Theoretical Investigation of Rate Rules for H-Intermigration Reactions for Cyclic Alkylperoxy Radicals

Abstract

As a starting channel, the H-intermigration reaction of alkylperoxy radicals (ROO radicals) that yields hydroperoxyl alkyl radicals (QOOH radicals) determines the low-temperature chemistry of alkanes. In this work, this type of reaction was investigated for typical cyclic alkanes, which are important fuel components and soot precursors, using theoretical ab initio methods. First, all the molecular geometries and vibrational frequencies were computed using the density functional theory method and the single point energies were refined using the post-Hartree fork method (M062X/6-311G(d,p)//DLPNO-CCSD(T)/CBS). Then, high-pressure limit rate constants were evaluated with tight transition state theory, with which tunneling effects were considered using the Eckart model and low-frequency torsion modes were modeled as hindered rotors. Pressure-dependent rate constants were also calculated for typical reaction channels. Rate expressions in the Arrhenius form for 91 reactions are proposed. All reactions were categorized into seven reaction types and the rate rule for each reaction type was estimated with uncertainty factors of three to six. These rules can be potentially used in the development of low-temperature kinetic mechanisms for cycloalkanes. A comparison between different reaction types was also performed and the favorable channels are discussed.

1. Introduction

Chemical kinetic mechanisms are currently the most widely used and effective tool for the description of the complicated oxidation characteristics of hydrocarbons. They play an essential role in our understanding of chemical reactive flows, especially transient combustion phenomenon, such as ignition, flame instability, and local extinction [1,2]. Although many quite reliable mechanisms have been developed for simple species and representative hydrocarbon fuels through decades of research [3,4,5], there are still many fuel components with unreliable oxidation models or which even remain unstudied. Usually, these species have large numbers of atoms and complex molecular structures, making it difficult to build kinetic models for them. Fortunately, hierarchical theory and analogy theory provide great help in constructing such mechanisms [6,7]. It has been proposed that the mechanisms employed by complicated hydrocarbons comply with sub-models of smaller intermediate species and radicals, and the rate constants of unknown elementary reactions are similar to those of reactions of the same type. Hence, two aspects are essential significance in the determination of mechanisms. On the one hand, the possible reaction pathways need to be analyzed. On the other hand, the reaction types for which reliable rate constants have been acquired must be categorized. Rate coefficients for reaction types are referred to as rate rules. Thanks to rate rules, it is possible to easily obtain reliable rate constants for new reactions without time-consuming ab initio computation. Additionally, rate rules also make great contributions to automatic optimizations of large-scale mechanisms [8,9,10].

Currently, over 30 reaction types have been proposed by various researchers, which can be roughly divided into high-temperature and low-temperature chemistry [6,7]. Recently, the low-temperature chemistry of hydrocarbons has received increasing attention because it is believed to be of great significance for the development of low-temperature combustion techniques, such as homogeneous charge compression ignition (HCCI), reactivity-controlled compression ignition (RCCI), and partially premixed compression ignition (PCCI) [11,12,13]. These are believed to be promising, clean, efficient combustion technologies. Unlike traditional combustion modes, which are mainly controlled by transport and fluid dynamics, these new combustion strategies are influenced by the low-temperature combustion chemistry of fuels.

Additionally, as is widely known, the low-temperature chemistry of saturated alkanes is much more complicated than the high-temperature chemistry. Figure 1 shows the main reaction pathways at low temperatures for alkanes [14,15]. Initiated with O₂ or through decomposition, alkanes decompose to alkyl radicals, which then undergo O₂ addition to form alkylperoxy radicals (ROO radicals). Then, the ROO radicals undergo isomerization to yield hydroperoxylalkyl (QOOH) radicals. Dissociation of QOOH radicals produces cyclic ethers, regenerates active hydroxyl radicals, and initiates secondary O₂ addition reactions. The active hydroxyl radical delivered by the QOOH radicals plays an important role in enhancing the low-temperature oxidation of hydrocarbons. As the first reaction step, ROO radicals’ transformation into QOOH radicals becomes the starting channel for the initiation of the low-temperature pathways.

In fact, many researchers have conducted theoretical studies on the rate rules of the alkyl plus O₂ system, including regarding the reaction type of H-intermigration of ROO to QOOH radicals. Miyoshi [16] investigated rate rules for the unimolecular isomerization and decomposition reactions of alkylperoxy (RO₂), hydroperoxyalkyl (QOOH), and hydroperoxyalkylperoxy (O2QOOH) radicals using a CBS-QB3 quantum chemical method. They found that the alkyl groups had effects on the rate constants and attempted to propose general principles for the effects. However, there are also cases in which the principles are not strictly obeyed and tabulated rate expression should be used, as they declared. Villano et al. [17,18] selected typical C3–C7 alkanes and calculated rate constants for alkyl plus O₂ systems using the CBS-QB3 method. They presented high-pressure rate rules for dissociation reactions, concerted elimination reactions of alkylperoxy radicals based on carbon sites, and isomerization reactions based on H-transfer types. Then, they also discussed rate constants for the unimolecular decomposition of QOOH radicals, including isomerization to RO₂, formation of cyclic ether, and the important β-scission. Pressure dependences were also discussed, and the authors concluded that the high-pressure rate rules could be directly used to describe the chemistry of RO₂ and QOOH. It can be observed that the rate rules of Villano et al. and Miyoshi et al. are very consistent. There are also studies on typical alkane fuels. Ye et al. [19] computed the potential energy surfaces and rate constants of iso-pentyl radical plus O₂ systems at the B3LYP/6-311++G(d,p)//QCISD(T)/CBS level. They pointed out that the 1,5 H-transfer reactions play significant roles in RO₂ = QOOH channels. The rate constants for the formation of ROO radicals from pentyl plus O₂ were found to agree well with the rate rule for the O₂ addition reaction described by Villano et al., while those for the H-transfer reactions described by Ye et al. showed discrepancies with those described by Villano et al.

In addition to chain alkanes, cycloalkanes have also received some attention regarding their low-temperature O₂ addition pathways. Cycloalkanes are important fuel components of jet fuels, diesel fuel, and new biofuels. They are believed to be important precursors of soot formation, especially at low temperatures [20,21,22]. Understanding the low-temperature chemistry of cycloalkanes can help in producing low-temperature designs for biofuels, jet fuels, and diesel fuels. It can also contribute to determining a soot formation mechanism in order for better soot control. Recently, Xing et al. [20,21] calculated the potential energy surfaces for methylcyclohexyl isomers plus O₂ and provided pressure-dependent rate constants for major decomposition channels. They pointed out that the rate rules developed for chain alkyl radicals probably do not apply to cycloalkanes. As H-transfer reactions are of great importance for O₂ addition systems, Yao et al. [22] chose typical alkylated cyclohexyl peroxy radicals to determine the rate rules. H-migration reaction types in the side chain, from cycle to chain, from chain to cycle, and in the cycle were investigated. In a comparison to the rate rules for chain alkyl radicals described by Villano et al. [17], they declared that the rate rules are not suitable for cycloalkanes. However, it was found that the rate constants provided by Xing et al. [20] and the rate rules described by Yao et al. [22] have clear discrepancies.

Due to the inadequate and inconsistent rate constants provided in the previous studies, theoretical investigation of the H-transfer reactions of cycloalkylperoxy radicals in the formation of hydroperoxylcycloalkyl was here performed. Based on the potential energy surfaces for typical cycloalkylperoxy radicals and the corresponding rate constants, the features of the reaction types and rate rules are discussed in detail.

2. Computational Method

All the electronic geometries of stable species and transition states related to cyclic H-transfer reactions were optimized at the M062X/6-311G(d,p) level. The M062X method proposed by Trular et al. [23,24,25,26] is a commonly used density functional theory (DFT) method and has been proved to be suitable for optimizing the structures of C/H/O molecules at the small to middle scale. Frequencies were also computed at the same level for all optimized geometries. Transition states were recognized as those molecules with a single imaginary frequency. Although DFT methods are considered convenient for geometry optimization, post-HF methods are usually employed to refine single point energies. Coupled cluster methods, such as CCSD(T) and QCISD(T), are desired for energy refinement [19,20,21,27,28], but they are very computationally time-consuming for large molecules. Alternatively, Liakos et al. [29] proposed a composite method called DLPNO-CCSD(T), which has been tested and found to be able to refine energy with comparable accuracy while simultaneously saving much computation time [30,31]. Additionally, basis sets of cc-pVDZ and cc-pVTZ are often used in energy refinement. Hence, single point energies of all stationary points are refined at the DLPNO-CCSD(T)/cc-pVDZ and DLPNO-CCSD(T)/cc-pVTZ levels in this work. The values of the T1 diagnostic for the reactants and transition states listed in Tables S1 and S2 in the Supplementary Material were smaller than 0.025 for all species, suggesting that the single reference method was reliable [19,30]. Then, as with other couple cluster methods, the energies were extrapolated to the complete basis set (CBS) using the following equation [32]:

$$E_{CBS}=E_{TZ}+\left(E_{TZ}-E_{DZ}\right)\ast 0.4628$$

(1)

where E_DZ and E_TZ represent the energies computed with the basis sets of cc-pVDZ and cc-pVTZ, respectively, and E_CBS denotes the extrapolated energy. The geometry optimization and frequency calculation were carried out using the open source suite PSI4 [33] and the DLNPO-CCSD(T) computation was conducted using the ORCA 4.0.0 package [34].

The rate constants at high-pressure limit were computed using transition state theory for all reactions with tight transition states [35]. The equation is as follows:

$$k^{TST}\left(T\right)=\mathsf{\kappa }\left(T\right)ℒ\frac{k_{B}T}{h}\frac{Q_{TS}\left(T\right)}{Q_R\left(T\right)}\exp \left[-\frac{E^{†}}{RT}\right]$$

(2)

In Equation (2), $ℒ$ represents the degeneracy of the reaction channel. $\mathsf{\kappa }\left(T\right)$ is the tunneling effect coefficient estimated with the Eckart model [36]. k_B and h represent the Boltzmann constant and Plank constant, respectively. $E^{†}$ denotes the barrier height of the ground state energy. Q_TS(T) and Q_R(T) are partition functions of the transition states and reactants, which require additional care during treatment with vibrational modes. High-frequency vibrations—mainly of stretching and bending modes—can be analyzed with rigid harmonic oscillator approximation, while low-frequency vibrations of bond rotation modes have low torsion barriers and can be dealt with as one-dimensional hindered rotors using Pitzer and Gwinn approximation [37]. In this work, the potentials for internal rotations around the -CH₃, -C₂H₅, -C₃H₇, -C₄H₉, and -OO single bonds were calculated using a relax scan of the dihedral angle with an interval of 10° at the M062X/6-311G(d,p) level. Figures S1 and S2 in the Supplementary Material exhibit the rotation potentials of all rotors. Most of the methyl rotors had three approximately equal wave troughs and crests. Other rotors around C-C bonds had three wave troughs but they were asymmetrical. The rotors of O-O bonds, however, showed obvious variety for different species. Therefore, they had to be treated one by one. It has been indicated that the introduction of hindered rotor approximations for species and transition states causes differences of about 1.8–3.5 times in the rate constants compared to those with low-frequency vibration modes treated as harmonic oscillators.

The H-transfer reactions studied in this work belong to pressure-dependent channels. The dependence of rate constants on pressure were analyzed using RRKM/ME theory [38,39]. Argon was used as the bath gas. The collisional energy transfer probability was treated with a single-exponent down model and the average transfer energy was estimated with a temperature-dependent model of <ΔE_down> = 150(T/300)^0.85 [40,41]. The Lennard–Jones (L-J) pairwise potential was employed to estimate the collisional frequency between reactants and bath gas. The L-J parameters were estimated using the method developed by Constantinou and Gani for reactants [42,43]. The pressure-dependent rate constants for all channels were computed using the computer code MULTIWELL [44,45].

3. Results and Discussion

3.1. Potential Energy Surfaces

Figure 2 depicts the molecular structures associated with the process whereby cycloalkylperoxy yields hydroperoxylcycloalkyl radicals via H-transfer studied in this work. The molecular structures of chain-alkylated cyclohexanes, including methylcyclohexane (MCH), ethylcyclohexane (ECH), n-propylcyclohexane (PCH), i-propylcyclohexane (IPCH), and butylcyclohexane (BCH), are represented as a combination of a side chain (-C-R) and a six-membered ring in Figure 2a, and the carbon sites taken into account are labeled. H atoms at those carbon sites are replaced with an -OO group to yield peroxy radicals. The ROO radicals use the notation XCHRNOO, where XCH represents the alkylated cyclohexanes, such as MCH, ECH, and PCH, and the letter N denotes the carbon sites of the H atom that is substituted with -OO. Then, ROO radicals isomerize to QOOH radicals, which use the notation XCHNOOH-Myl, where XCH and N have the same meaning as above, and the letter M represents the carbon site at which the H atom is transferred. These species include H-transfer from cycle to side chain and H-transfer in the carbon cycle. In addition, cyclohexylperoxy, cyclopentylperoxy, and cyclohexenylperoxy, as well as several dialkylated cyclohexylperoxy radicals, were included in the present study in order to enhance its generality, as depicted in Figure 2b–g. The whole structures and their names can be found in Figures S3–S9 in the Supplementary Material.

3.1.1. Reaction Types

The potential energy surfaces of H-transfer reactions for alkylated cycloalkylperoxyl radicals were computed at the DLPNO-CCSD(T)/CBS//M062X/6-311G(d,p) level with the energies of the saddle and stable points corrected with zero point energies. The potential energy surfaces of all investigated cycloalkylperoxy radicals are shown in Figure 3. All the reactions are shown in Figures S3–S9 in the Supplementary Material.

All the channels can be categorized based on intramolecular H-migration types and barrier energy heights. First, H-transfer from cycle to chain yields bicyclic transition states in which the newly formed cycle shares two or three carbon atoms with the original carbon cycle. Taking cyclohexyl-methylperoxy as an example, there are four types of H-atoms on the main carbon ring to transfer to the side peroxyl chain. As shown in Figure 4a, H-transfer from carbon site #1 yields hydroperoxylmethylcyclohex-1-yl (MCHR0OOH-1yl) via a five-membered-ring TS structure. This is referred to as 1,4-H transfer and it has an energy barrier of 30.4 kcal. Xing et al. [20] reported the barrier height to be 30.1 kcal/mol, and Yao et al. [22] reported 28.7 kcal/mol. The corresponding reactions of the radicals ECHR0OO, PCHR0OO, BCHR0OO, and IPCHR0OO have comparable energy barriers of 28.2–30.8 kcal/mol. This reaction class was categorized as type one. H-transfer from carbon site #2, shown in Figure 4b, forms bicyclic TS with two common carbon atoms, which is referred to as 1,5-H transfer. The bicyclic TS of the H-transfer from site #3 involves three common carbon atoms for two cycles and is referred to as 1,6-H transfer, as illustrated in Figure 4c. It was found that these two reactions have the same energy barriers of 24.3 kcal. This value is very close to the results computed by Xing et al. [20], who provided 23.97 kcal for these two reactions, while it is 2.5 kcal higher than that provided by Yao et al. [22]. Similar reactions for the ECHR0OO, PCHR0OO, BCHR0OO, and IPCHR0OO radicals were computed and it was found that the 1,5-H and 1,6-H transfers from cycle to side chain have comparable energy barriers at 23.2–24.7 kcal/mol. Hence, they were categorized as one reaction type, denoted type two. H-transfer also occurs from the opposite carbon site (site #4) with an energy barrier of 26.9 kcal/mol, as shown in Figure 4d. In ideal terms, this reaction is of the 1,7-H transfer type and its bicyclic transition state has four common carbon atoms. However, the self-held atoms of the newly formed cycle and the C atom at site #0 form a plane that is nearly vertical with respect to the original carbon ring, turning out to be a lantern-shaped TS structure. It was found that the structure of the side chains only exerts a small influence on the energy barriers. This type of reaction was denoted type three and has barrier heights of 26.9–28.2 kcal/mol.

Figure 5 presents representative H-transfer reactions occurring on the carbon ring. As discussed above, there are two or three common carbon atoms in the two cycles of the transition state molecules. First, peroxyl obtains an H atom from its neighbor carbon site via 1,4-H transfer with a TS, and the newly formed cycle shares two carbon atoms with the original carbon ring. Although the 1,4-H transfer occurs from the secondary site to the tertiary site (Figure 5a) and from secondary/tertiary sites to the secondary site (Figure 5b), no differences were distinguishable in the energy barriers for these types. All of them were in the range of 32.7 kcal/mol to 35.5 kcal/mol. The energy barriers were greater than for the 1,4-H transfer type from the carbon ring to the side chain discussed above. According to a previous study, the energy barrier of 1,4-H transfer for alkane radicals amounts to 23–31 kcal [17]. It can be suggested that the inner strain energy of the bicyclic ring in the TS structure elevates the energy barrier of the 1,4-H transfer reactions in the carbon cycle. This reaction type is denoted type four in this work. Figure 5c,d depict representative reaction channels for 1,5- and 1,6-H transfer from secondary to secondary sites. The 1,5-H transfers to secondary sites occurring on the carbon ring, denoted type five, have obviously lower energy barriers, ranging from 26.2 kcal/mol to 28.6 kcal/mol. There are three common carbon atoms shared between the original cycle and the newly formed cycle in the transition states. In comparison, the 1,5-H transfer for alkane radicals has an energy barrier of less than 24 kcal [17], which is lower than that of the H-migration reaction in the carbon ring. The 1,6-H transfer that occurs between two opposite carbon atom sites on the ring has a higher energy barrier than the 1,5-H transfer, with a range of 31.0–33.1 kcal/mol. This type of reaction is denoted type 6 and includes 1,6-H transfer to secondary carbon sites from secondary or tertiary sites. Figure 5e shows the 1,5-H and 1,6-H transfers from the secondary to tertiary sites, which were picked out as another reaction type denoted type seven. Due to the tertiary C-H bond being almost vertical with respect to the main carbon ring, the ring strain energy of the 1,5-H transfer becomes greater, albeit smaller than that of the 1,6-H transfer smaller, resulting in energy barriers in the range of 29.6–30.7 kcal.

Figure 6 summarizes the activation energies for all reaction classes. It can be clearly seen that 1,5-H transfer reaction types have lower activation energies than other reaction types, while 1,4-H transfer reactions have 6–8 kcal/mole higher energy barriers than the 1,5-H transfer types for the two categories of H-migration from cycle to chain and in cycles. On the other hand, H-migration on the carbon ring tends to show a higher energy barrier than H-transfer from cycle to chain for similar transition-state molecular structures.

3.1.2. PES Features

The minimum energy path (MEP) is the steepest descent from the saddle point down to reactants and products in the mass-scaled Cartesian coordinate system. It is calculated point by point with a step size of 0.01 amu/bohr. In Figure 7, the MEPs for the reactions for radical MCHER2OO are illustrated as an example. In each panel, the single point energies relative to that of the saddle point are plotted along the MEP combined with the relative ground state energy. The ground state energy is computed as the sum of the single point energy and zero point energy. It can be seen that, for every reaction, the peak ground state energy and single point energy are almost identical with the saddle point. This suggests that the adoption of the saddle point as the bottleneck of the pathway is reliable. There is a sharp decrease and increase in the zero point energy along the MEP, leading to paths that can be observed well in the curves of the ground state energy. However, the peak is not altered.

3.2. Reaction Rate Coefficients

The rate constants for all reactions were computed based on tight transition state theory and fitted using the Arrhenius formula, as shown in Equation (3).

$$k=A{T}^{n}exp\left(-\frac{E}{RT}\right)$$

(3)

The reaction labels corresponding to the transition state labels are presented in Figure 3 and the coefficients of A, n, and E for all reactions are listed in

Table 1. The rate constants for all reactions; units are s⁻¹.

Reaction	A	n	E/cal	Reaction	A	n	E/cal
R1	5.33 × 10−2	3.799	22,117.3	R2	1.64 × 102	2.664	18,747.0
R3	1.78 × 101	2.846	18,042.8	R4	1.35 × 102	2.591	21,383.4
R5	2.94 × 10−1	3.635	22,149.9	R6	1.41 × 103	2.462	18,377.6
R7	1.32 × 102	2.671	18,181.3	R8	6.80 × 102	2.483	21,061.5
R9	1.96 × 100	3.312	20,018.2	R10	3.02 × 103	2.324	17,491.7
R11	2.42 × 102	2.531	17,444.0	R12	5.55 × 10−2	3.632	19,672.0
R13	7.97 × 10−2	3.646	22,334.5	R14	3.24 × 102	2.597	18,408.8
R15	3.09 × 101	2.775	18,135.4	R16	8.39 × 101	2.587	22,226.7
R17	1.79 × 10−3	4.567	21,782.5	R18	8.39 × 100	3.491	17,772.9
R19	5.43 × 10−1	3.652	17,851.8	R20	3.05 × 100	3.417	21,924.4
R21	1.79 × 10−6	5.107	24,557.6	R22	1.28 × 10−2	4.163	24,748.5
R23	9.09 × 103	2.460	22,315.8	R24	2.27 × 101	3.232	25,350.7
R25	1.93 × 10−2	4.092	23,930.3	R26	5.71 × 103	2.468	22,199.5
R27	1.28 × 102	2.985	25,810.1	R28	6.52 × 10−4	4.476	24,158.1
R29	2.33 × 103	2.549	21,565.1	R30	3.26 × 101	3.099	24,217.7
R31	8.63 × 10−3	4.191	23,689.1	R32	5.64 × 102	2.594	21,947.8
R33	1.10 × 102	3.020	25,874.7	R34	3.06 × 102	2.614	20,375.1
R35	9.95 × 10−2	3.881	24,411.8	R36	2.16 × 10−3	4.345	24,748.4
R37	1.40 × 103	2.593	22,767.5	R38	1.24 × 101	3.199	26,173.7
R39	1.53 × 103	2.575	22,879.6	R40	7.11 × 10−3	4.224	23,899.1
R41	1.46 × 103	2.616	21,608.2	R42	2.15 × 101	3.188	24,879.0
R43	1.42 × 103	2.618	21,327.4	R44	6.90 × 10−3	4.246	23,479.5
R45	8.61 × 102	2.675	21,234.0	R46	1.07 × 101	3.279	24,422.2
R47	1.03 × 103	2.693	21,021.8	R48	8.81 × 10−3	4.287	23,754.1
R49	4.16 × 103	2.529	21,630.1	R50	2.76 × 101	3.146	24,944.7
R51	3.97 × 103	2.606	21,250.6	R52	1.14 × 10−1	3.874	24,261.3
R53	2.17 × 10−3	4.296	24,899.8	R54	2.67 × 103	2.506	21,271.5
R55	7.81 × 102	2.629	22,611.7	R56	5.06 × 10−1	3.661	24,562.7
R57	5.26 × 10−3	4.226	24,725.1	R58	3.81 × 103	2.466	20,956.4
R59	3.00 × 103	2.527	22,605.4	R60	1.53 × 100	3.213	22,847.1
R61	5.90 × 100	3.338	23,189.0	R62	1.56 × 10−4	4.695	25,028.5
R63	7.71 × 10−4	4.479	24,164.7	R64	1.89 × 102	2.850	22,302.2
R65	1.70 × 10−4	4.684	25,052.7	R66	2.18 × 101	3.209	22,853.4
R67	6.69 × 10−3	4.243	24,031.6	R68	3.63 × 10−3	4.312	24,012.9
R69	7.68 × 102	2.683	22,153.3	R70	1.28 × 101	3.301	25,423.8
R71	7.01 × 100	3.293	23,179.9	R72	5.85 × 10−4	4.525	25,199.4
R73	3.05 × 10−4	4.555	23,856.8	R74	4.22 × 102	2.780	22,153.4
R75	2.02 × 102	2.922	23,838.1	R76	5.67 × 10−4	4.503	25,440.5
R77	1.18 × 10−3	4.424	22,862.0	R78	7.15 × 102	2.700	23,045.3
R79	1.54 × 10−4	4.654	25,172.1	R80	3.27 × 102	2.769	22,732.4
R81	2.07 × 100	3.452	25,812.8	R82	6.22 × 10−6	4.923	21,781.1
R83	2.26 × 10−2	3.812	24,907.6	R84	7.38 × 100	3.030	28,626.4
R85	3.53 × 101	3.081	24,096.7	R86	3.86 × 102	2.742	21,915.3
R87	4.97 × 10−4	4.509	24,852.8	R88	1.93 × 101	2.904	25,569.5
R89	1.63 × 101	3.148	24,139.8	R90	6.56 × 102	2.695	21,798.9
R91	2.32 × 10−3	4.337	23,824.0

. They were first compared with previous rate constants for identical reactions published in the literature. Then, rate constants were categorized based on the reaction classes discussed in Section 3.1, and rate rules for all reaction classes are proposed.

3.2.1. Comparison with Previous Data

Since it is very difficult to obtain experimental data for rate constants for H-transfer reactions, the theoretically obtained rate constants published in the literature were employed for a comparison with those computed in this work. Figure 8a demonstrates the high-pressure limits obtained in this work for the reaction whereby MCHR0OO yields hydroperoxy methylcyclohexyl radicals, as well as the rate constants for the same reactions at p = 100 atm computed by Xing et al. [20]. Owing to the use of similar quantum chemistry methods, as discussed in Section 3.1, Xing et al. obtained very similar energy barrier for each pathway as this work. As a result, the rate constants provided by Xing et al. show good agreement with those from this work at low temperatures and become slightly lower than the latter as the temperature increases. The rate constants for the most favorable channel, MCHR0OO = MCHR0OOH-2yl, match very well with each other at temperatures lower than 1300 K and only exhibit a difference of a factor of 1.5 at 1500 K. The present rate constants for the reaction whereby MCHR0OO yields MCHR0OOH-1yl begin to diverge from those provided by Xing et al. from 800 K and become about 1.5 times greater than the latter at 1500 K, and this was the least similar channel. Figure 8b presents the rate constants for six reaction channels for the radical MCHR3OO obtained in this work and those from the study by Yao et al. [22]. As the energy barriers reported by Yao et al. are about 2–3 kcal lower for each reaction, the rate constants are seven to ten times greater for the corresponding reactions. According to the present results, the reactions MCHR3OO = MCHR3OOH-1yl and MCHR3OO = MCHR3OOH-5yl have the greatest rate constants, followed by MCHR3OO = MCHR3OOH-0yl. The rate constants for the reactions whereby MCHR3OO yields MCHR3OOH-2yl, MCHR3OOH-4yl, and MCHR3OOH-6yl are the lowest. Yao et al. [22] showed similar ranks for all reactions as the results in this work. Additionally, Figure 8c illustrates a comparison of the rate constants for H-transfer reactions in cycles for ECH obtained in this work and in the studies by Yao et al. [22] and Ning et al. [46]. There are notable discrepancies between the rate constants. Yao et al. [22] reported rate constants two to six times greater than the present work for the reaction ECHR2OO = ECHR2OOH-4yl and three to ten times greater for the reaction ECHR3OO = ECHR3OOH-4yl, while the rate constants for these reactions provided by Ning et al. [46] are over two orders of magnitude greater than the present data. The rate constants for ECHR3OO = ECHR3OOH-5yl in the present work are two to five times greater than those provided by Ning et al. [46] and two to six times lower than those provided by Yao et al. [22]. The rate constants for ECHR3OO = ECHR3OOH-1yl in the literature are four to ten times greater than the present data. Comparisons of the rate constants of H-transfer reactions for other species also showed notable differences between the present work and the literature data. The figures are given in Figure S10 in the Supplementary Materials.

In addition, the pressure-dependent rate constants at pressures of 0.01–10 atm for the decomposition of the MCHR0OO radical are shown in Figure 9. As expected, rate expressions decrease with decreases in pressure. The influence of pressure on the rate constants increases as the temperature increases. It should be pointed out that the rate constants at p = 10 atm approximate the high-pressure limits. The close lines of the rate expressions between pressures of 10 and 1 atm indicate quite weak pressure effects. This suggests that the pressure influence on such H-transfer reactions is not very strong in comparison with other β-scission channels. Furthermore, Xing et al. [20] computed pressure-dependent rate constants for the reaction channels of the MCHR0OO radical, including OH elimination and subsequent decomposition, in addition to the H-transfer. Their data were employed for a comparison with the present data. Fairly acceptable agreement is shown in Figure 9, especially for the reactions forming MCHR0OOH-2yl and MCHR0OOH-4yl, in which the scatter point (data from Xing et al.) overlaps with the solid lines (present data). The difference for the reaction yielding MCHR0OOH-3yl is within a factor of 1.4, while it is greater than a factor of 3 for the reaction forming MCHR0OOH-1yl. In general, it is safe to conclude that the pressure-dependent rate constants are comparable.

3.2.2. High-Pressure Limit Rate Rules

In accordance with the findings discussed above, the rate constants for all reactions were computed from temperatures of 600 K to 2000 K using the TST method. Then, the data were fitted in the form of the Arrhenius formula, which is shown in Equation (4).

$$k=A\exp \left(-\frac{E}{RT}\right)$$

(4)

In Section 3.1, the reactions were categorized into seven classes based on the H-transfer types and the values for the energy barrier. Next, the rate rules were derived for each energy reaction class with the following method: the logarithmic average value for the rate constant for each reaction in the same class was calculated at discrete temperature points. The equation is as follows:

$$\ln \left(k_{\mathrm{avg}}^T\right)=\frac{1}{N}{\displaystyle \sum }_{i=1}^{i=N}\ln \left(k_i^T\right)$$

(5)

where N denotes the number of reactions in the same class, $k_i^T$ represents the rate constant at T for the ith reaction, and $k_{\mathrm{avg}}^T$ is the average value at temperature T. In this way, a set of average rate constants were obtained for each reaction class over the temperature range of 600–2000 K. Then, they were fitted based on Equation (3) to obtain the average rate coefficients of (A,E), which represented the rate rule for the concerned reaction type. Furthermore, the uncertainty for each rate rule was estimated as the square root of the ratio of the largest to the smallest rate constants for reactions in the same class.

The rate rules for reaction classes involving H-migration from the cycle to side chain are shown in Figure 10. In each panel, the largest and smallest rate constants in each rate class are also depicted. Rate rules proposed in the literature are plotted for comparison. Figure 10a shows the rate rule for 1,4-H transfer from the cycle to side chain. The differences between the largest and smallest rate constants have factors of 3–20, suggesting that the uncertainty for the rate rule is 5. Since no rate rule for this class has been provided, the rate rules for 1,4-H transfer of the alkylperoxy radical proposed by Villano et al. were used for comparison with the present data. It can be seen that the rate rule for 1,4-H transfer to secondary sites provided by Villano et al. shows good agreement with the present rate rule, although the rate rule for 1,4-H transfer to tertiary sites provided by Villano et al. is two to five times greater. The agreement is likely coincidental, as most rate rules for chain alkylperoxy are not applicable to cyclic alkylperoxy. Figure 10b shows the rate rule for the 1,5-/1,6-H transfer reaction class. In this class, the 1,6-H transfer MCHR0OO to MCHR0OOH-3yl has the lowest rate constants, while the 1,5-H transfer BCHR0OO to BCHR0OOH-2yl has the greatest rate constants. The ratio between the greatest and lowest rate constants is about 25 and, therefore, the uncertainty for the rate rule can be estimated as 5. Yao et al. [22] proposed separate rate rules for 1,5- and 1,6-H transfers from the cycle to chain, which are also plotted. The rate constants for the 1,5-H transfer rate rule provided by Yao et al. are 1.5–3 times greater than those of the same authors’ 1,6-H transfer class, but they are 4–60 times larger than those of the present rate rule. Figure 10c shows the range of the rate constants for 1,7-H transfer from the cycle to chain. The greatest rate constant for IPCHR0OO = IPCHR0OOH-4yl is three to five times that of the smallest values for PCHR0OO = PCHR0OO H-4yl. Hence, the uncertainty for the rate rule for this class can be estimated as 2.5. Due to the evident differences in the molecular geometries of the transition states of 1,7-H transfer for the chain alkylperoxy radical and those for the cycle to chain in the present work, significant disagreement can be noted between the present rate rule and those provided by Villano et al. [17]. This shows that the rate rules for chain alkylperoxy cannot be used for cyclic alkylperoxy radicals, as pointed out in the literature.

The rate rules for H-transfer for cycles are shown in Figure 11. Figure 11a plots the rate rule for 1,4-H transfer for a cycle, for which the uncertainty was estimated as a factor of 5. In this work, we did not distinguish between the reactions involving H-transfer to secondary sites and those to tertiary sites. Yao et al. [22] proposed a rate rule for 1,4-H transfer to secondary sites for cycles and Villano et al. [17] developed two sets of rate rules for 1,4-H transfer to secondary and tertiary sites for chain alkylperoxy, respectively. The rate rule for 1,4-H transfer to secondary sites provided by Villano et al. [17] coincidently falls into the range of uncertainty for the present rate rule. The rate rules provided by Yao et al. [22] are five times greater than the present results at high temperatures and become ten times greater at 600 K. The rate rule for 1,5-H transfer to secondary sites is plotted in Figure 11b, accompanied by the rate rule for the same reaction class provided by Yao et al. [22] and those for 1,5-H transfer to secondary and tertiary sites for alkylperoxy radicals provided by Villano et al. [17]. Since the uncertainty for the present rate rule was estimated as a factor of 4, the rate rule provided by Yao et al. [22] was found to agree fairly well with the present rate rule. Figure 11c presents the rate rule for 1,6-H transfer to secondary sites with uncertainty estimated as a factor of 3. It can be seen that the rate rule proposed by Yao et al. [22] for the same reaction class is four times greater than the present results. Finally, Figure 11d shows the rate rules for 1,5-H transfer and 1,6-H transfer to tertiary sites. The greatest rate constants are about seven times the lowest value and, hence, the uncertainty can be estimated as a factor of 3. The value of the rate rules provided by Villano et al. [17] again demonstrate significant disagreement with the present results.

The rate rules and their uncertainties presented above are also collected in

Table 2. The rate rules and uncertainties for all reaction types.

Reaction Type	A	E (Cal)	Uncertainty
1,4-H transfer from the ring to the side chain (type one)	1.08 × 1012	29,575.5	5
1,5-/1,6-H transfer from the ring to the side chain (type two)	3.86 × 1011	23,790.6	5
1,7-H transfer from the ring to the side chain (type three)	2.18 × 1011	27,280.7	2.5
1,4-H transfer to secondary site on the ring (type four)	3.38 × 1012	33,676.1	5
1,5-H transfer to secondary site on the ring (type five)	1.91 × 1012	27,573.3	4
1,6-H transfer to secondary site on the ring (type six)	3.10 × 1012	32,162.0	3
1,5-/1,6-H transfer to secondary site on the ring (type seven)	2.30 × 1012	30,377.6	3

. Moreover, the rate rules in this work are compared in Figure 12. It can be seen that the differences in rate constants between reaction types are prominent at low temperatures but are reduced at high temperatures. Type-two and type-five reaction classes represent the most favorable channels. Rate constants of type five are about two times those of type two at 2000 K. After a crossover point at 1200 K, rate constants of type two reach values nearly five times those of type five. The rate rule for 1,5-/1,6-H transfer to tertiary sites represented as type seven has comparable values to those of type six but gradually increases to four times the latter at a temperature of 600 K. The type-one reaction class, the 1,4-H transfer from the cycle to the chain, has comparable values to those of type seven and shows much lower rate constants than those of 1,5-H transfer. It is similar to the 1,4-H transfer for cycles represented as type four. Its rate constants are even ten times lower than those of type one at T = 600 K. The type-three rate constants, which represent 1,7-H transfer from the cycle to chain, are comparable with those of type one at T < 1000 K and gradually drop to one third of the value of the latter at T = 2000 K. In conclusion, the rate constants of 1,5-H transfer are clearly greater than those of 1,4-H transfer and the other reactions are in-between them. For the same H-transfer type, intermigration from cycle to chain has greater rate constants than that in the carbon cycle.

4. Conclusions

In this paper, the potential energy surfaces of intermolecular H-migration for a set of typical cycloalkylperoxy radicals were computed using the M062X/6-311G(d,p)//DLPNO-CCSD(T)/CBS dual-level quantum chemistry method. Considering tunneling effects and treating low-frequency vibration modes as hindered rotors, the high-pressure limits of all reactions were calculated using tight transition state theory. As the rate constants reported by different authors exhibit clear discrepancies, the present rate constants were compared with previous data from several literature resources. Acceptable agreement was found between the present rate constants and those in the study by Xing et al. for the MCHR0OO radical, probably owing to the similar quantum chemistry methods adopted. It should also be pointed out, however, that the rate constants in this work clearly deviated from those provided by Yao et al. and Ning et al., with differences amounting to factors in the tens. This was ascribed to the different energy barriers evaluated at different ab initio levels. Thus, it can be concluded that the choice of ab initio quantum method needs to be taken into account in the estimation of the uncertainty of results.

Additionally, all the reactions were separated into seven reaction classes according to the H-transfer type and energy barriers. H-transfer from the cycle to chain had three reaction classes: 1,4-H transfer with barrier energies of 28.2–30.8 kcal/mol, 1,5-/1,6-H transfer with barrier energies of 23.2–24.8 kcal/mole, and 1,7-H transfer with barrier energies of 26.9–28.2 kcal/mole. There were four reaction classes for H-transfer in the cycle. They were 1,4-H transfer with energy barriers of 32.7–35.5 kcal/mole, 1,5-H transfer to secondary sites with energy barriers of 26.2–28.6 kcal/mole, 1,6-H transfer to secondary sites with energy barriers of 31.0–33.1 kcal/mole, and 1,5-/1,6-H transfer to tertiary sites with energy barriers of 29.6–30.7 kcal/mole. The rate rule for each reaction type at the temperature range of 600–2000 K was calculated and expressed in the Arrhenius form. The uncertainties for rate rules were estimated as factors of three to six. The rate rules and rate expressions for all 91 reactions will be useful for the development of detailed low-temperature kinetic mechanisms for cycloalkanes. It was also found that the 1,5-H transfer reaction type was more favorable than the 1,6-H transfer type, and 1,4-H transfer was the least favorable. Moreover, H-transfer reactions from the cycle to side chain seemed to be more favorable than H-transfer in the carbon cycle. There were differences between the current rate rules and previous rules proposed in the literature. The reliability must be validated with experimental data in the future.