxy
1,3 1,3 1,3 1,3 1,3 1,3 1,4 1,4 1,4 1,4 1,4 1,4 1,4 1,4 1,4 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,6 1,6 1,6 1,6 1,6 1,7 1,7 1,7 1,8
4pp 4sRp 4sβp 4sβp 4sβp 4sβp 5pp 5sREp 5sβEp 5sβEp 5sβEp 5sRAp 5sβAp 5sβAp 5sβAp 6pp 6sREp 6sβEp 6sβEp 6sRAp 6sβAp 6sβAp 7pp 7sREp 7sβEp 7sRAp 7sβAp 8pp 8sREp 8sRAp 9pp
∆H‡ CBS-Q
∆H°rxn CBS-Q
42.6 42.3 42.3 42.5 42.6 37.4 33.4 33.8 33.2 32.8 33.3 33.2 32.3 31.4 24.2 21.6 21.6 20.0 21.7 21.7 20.3 24.5 21.2 20.1 21.8 20.7 25.6 21.7 21.2 26.3
12.9 11.4 11.6 11.6 11.9 11.8 17.6 14.5 15.4 15.1 14.8 14.5 15.4 15.1 14.8 16.4 14.1 14.8 13.5 14.1 14.8 13.5 16.5 13.5 13.7 13.5 13.7 16.6 13.4 13.4 16.8
∆H‡ G2
∆H°rxn G2
43.6 43.3 43.2 43.2 43.2 38.4 34.4 34.3 34.3 34.2 34.2 33.9 33.9 33.8 25.6 23.0 22.7 22.6 22.9 22.4 22.3 24.9 21.7 21.4 22.0 21.5 25.3 22.2 22.0 26.1
13.1 12.2 12.3 12.3 12.3 12.3 17.7 15.1 15.4 15.4 15.4 15.1 15.4 15.4 15.4 16.6 14.2 14.5 14.5 14.2 14.5 14.5 16.8 14.2 14.5 14.2 14.5 16.7 14.1 14.1 16.6
∆H‡ rms 1.0 1.0 0.9 0.8 0.6 1.0 1.0 0.5 1.1 1.4 0.8 0.7 1.6 2.4 1.4 1.5 1.1 2.6 1.2 0.7 2.0 0.4 0.5 1.4 0.1 0.8 0.3 0.5 0.7 0.2
∆H‡ literature computational
experimental
cycloalkane model
42.330 c 41.8,8 39.730 c 39.730 c
38.2,7 37.4,8 34.230 c 34.9,1 30.1,30 c 32.87 d 33.07
34.320 d
23.6,34,58 b 29.7,4 29.45 20.6,34,58 b 27.9,4 26.95
25.2,7 21.030 c 22.3,7 18.230 c
23.5
23.9,7 20.030 c 16.630 c
21.5,17 23.423 19.223
23.6,34,58 b 21.1,4 22.45 20.6,34,58 b 19.4,4 19.15
19.930 c 18.230 c
19.623
27.1,34,58 b 23.9,4 25.65 24.1,34,58 b 22.2,4 22.15
17
17.1,34,58 b 23.9,4 24.45 14.5,34,58 b 22.2,4 20.95
a
Energies are relative to the primary alkylperoxy radical for each molecule. b Benson’s cycloalkane estimation model with abstraction values from Carstensen et al. c Calculation at 298 K. d Concerted elimination.
H-migration reactions (rxn 3 in Figure 1) are in much better agreement (Table 3). For both the CBS-Q and G2 data sets, the enthalpy of reaction (rxn 3 in Figure 1) depends on the location of the radical site relative to both the peroxy group and terminal carbon, and they appear to be affected by the presence of substituents on either the same carbon or the adjacent one. For example, there is a 3.0 and 3.1 kcal mol-1 increase in reaction enthalpy going from the C1 to C2 sites for the CBS-Q and G2, respectively (Figure 2). When going from a C2 to C3 radical site (R to β from the peroxy group), there is a 1.3 and 0.9 kcal mol-1 decrease for the CBS-Q and G2, respectively. The presence of an alkyl group on the abstraction site appears to have a similar effect to the peroxy group, although lower and in the opposite direction. A terminal adjacent secondary site, Cn-1, has a reaction enthalpy that is 2.4 and 2.5 kcal mol-1 lower than a primary radical site Cn, for the CBS-Q and G2, respectively. And going from the Cn-1 to Cn-2 site results in a consistent 0.3 and 0.4 kcal mol-1 increase. The effect of the location of both the peroxy group and terminal carbon relative to the abstraction site is limited to two carbons away. This suggests that for other hydrocarbon radical systems, substituents on either the R or β site will affect the reaction enthalpies that produce those radicals and that the sign of the effect will be opposite for the R and β position. C. Transition States. In order to assess the reliability of the computational methods for predicting kinetic parameters, several predicted rate coefficients over a wide range of temperatures are compared to existing literature values (Figures 3A-E). The predicted rates, which are a combination of the A-factor,
tunneling transmission coefficient, and activation energy, also show excellent agreement with available experimental values. The large deviation observed in Figure 3A is the result of the experimental value being determined prior to the elucidation of the dominant, concerted elimination path.23 The 1,4 Hmigration rate reported by Baldwin et al. and Curran et al., who used these experimental values to help develop their combustion model, should therefore be interpreted as rates for the concerted elimination path that gives better agreement with the relative rate of olefin and HO2 formation, previously attributed to the 1,4 H-migration.4,20 As a result, one would expect the rate of the 1,4 H-migration reaction to be significantly lower than the experimental value.4 Although not determined here, the recent study by Wilke et al. demonstrated the reliability of both the G2 and CBS-Q methods for determining the concerted elimination barrier, reporting deviations of less than 0.2 kcal mol-1, compared to their high level calculations for the ethylperoxy radical.6 The ability of the computational methods used in this study to accurately reproduce experimental findings suggests that they should also produce reliable predictions for those systems that either do not yet have experimental values or that cannot be directly determined experimentally. Effect of Transition State Ring Expansion. The cycloalkane model predicts that the activation energy of the H-migration reaction will decrease with the expansion of the transition state ring for the 1,3 through 1,5 H-migration reactions, after which the activation energy should begin to increase, with the 1,6 H-migration being ∼6.5 kcal/mol above the 1,5 barrier. The values obtained for both the G2 and CBS-Q data set (Table 3)
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Figure 3. (A) A comparison of the alkylperoxy 5pp H-migration reaction rate coefficients calculated in this study and those reported in other studies. Black lines and symbols represent experimentally derived values. (B) A comparison of the alkylperoxy 6sp H-migration reaction rate coefficients calculated in this study and those reported in other studies. Black lines and symbols represent experimentally derived values. (C) A comparison of the alkylperoxy 7pp H-migration reaction rate coefficients calculated in this study and those reported in other studies. Black lines and symbols represent experimentally derived values. (D) A comparison of the alkylperoxy 7sp H-migration reaction rate coefficients calculated in this study and those reported in other studies. Black lines and symbols represent experimentally derived values. (E) A comparison of the alkylperoxy 8pp H-migration reaction rate coefficients calculated in this study and those reported in other studies. Black lines and symbols represent experimentally derived values.
do decrease when going from the 1,3 to 1,4 H-migration and again from the 1,4 to 1,5. However, the 1,6 H-migration activation energies are not higher than the 1,5. For the CBS-Q data the 1,5 and 1,6 H-migration reactions are roughly equivalent for both the Npp and NsRp. The 1.0-1.5 kcal mol-1 reduction going from the 6sβp to 7sβp is most likely be due to the previously described size dependence of the CBS-Q values. In
the case of the G2 values, the 1,6 H-migration barriers are 0.7, 0.9, and 1.3 kcal mol-1 lower than the 1,5 barriers for the Npp, NsAp, and NsEp, respectively. This result, of the 1,6 barrier being lower than the 1,5 H-migrations, when comparing similar abstraction sites (i.e., primary vs secondary), is also seen in the reported data of DeSain et al. and Sharma et al., although neither study discuss it.7,30 On the other hand, both the experimental
Hydrogen Migration in 1-Alkylperoxy Radicals work of Baldwin, Walker, and co-workers and the combustion modeling work of Curran et al. both specifically mention that, contrary to the cycloalkane model, the 1,6 H-migration reactions have a lower barrier than the 1,5 H-migration reaction, however neither group describes the underlying reason for this trend.4,5,17,23 After the 1,6 H-migration, further increases in transition state ring size lead to higher activation energies, although the rate of increase is less than predicted by the cycloalkane model, with the 8pp barrier being 0.4 and 0.9 kcal mol-1 higher than the 7pp for the G2 and CBS-Q results, respectively, instead of the predicted 3.5 kcal mol-1. Both the CBS-Q and G2 activation energies demonstrate excellent agreement with the higher level calculations of Taatjes and co-workers and Rienstra-Kiracofe et al., as well as the experimental work of Baldwin, Walker, and co-workers (Table 3).1,7,8,17,23 The general trends in barrier height reported here are consistent with the recent work of Sharma et al.30 However, their results, which use a similar composite method, CBS-QB3, yield values that are consistently 3-4 kcal mol-1 lower than the CBS-Q and 4-5 kcal mol-1 lower than the G2. Although some of this discrepancy is attributable to differences in the reported temperature for the calculation, 0 K versus theirs at 298 K, the discrepancy is still significant at 298 with differences of 2-4 kcal mol-1 and 3-4 kcal mol-1 for CBS-Q and G2, respectively (Table S6 of the Supporting Information). The values reported here are in better agreement with the majority of both previous experimental and computational studies (Table 3). Within their study they did compare the barrier heights of the CBS-QB3 and G2 for the 4pp and 5pp H-migration reactions and reported a 2.5 and 2.2 kcal mol-1 difference, respectively. Although better than the 3-5 kcal mol-1 observed in Table 2, 2.2 kcal is still relatively high. Within the activation energies reported here there is a significant difference between the CBS-Q and G2 results, which tend to increase with molecule size, with the largest differences occurring in the H-migrations for pentyl and hexylperoxy radicals. This difference is attributed to an error within the CBS-Q method, which when it was developed was intended for systems with less than six non-hydrogen atoms.41 As mentioned above, the ∆Hrxn values for CBS-Q also demonstrate a strong size dependence, resulting in increasing discrepancies between the G2 and CBS-Q values with increasing molecule size. The validity of the cycloalkane model for estimating the activation energy of peroxyalkyl radicals relies on two significant assumptions. First, that the ring strain energy of the transition state is well described by an (n + 1)-cycloalkane, where n is the location of the abstracted hydrogen. Second, the potential energy surface and path of the migrating hydrogen are very similar to the bimolecular abstraction of a hydrogen from an alkane, by an alkylperoxy radical.34,35 The problem is that both assumptions contradict each other. Looking only at the O-H-C angle formed by the terminal oxygen on the peroxy group, the migrating hydrogen, and the carbon of the abstraction site, Aβ (See Figure S2 and Tables S8B and S10B), the problem becomes apparent. The reaction path of the hydrogen in the bimolecular H-abstraction between the chemically similar alkane and alkyl radical occur via a ∼180 °C-H-C angle.51 However, in the cycloalkane model, in order for the hydrogen to take the place of a methyl group within the ring, it would need to adopt an angle of between 90° for the 1,4 and 118° for the 1,8 H-migration. By constraining the movement of the migrating hydrogen to a narrower angle, the hydrogen is forced to traverse a different, higher energy path along the potential energy surface described by the bimolecular reaction. At the same time, if the
J. Phys. Chem. A, Vol. 114, No. 43, 2010 11499 hydrogen moves along a 180° angle then the ring structure would need to deviate from the cycloalkane model, increasing the ring strain component of the transition state. For the H-migration reactions, as the transition state ring increases in size, thereby increasing the flexibility of the ring structure, Aβ increases rapidly. For the 1,5 H-migration the values is ∼150°, which implies that there is still sufficient strain within the transition state to affect the path of the migrating hydrogen. Further expansion of the transition state to the 1,6 H-migration increased Aβ to 160°, at which point the position of the migrating hydrogen begins to more closely resemble the middle point of a bond in an n-cycloalkane rather than a methyl position in an (n + 1)-cycloalkane. Furthermore, the dihedral angles within the transition state structure of a 1,6 H-migration reaction resemble the “strain free” cyclohexane (See Supporting Information for a more in-depth comparison of the transition state structures to the cycloalkane model system). The notion of the Eabs (eq 1) component of the overall activation energy reducing as the flexibility in the ring structure increases, allowing the hydrogen to move along a wider angle, also helps to explain why the observed increase in activation energy with increasing rings size following the 1,6 H-migration is less than expected. The increase in Estrain is partially compensated by reductions in Eabs, resulting in lower than expected alterations in the activation energy. Effect of Chain Substituents on ActiWation Energy. It has been well established that there is a 2-4 kcal mol-1 reduction in activation energy going from an abstraction off of a primary site to a secondary site.4,30,35 The majority of computational studies that look at H-migration reaction for multiple alkylperoxy radicals use a limited number of reactions for each system. They focus on the difference between primary, secondary, and tertiary abstraction sites, but only for the NsRp reactions and tertiary sites where the methyl group is adjacent to a terminal carbon, even though based on the group additivity model, substituents that are β to a H-abstraction site, for bimolecular reactions, have an impact on ∆Hrxn.52 Cohen, while investigating hydrogen abstraction reactions, suggested that although abstraction reaction are commonly classified as primary, secondary, or tertiary, depending on the abstraction site, there are likely differences in C-H bond strengths between systems that will lead to differences between primary, secondary and tertiary sites, resulting in different activation energies for these reactions.53 This same effect holds true for the H-migration reactions. Evidence for this can be seen in both previous computational and experimental results, where such a comparison is possible. Within the 1,4s data set in DeSain et al., the activation energy for n-propyl is 0.2 kcal mol-1 higher than n-butyl.7 It is interesting to note that there is also a 0.5 kcal mol-1 reduction for the concerted elimination, between n-propyl and n-butyl.7 Also, in the experimental work presented in Baldwin et al. there is a ∼1 kcal mol-1 difference between the butyl-2-peroxy and pentyl-2-peroxy 1,4 H-migration activation energies.23 It should be noted that in the recent work by Sharma et al. the 1,3 H-migration activation energies for ethylperoxy and propylperoxy are both 39.7 kcal mol-1. However, when more methyl groups are added to the carbon β to the abstraction site a reduction of 0.1 and 0.4 kcal mol-1 is observed for one and two additional methyl groups, respectively.30 Within both the G2 and CBS-Q data set there is between a 2.6 and 4.0 kcal mol-1 reduction in activation energy going from an Npp to Nsp H-migration. In each case the reduction in activation energy is larger than the reduction in ∆Hrxn for the corresponding process, with both alterations representing a
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reduction in energy. However, the alteration in activation energy between the NsRp and Nsβp is opposite in sign to change in ∆Hrxn. For each pairing of NsRp and Nsβp the activation energy reduces by between 0.1 and 0.5 kcal mol-1 for the G2 data set, the pattern is less clear for the CBS-Q values. Due to the observed size dependence of the CBS-Q calculations described above, it is unclear if the elongation of the alkyl chain has an impact on the activation energy. This size dependence is absent from the G2 data set. Effect of an Axial Ws Equatorial Nsp Transition State Geometry. Given that cycloalkanes are commonly use to describe the transition state of H-migrations, it is reasonable to expect that for Nsp reactions the position of the remaining alkyl group relative to the ring, as either axial or equatorial, may influence the activation energy of the transition state. In a recent study Huynh et al. noted a 0.7 kcal mol-1 difference in Ea for the 1,4 H-migration reaction in CH3CH(OO•)CH3, resulting in C•H2CH(OOH)CH3, on based on the directionality of the other methyl substituent on the peroxy carbon.31 However, because they were looking at a 5pp reaction there would be no difference on a per hydrogen basis at the abstraction site. Although the effects of axial versus equatorial geometries are indiscernible experimentally, because both pathways result in the same end product, and it is through end product yields that rates are derived, computational studies are sensitive to the transition state geometry and will need to address the effect these systems have on the overall rate. In the transition state, the abstraction of a secondary hydrogen results in the formation of a temporary chiral center with the migration of each hydrogen at the abstraction site forming a different diastereomer. For nonplanar rings, the different configurations should have different ring strains, which would be exacerbated by the presence of an additional substituent on the alkylperoxy radical. This effect can be partially evaluated through the interaction between the hydrogen atoms within the ring and the remaining alkyl chain. Since the 1,3 diaxial interaction in methylcyclohexane is 1.8 kcal mol-1, any effect is expected to be less than this value.54 For the 5sp, 7sp, and 8sp groups both the G2 and CBS-Q display similar results, though different in magnitude. For the 1,4 H-migration reactions the 5sAp have an activation energy that is higher than the 5sEp by between 0.1 and 1.4 kcal mol-1 for CBS-Q and between 0.2 and 0.4 kcal mol-1 for G2. In the case of the 1,5 reactions the G2 and CBS-Q show similar magnitudes, but they report a different directionality for the change in activation energy between the 6sAp and 6sEp configurations. For the G2 data set, the 6sAp has a barrier that is between 0.1 and 0.3 kcal mol-1 lower than the 6sEp, while for CBS-Q the 6sAp is higher by between 0.1 and 0.3 kcal mol-1. For the two remaining systems, 7sp and 8sp, both methods agree on the directionality of the change. For the 1,6 H-migration reactions the 7sAp has a barrier that is 0.6 and between 0.1 and 0.3 kcal mol-1 higher than the 7sEp, for CBS-Q and G2 respectively. Finally, the 8sAp is 0.2 and 0.5 kcal mol-1 lower than the 8sEp, for G2 and CBS-Q, respectively. Due to the magnitude of the 1,3 diaxial interaction in methylcyclohexane, the difference between the two configurations of the 6sp are expected to be especially large, meaning that both methods should at least agree on the sign of the energy difference. This, combined with the fact that the 6sEp dihedral angle that describes the directionality of the remaining alkyl chain relative to the ring, D3, should have a value of ∼180° but instead they are -144° and +143° for G2 and CBS-Q, respectively (Tables S8C and S10C in the Supporting Information), further supports the breakdown of the (n + 1)-cycloalkane
Davis and Francisco model described previously. Interestingly, the angle describing the directionality of the chain in the 7sEp, D4, should be approximately -150° but is instead ∼173° for both G2 and CBSQ, which agrees with an n-cycloalkane model. In general, the axial geometries, as expected, have activation energies that are between 0.1 and 0.4, and between 0.1 and 1.4 kcal mol-1 higher than the equatorial geometries for the G2 and CBS-Q, respectively. At 1000 K these differences mean that the NsAp will have a rate that is between 5 and 15% and between 5 and 50% lower than the NsEp for G2 and CBS-Q, respectively. The difference in ranges between the G2 and CBS-Q are again attributable to the observed size dependence of the CBS-Q method. It is interesting that the CBS-Q and G2 methods disagree on the directionality of the energy difference between the axial and equatorial geometries. Arrhenius Pre-Exponential A-Factor. As the size of the transition state increases additional hindered rotors are introduced into the system as part of the expanding ring. As a result, the entropy of the transition state, which is inversely proportional to the A-factor, increases. Figure 4A and B show the Npp and Nsp A-factors (See Table S2 for numerical values). As expected, for both groups the A-factor decreases significantly between each successive increase in ring size. However, because both the Qvib and Qrot partition functions that are used to obtain the A-factor are temperature dependent, the A-factors also exhibit temperature dependence. Table 4 shows the rate of decrease in A-factor with each increase in ring size. At 400 K the A-factor decreases by 10-0.30 or 50% for each new methyl group added to the ring; this rate decline increases to 10-0.34 (54%) at 800 K. For the Nsp reactions the rate of reduction in A-factor is higher, 10-0.35 (55%) at 400 K, and increases further to 10-0.41 (61%) by 900 K. Although the difference in activation energy between the NsAp and NsEp was relatively low, a clear, larger trend emerges in the A-factors. Starting with the 1,4 H-migration, the first nonplanar transition state, there is a observable splitting between the equatorial and axial values. This separation becomes more pronounced as the transition state ring expands and, for the larger transition states, can lead to over a 40% reduction in rate. Huynh et al. demonstrate a similar 22% decrease in A-factor between an axial and equatorial methyl substituent on the peroxy carbon.31 There is also a much less dramatic split between the NsRp and Nsβp, which is practically negligible in all but the 1,5 H-migrations. Within the 1,5 H-migration data set the divergence between both the R and β and between the axial and equatorial configurations is observed (Figure 4C). Although the splitting between axial and equatorial is more important, there is still a 12% difference between the 6sRAp and 6sβAp. Because the rate of decrease in A-factor with increasing ring size alters with temperature, H-migration abstraction site, and the geometry of the transition states, and the use of a single constant value for the reductions in the A-factor is not recommended for combustion modeling studies that are intended to predict rates over a wide range of temperatures. The Effect of Tunneling. Tunneling is an essential component in the determination of reaction rates, in the temperature range where alkylperoxy radical H-migration reactions are important (1000 K). However, at 800 K, S&T predicts a tunneling coefficient twice that of the Wigner method for the 5sp reactions, and by 600 K the difference is over an order of magnitude. For the larger transition states the agreement is better but still troubling: 25% difference at 800 K and two times larger at 600 K. Figure 5C shows the tunneling coefficients for each of the H-migrations available to the hexylperoxy radical at 600 K. The results for S&T using the energies of both G2 and CBS-Q show excellent agreement with each other as well as a reasonable agreement with the Wigner method for the 6sp and larger reactions. Both the S&T and Wigner produce the same general trend, although it is more pronounced in the S&T values, with the 1,5 H-migration being the least affected followed closely by the 1,6 through 1,8 H-migrations. There is also some splitting of the transmission coefficient, that can be
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Figure 5. (A) Plot of transmission coefficient for the alkylperoxy Npp reactions using the Wigner tunneling method. (B) Plot of transmission coefficient for the alkylperoxy Npp reactions using the S&T tunneling method for G2 data set. (C) Comparison of transmission coefficient for the octylperoxy H-migration reactions at 600 K. S&T represents the use of the Skodje and Truhlar tunneling method and the/G2 and/CBS-Q signify the use of G2 and CBS-Q energy values in the tunneling calculation.
ascribed to the differences in barrier height for the axial and equatorial geometries, for the CBS-Q 5sp and both the CBS-Q and G2 8sp values. The large differences in predicted transmission coefficient between the Wigner and S&T for the 4sp and 5sp reactions suggests that either the S&T method overestimates tunneling for large barriers, or that because the Wigner method does not take into account the height of the barrier it should not be used on systems with large barriers. In their recent study on H-migrations in n-hexyl and n-heptyl radicals, Sirjean et al. reported on the S&T, Wigner, and Eckart barriers for the 1,4 through 1,6 H-migrations.47 Their results showed good agreement between the Eckart and S&T methods, which at 600 K had nearly identical values for the four H-migrations studied. Below 600 K the two methods diverge, with the S&T increasing rapidly with decreasing temperature. This means that the large deviation observed between the S&T and Wigner methods for the 4sp and 5sp values is likely due to the importance of the height of the barrier on tunneling. However, in order to better assess the role tunneling plays in the 1,3 and 1,4 H-migrations more elaborate treatments are required. Rate Coefficients. As demonstrated by the calibration calculations, the methods used in this study predict rate coefficients that are in excellent agreement with the available experimental data. The only large deviation is for the 1,4 H-migration reactions, which are the result of the experimental values utilizing
HO2 and olefin formation as indicators of 1,4 H-migrations instead of concerted elimination reactions. Figures 6A-D. show the impact of tunneling, location of the abstracted hydrogen, and the directionality of the remaining alkyl chain on the rate coefficient. By comparing Figures 6A and B it becomes clear that the choice of tunneling calculation method has a dramatic effect on the rate coefficient and the curvature of the Arrhenius plots, particularly at the lower temperatures (
