ARTICLE pubs.acs.org/JPCA
Tunneling in Hydrogen-Transfer Isomerization of n-Alkyl Radicals Baptiste Sirjean,*,†,§ Enoch Dames,† Hai Wang,*,† and Wing Tsang‡ †
Department of Aerospace and Mechanical Engineering, University of Southern California, Los Angeles, California 90089-1453, United States ‡ National Institute of Standards and Technologies, Gaithersburg, Maryland 20899, United States
bS Supporting Information ABSTRACT: The role of quantum tunneling in hydrogen shift in linear heptyl radicals is explored using multidimensional, smallcurvature tunneling method for the transmission coefficients and a potential energy surface computed at the CBS-QB3 level of theory. Several one-dimensional approximations (Wigner, Skodje and Truhlar, and Eckart methods) were compared to the multidimensional results. The Eckart method was found to be sufficiently accurate in comparison to the small-curvature tunneling results for a wide range of temperature, but this agreement is in fact fortuitous and caused by error cancellations. High-pressure limit rate constants were calculated using the transition state theory with treatment of hindered rotations and Eckart transmission coefficients for all hydrogen-transfer isomerizations in n-pentyl to n-octyl radicals. Rate constants are found in good agreement with experimental kinetic data available for n-pentyl and n-hexyl radicals. In the case of n-heptyl and n-octyl, our calculated rates agree well with limited experimentally derived data. Several conclusions made in the experimental studies of Tsang et al. (Tsang, W.; McGivern, W. S.; Manion, J. A. Proc. Combust. Inst. 2009, 32, 131138) are confirmed theoretically: older low-temperature experimental data, characterized by small preexponential factors and activation energies, can be reconciled with high-temperature data by taking into account tunneling; at low temperatures, transmission coefficients are substantially larger for H-atom transfers through a five-membered ring transition state than those with six-membered rings; channels with transition ring structures involving greater than 8 atoms can be neglected because of entropic effects that inhibit such transitions. The set of computational kinetic rates were used to derive a general rate rule that explicitly accounts for tunneling. The rate rule is shown to reproduce closely the theoretical rate constants.
1. INTRODUCTION Understanding the thermal decomposition reactions of alkyl radicals is critical to the development of combustion models of hydrocarbon fuels. In n-alkane oxidation at high temperatures, alkyl radicals are initially formed by CC or CH fission in the fuel molecule or H-abstraction reaction by a free radical. Small alkyl radicals (Ce3) decompose mainly by CC β-scission, as the 12 and 13 hydrogen shifts can occur only through threeor four-membered ring transition state structures with large energy barriers due to strain energies.1 For large alkyl radicals, the thermal decomposition is complicated by internal hydrogen shifts.27 These isomerization processes occur through cyclic transition state structures. Internal isomerization can occur through five-, six-, or seven-member ring transition state structures, which may be denoted as 14, 15, and 16 hydrogen shifts, respectively. Some of these transition state structures are associated with ring strain energies, while others are practically unstrained. H-atom shifts are the predominant unimolecular reactions under lower temperature conditions. As the temperature is increased, β-scissions become increasingly important, but H-atom shift remains critical to the dissociation of alkyl radicals and especially the product distributions resulting from the β-scission processes. The combustion chemistry of olefins r 2011 American Chemical Society
and smaller alkyl radicals, formed in the decomposition process, play a key role in the reactivity of the fuel and formation of pollutants and polycyclic aromatic hydrocarbon (PAH) and soot.812 The basic problem with studying alkyl isomerization processes is that they involve large organic moieties (C5 or above). Except at the lowest temperatures, β-scission will always make contributions. Unless careful attention is paid to setting the reaction conditions to minimize mechanistic artifacts, it is usually difficult to derive truly quantitative kinetic parameters. In the case of single pulse shock tube experiments, the isomerization rate constants derived are all dependent on the initial β-scission rate constants, which, in turn, are derived from literature values. Thus, the uncertainty in the β-scission rate constants inevitably propagates into the isomerization rate constants. For this type of tightly coupled unimolecular reaction involving competitive reaction paths, theoretical reaction kinetics is often necessary to interpret the experimental data. Received: September 28, 2011 Revised: November 22, 2011 Published: November 30, 2011 319
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Figure 1. Arrhenius plots of (a) 14 hydrogen shift in n-pentyl radical, and (b) 15 hydrogen shift in n-hexyl radical.
Reactions involving the transfer of a hydrogen atom are inherently subject to quantum tunneling effect.13 The role of tunneling in n-alkyl radical isomerizations was recently highlighted by Tsang and co-workers.47 They showed that tunneling must be considered to reconcile older data obtained at low temperatures with measurements made in their high-temperature single-pulse shock tube. Low-temperature kinetic data have been subject to controversies because of the unusually small pre-exponential factors (A) that were difficult to rationalize with the thermochemical kinetic theory.14 Direct measurements of the isomerization reaction rates of alkyl radicals are difficult, and kinetic data were derived usually from analyses of complex reaction mechanisms. Moreover, most experiments were performed in the falloff region and/or involved a chemically activated process. In such cases, the highpressure limit rate constant must be extrapolated from the Rice RamspergerKasselMarcus (RRKM) theory and, for the latest studies, from Master Equation (ME) analysis. Among all n-alkyl radicals, the unimolecular reactions of 1-pentyl and 1-hexyl have been studied most extensively in the past. Figure 1 presents selected literature rate constants. Endrenyi and Le Roy15 proposed the first experimental rate constant for the gas-phase 14 hydrogen migration in n-pentyl radicals: 1 C5 H11 / 2 C5 H11
based on a new mechanistic interpretation of the data of Endrenyi and Le Roy and proposed: k1 ðs1 Þ ¼ 5:0 1011 e10600=T
Similar low A factor values were observed from low-temperature experiments on n-hexyl radicals. Watkins and Ostreko18 proposed a rate expression for the 15 hydrogen migration in n-hexyl radicals: 1 C6 H13 / 2 C6 H13
k2 ðs1 Þ ¼ 2:0 107 e4180=T
ð4Þ
The small A factor has been a subject of debate in the literature.19,20 In the late 1980s, Dobe et al.21 reported an experimental investigation of k2 for 300 e T e 453 K and pressures from 100 to 200 Torr and proposed a rate expression that again features a small A factor: k2 ðs1 Þ ¼ 3:16 107 e5840=T
ðR1Þ
ð5Þ
Using the same experimental approach, these authors also proposed rate expressions for n-octyl radicals isomerization with similar A factors. Marshall22 examined the thermal decomposition of n-pentane in the temperature range of 737923 K and pressures below 200 Torr. The rate expression for the 14 hydrogen shift of reaction R1 was derived from the distribution of major products as
ð1Þ
for temperatures 439 e T e 503 K at low pressures. Within the framework of transition state theory, they concluded that the unusual low pre-exponential factor can be explained only by considering a quantum tunneling effect. Watkins16 derived a rate expression from experiments performed 297 e T e 435 K and pressures from 1 to 30 Torr: k1 ðs1 Þ ¼ 3:3 108 e7600=T
ðR2Þ
for temperatures from 352 to 405 K and a pressure of 46 Torr:
They reported: k1 ðs1 Þ ¼ 1:4 107 e5440=T
ð3Þ
k1 ðs1 Þ ¼ 9:1 1011 e11800=T
ð6Þ
By considering earlier low-temperature measurements, he proposed that
ð2Þ
k1 ðs1 Þ ¼ 1:2 1011 e10100=T
Watkins also noted the unusually low A factor of rate constant and proposed that errors resulting from photochemical activation could lead to an underestimation of the A factor. In a follow-up paper,17 Watkins proposed a rate expression with a larger A factor
ð7Þ
for 438 e T e 923 K. Imbert and Marshall23 followed a similar approach to determine the high-pressure limit rate constant 15 hydrogen transfer in n-hexyl radicals by n-hexane pyrolysis 320
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and proposed: k2 ðs1 Þ ¼ 3:16 1010 e8560=T
the framework of strain energy in the transition state structure. Using B3LYP/ccpVDZ calculations and transition state theory (TST) on prototypal reactions, Matheu et al.26 proposed rate rules for 12, 13, 14, 15, and 16 hydrogen shifts in alkyl radicals. Hayes and Burgess27 calculated the energy barriers of hydrogen transfer in alkyl, allyilc, and oxoallylic radicals using several composite methods and showed that an EvansPolanyi correlation can be developed in the case of linear alkyl radicals. Quantum tunneling effects were not considered in these studies. Truong and co-workers28 studied the 1,4-intramolecular hydrogen migration in linear alkyl radicals using the class transition state theory. They calculated the rate constant for the reference reaction of n-C4H9 using canonical variational transition state theory (CVTST) with the small curvature tunneling (SCT) correction. It is important to note here that SCT approximation allows for a multi-dimensional treatment of tunneling through computationally expensive calculation of the Hessian for numerous points along the reaction path. In their study, the potential energy surface (PES) was computed at the CCSD(T)/ cc-pVDZ//BH&HLYP/cc-pVDZ level of theory. They found a significant tunneling contribution at temperatures below 1000 K. Following their study on 14 hydrogen shift, they recently proposed high-pressure limit rate expressions for 13 to 16 hydrogen migrations in linear alkyl radicals.29 They described the isomerization reactions of n-propyl to n-hexyl radicals using the same methodology. The rate constants for analogous reactions were proposed on the basis of these prototype reactions within the framework of reaction class TST. Again, the tunneling effect was found to be less prominent above 1000 K. Additionally, they compared transmission coefficients obtained from multidimensional SCT tunneling with those obtained from the one-dimensional Eckart function and showed that the Eckart tunneling effect is more pronounced than SCT tunneling below 400 K. Jitariu et al.30 carried out direct dynamic calculations with CVTST/SCT to study the decomposition and isomerization pathways of n-pentyl radicals. They reported dual-level calculations at the PUMP2-SAC/6-311G**///AM1 level of theory. Here, the triple slash (///) denotes interpolating optimized corrections (IOC) in the VTST calculations.31 In this method, the PES along the reaction path are corrected using higher-level values at selected points. The Hessians, the potential energies, and moment of inertia along the reaction path were computed at the AM1 level of theory. UMP2/6-311G** geometries, vibrational frequencies, moments of inertia, and PUMP-SAC2/ 6-311G** energies of the stationary points were then used to scale the low-level AM1 results. This rather elaborate approach led them to conclude that tunneling is pronounced at low temperatures in the 14 hydrogen shift of 1-pentyl. At 1000 K, however, their proposed rate constant is about a factor 3 larger than experimental values of Tsang et al.6,24 and Yamauchi et al.6,24 Zheng and Truhlar studied the 14 H shift in the 1-pentyl radical and the 14 and 15 hydrogen transfers in the 1-hexyl radical using CVTST/SCT theory.32 They calculated the PES with several levels of density functional theory (DFT) (e.g., M062X/MG3S and M08-HX/cc-pVTZ+) and multilevel methods for the stationary points. They used the interpolated variational transition state theory by mapping (IVTST-M) in all of their dynamics calculations. This method gives PES data (energies, gradients, and Hessian) along the full reaction path based on a limited number of points calculated near the saddle point. They compared several one-dimensional approximations (Wigner, zero curvature tunneling, and parabolic tunneling approximation) to
ð8Þ
for 723 e T e 823 K. Miyoshi and co-workers derived the expressions for k1 and k2 from an analysis of shock-tube experiments for 900 e T e 1400 K near the atmospheric pressure using a complex reaction mechanism. They proposed expressions for the high-pressure limit rate constants k∞ from an RRKM analysis of their experimental results along with earlier, low temperature data. They included the tunneling effect via a Wigner correction with the imaginary frequencies taken from HF/6-31G(d) calculations. They proposed: 24
k1, ∞ ðs1 Þ ¼ 4:9 108 T 0:846 e9830=T
ð9Þ
and k2, ∞ ðs1 Þ ¼ 6:7 107 T 0:823 e6570=T
ð10Þ
for 350 e T e 1300 K. The somewhat strong and positive temperature exponents point to an apparent upward curvature of the Arrhenius form largely because of the tunneling effect being considered in the rate calculation. Miyoshi and co-workers25 reported the first direct measurement of the rate coefficients of 14 hydrogen shift in 1-pentyl radical in the temperature range of 440520 K and pressures from 1 to 7 Torr. From an RRKM/ME analysis including Eckart tunneling, they evaluated the high-pressure limit rate constant to be k1, ∞ ðs1 Þ ¼ 1:9 1010 e8870=T
ð11Þ
for 440 e T e 520 K and k1, ∞ ðs1 Þ ¼ 2:4 103 T 2:324 e8180=T þ 9:1 105 e5300=T
ð12Þ
300 e T e 1300 K. The use of a bi-Arrhenius form is the result of a large tunneling effect, leaving a high Arrhenius curvature. More recently, Tsang and co-workers carried out a series of systematic studies on the thermal decomposition of n-pentyl,6 n-hexyl,7 n-heptyl,4 and n-octyl5 radicals in single-pulsed shock tube for temperatures from 850 to 1050 K and pressures between 1.5 and 7 bar. From experimentally determined product branching ratios, they deduced high-pressure limit rate expressions for 14, 15, and 16 hydrogen transfers using an RRKM/ME analysis. In the case of n-octyl, they also explored a hydrogen migration channel through the eight-membered ring transition state structure, but found that such a transition is unimportant compared to other reaction channels. Earlier, lower temperature data on n-pentyl and n-hexyl were included in their theoretical analysis considering Eckart tunneling. Notably, they proposed the high-pressure limit rate constant: k1, ∞ ðs1 Þ ¼ 1:0 1012 e11300=T
ð13Þ
for 14 hydrogen shift in 1-pentyl from 850 to 1000 K, and k2, ∞ ðs1 Þ ¼ 1:8 102 T 2:55 e5550=T
ð14Þ
for 15 hydrogen migration in 1-hexyl for 500 e T e 1900 K. Theoretically, Viskolcz et al.1 computed the energy barriers of several isomerization reactions of linear alkyl radicals and the barrier of the 14 hydrogen shift in 1-pentyl at the MP-SAC2// UHF/6-31G* level of theory. They discussed their results within 321
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(ZPE) were quite close at the B3LYP/6-311G(2d,d,p) and CBSQB3 levels of theory (Table 1). Hindered Internal Rotations. The description of isomerization channels of large aliphatic chains requires an accurate treatment of low-frequency internal rotations. Using the harmonic oscillator (HO) approximation to describe these torsion modes can lead to large errors in the partition function. Vansteenkiste et al.38 calculated the thermodynamic properties of n-alkanes using a quantum-mechanical treatment and showed that treating hindered internal rotation is critical to obtain accurate entropies and heat capacities values. As hydrogen transfers occur through cyclic transition state structures, most of the internal rotations of the initial reacting radicals are locked in the ring, and the rate coefficients will be strongly dependent on the entropy variation. As an example, Vansteenkiste et al. reported for n-heptane a difference of 9.3 cal/(mol K) at 1000 K in entropy between calculations using the assumption of harmonic oscillator (HO) and a more precise quantum mechanical treatment. The HO approximation underestimates the entropy change. In the present work, hindered internal rotations were treated using the following procedure. First, the potentials of each internal rotation of 1-pentyl, 2-pentyl, and 3-pentyl radicals were calculated at the B3LYP/6-311G(2d,d,p) level of theory using a relaxed energy scan. The energy barriers for these hindered internal rotors were then calculated at the CBS-QB3 level of theory. The characteristics of the rotational potentials and the barriers of rotation obtained were used to correct the HO partition function using Pitzer and Gwinn tabulations.39 Rotational potential functions and barriers for each internal rotor in n-pentyl were used for similar internal rotors found in largest alkyl radicals. Reduced moments of inertia for internal rotations were calculated for each species using B3LYP/6-311G(2d,d,p) geometries with the method of Pitzer implemented in ChemRate.40,41 For cyclic transition states, the vibrational modes of the cyclic part of the molecular structure were described within the HO approximation, and the lateral alkyl group internal rotations were treated as hindered rotors (HR) via relaxed scans (with cyclic bond lengths frozen) in the critical geometries for 1-pentyl to 2-pentyl and 1-hexyl to 3-hexyl. Parameters for HR corrections for these two critical geometries were used for all other similar internal rotors found in the saddle point geometries of larger alkyl radicals. Hindrance potentials and barrier heights are given in the Supporting Information. Transmission Coefficient. Within a canonical TST or VTST framework, quantum tunneling is taken into account by a temperature-dependent transmission coefficient k(T):
Table 1. Critical Energies (kcal/mol) Computed for H-Atom Shifts in n-Heptyl Radicals at 0 K B3LYP/ 7ps (1-heptyl / 2-heptyl) 6ps (1-heptyl / 3-heptyl)
a
6-311G(2d,d,p)a
CBS-QB3a
G3MP2B327
14.1 14.6
14.7 15.0
15.8 16.1
5ps (1-heptyl / 4-heptyl)
21.3
22.0
23.5
5ss (2-heptyl / 3-heptyl)
23.2
22.8
23.9
This work.
their multidimensional SCT results. They noted that at low temperatures the discrepancies among these methods are rather large. An interesting point is that tunneling is more important in the case of 14 hydrogen transfer than for 15 shift, in agreement with the work of Tsang and co-workers. Unfortunately, the widely used Eckart method was excluded from their study, and its ability to reproduce results of higher dimensional tunneling treatments remains unclear. In the present study, we intend to address several issues regarding the role of tunneling in linear alkyl radical isomerizations. We compute the transmission coefficients using n-heptyl as the model system employing a multi dimensional treatment. The results allow us to assess accuracy of the various one-dimensional methods for the calculation of transmission coefficients. Beyond the n-heptyl radical, we use the rigid rotor harmonic oscillator (RRHO) approximation with corrections for hindered internal rotation (HR), the TST theory with a selected 1-D tunneling method, and we systematically determine the rate coefficients for 14 to 16 H-shifts in n-alkyl radicals ranging from n-pentyl to n-octyl. Note that as the size of the linear alkyl chain increases, most favorable isomerizations (five-, six-, and seven-member transition state structures) can involve hydrogen shift between two secondary radicals (e.g., 2-heptyl to 3-heptyl). It is noteworthy to mention that there are almost no kinetic data available for these types of processes. Finally, we discuss the role of tunneling within the framework of correlations between structure and reactivity, and its impact on the uncertainty of rate coefficients derived from theoretical calculations.
2. COMPUTATIONAL DETAILS Electronic Structure Calculations. Potential energy and molecular properties of stationary points were calculated using Gaussian 03 revision B.0333 and QChem version 3.1.34 For all stationary and critical structures, geometry optimizations were performed at the B3LYP/6-311G(2d,d,p) level of theory.35,36 Frequency calculations were performed at the same level of theory for all optimized geometries to determine the nature of stationary points. The composite CBS-QB3 method was applied for all stationary geometries and transition states.37 The CBSQB3 model involves a five-step calculation starting with a geometry optimization and a frequency calculation (scaled by a factor 0.99) at the B3LYP/6-311G(2d,d,p) level of theory, followed by single point energy calculations at the CCSD(T)/ 6-31G(d0 ), MP4SDQ/cbsb4, and MP2/cbsb3 and a complete basis set extrapolation to correct the total energy. Dynamic calculations along the reaction path were performed at the B3LYP/ 6-311G(2d,d,p) level of theory. We found that for reactions examined here, the energy barrier heights with zero-point energy
kðTÞ ¼ kðTÞAeE=T
ð15Þ
Within the framework of the RRKM theory and to account for tunneling, Miller42 proposed one-dimensional tunneling probability P(E,J) in the sum of states N(E,J) of the transition state: NðE, JÞ ¼
∑n ½PðE Eqn Þ, J
ð16Þ
where εnq is the nth vibrational energy level and J is the angular momentum. The standard microcanonical rate expression is calculated using N(E,J): kðE, JÞ ¼
NðE, JÞ hFðE, JÞ
ð17Þ
where h is the Planck constant and F(E,J) is the density of energy states of the reactant. Note that k(T) can be calculated from P(E,J) 322
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by integrating over the Boltzmann-weighted energy distribution and that k(T) also includes nonclassical reflection effects. For a one-dimensional system, the transmission coefficient may be calculated by solving the Schr€ odinger equation with an appropriate potential function. In this approach, the potential energy surface is fitted with a function for which the tunneling probabilities are known analytically. The simplest method is that of Wigner43 who assumed a parabolic function for the PES leading to the transmission coefficient: " #2 1 hImðνq Þ kðTÞ ¼ 1 þ 24 kB T
ð18Þ
where νq is the imaginary frequency corresponding to the reaction coordinate, and kB is the Boltzmann constant. Skodje and Truhlar44 proposed an improvement over the Wigner method by treating the energy barrier as a truncated parabola. The Eckart function is particularly useful as it provides realistic potential function that has the correct asymptotic properties of the one-dimensional PES. Moreover, the parameters of the Eckart potential can be fitted to reproduce the curvature at the saddle point and the exothermicity of the reaction.45 This onedimensional method is widely used in the theoretical combustion kinetics because they are rather inexpensive computationally yet accurate enough for the temperatures of interest. More sophisticated methods to calculate the transmission coefficient have been proposed by Truhlar and co-workers. Particularly, they developed semiclassical methods to calculate transmission coefficients that take into account the multidimensional nature of tunneling; that is, tunneling paths deviate dramatically from the reaction path. These methods are computationally expensive as they required detailed information (e.g., Hessians) along the reaction path. For reactions with a path curvature assumed to be small, the small curvature tunneling (SCT) method was proposed.46,47 It allows for corner-cutting approximately on the concave side of the turning points of the vibrations transverse to the minimum energy path. For larger curvatures, complications arise as contributions from tunneling into excited states may have to be considered. In that case, the large curvature tunneling (LCT) method can be used to calculate transmission coefficients.48 For the n-heptyl radicals, we carried out calculations of transmission coefficients with the SCT method. The minimum energy paths (MEP) were determined at the B3LYP/6-311G (2d,d,p) level of theory with direct dynamics calculations. MEP calculations were performed in Cartesian coordinates with a step size of 0.01 Å using the Euler steepest-descents integrator. This step size was found to be sufficiently small to converge the reaction path and the transmission probabilities in the range of reaction coordinates ranging from 2.0 to +2.5 Å for 16 and 15 hydrogen shifts, and 2.8 to 3 Å and 2.5 to +3 Å for 14 and 25 hydrogen migrations, respectively. All of the SCT transmission coefficients reported here were calculated using the POLYRATE 9.749 and GAUSSRATE 9.750 codes. LCT calculations were also performed to determine if a large curvature mechanism could be important for the range of reaction coordinate studied here. We do not expect to obtain accurate transmission coefficients from the LCT method as the reaction coordinate ranges may be too small for a proper description of a large curvature path, and it may be necessary to include tunneling in excited states to obtain accurate results.
Figure 2. Potential energy of H-atom shift in the n-heptyl radicals computed using the CBS-QB3 method at 0 K.
One-dimensional transmission coefficients were also computed using Wigner and Skodje and Truhlar approximations directly from CBS-QB3 results on stationary points of the PES. The one-dimensional Eckart transmission coefficients were calculated using ChemRate software where the characteristic length of the Eckart function is obtained from forward and reverse barrier heights (E1 and E1) at 0 K along with the imaginary frequency (νi) of the transition state using the equations reported by Johnston and Heicklen.51 Note that in Chemrate, canonical transmission coefficients are calculated by integrating microcanonical transmission probabilities over the Boltzmann-weighted energy distribution. Lennard-Jones parameters were taken from the JetSurF version 1.0 transport database.52,53 Calculated characteristic lengths, as well as parameters E1, E1, and νi, are given in the Supporting Information.
3. RESULTS AND DISCUSSION Potential Energy Surfaces. The potential energy for hydrogen shift in n-heptyl is presented in Figure 2. As the heptyl radicals allow for secondary to secondary H shift, we will follow the notation proposed by Hardwidge et al.19 that explicitly retains this information. Within this framework, a particular hydrogen transfer will be described as an iab process, where i is the ring size of the cyclic transition state structure, a refers to a primary or secondary radical (noted p or s) for a reactant, and b is p or s for a product. Within this nomenclature, the 14 hydrogen transfer of 1-pentyl radical would be referred to as a 5ps isomerization. It is known that transmission coefficients are sensitive to the barrier heights. Therefore, to be able to analyze the SCT transmission coefficients obtained with molecular parameters obtained at the B3LYP/6-311G(2d,d,p) level of theory and the Eckart k from CBS-QB3//B3LYP/6-311G(2d,d,p), the critical energies computed at these two level of energies were compared. From Table 1, it can be seen that the CBS-QB3 and DFT energy barriers computed for heptyl are very close to each other with the largest deviation being 0.7 kcal/mol. Consequently, comparisons can be made directly between the one-dimensional and SCT transmission coefficients. Comparisons are also made with the G3MP2B3 results of Hayes and Burgess27 (Table 1). The CBS-QB3 critical energies are found to be systematically lower than the G3MP2B3 values. The discrepancies range from 1.1 kcal/mol (5ss) to 1.5 kcal/mol (5 ps), with a mean absolute deviation of 1.2 kcal/mol. These differences are perhaps within the limits of the uncertainty of the 323
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300 and 400 K than the present results. In particular, their Eckart transmission coefficients are larger than our values. This fact highlights the strong sensitivity of Eckart k(T) to the barrier height and the imaginary frequency, an issue to be discussed later. Although the discrepancy between the Eckart and SCT methods is small, the fundamental cause for the apparent agreement is clear. The Eckart method inherently underestimates the tunneling probability by neglecting corner-cutting. In most cases, this underestimation is balanced by the fact that the Eckart function yields a potential energy curve narrower than the actual PES, as illustrated in Figure 4, thus leading to an overestimation of the tunneling probability. Hence, the good agreement between Eckart and SCT transmission coefficients is in many ways fortuitous because of error cancellation in the Eckart approximation. For the same reason, the ability of the Eckart method in reproducing the SCT results should not be generalized to other reaction systems. For hydrogen shift reactions studied here, however, the Eckart method is accurate. Zhang and Dibble60 studied the impact of tunneling on hydrogen-transfer isomerizations in n-propylperoxy radical using multidimensional SCT calculations as well as the onedimensional, Eckart, and Wigner transmission coefficients. They reached a similar conclusion for their system. The Eckart method works well as compared to the SCT result, but they cautioned that the agreement may not be generalized to other systems. Multidimensional tunneling calculations allow for the calculation of representative tunneling energies. This energy represents the path with the greatest tunneling probability at a given temperature. SCT representative tunneling energies for hydrogen transfers in n-heptyl radicals system are presented in Table 4. Below 800 K, the representative tunneling energies are well below the barrier top, and the tunneling paths can deviate dramatically from the reaction path. As an example, Figure 5 presents the PES for 1-heptyl / 4-heptyl and the representative tunneling energy at 300 K. Above 800 K, tunneling and reaction paths are converging, and a one-dimensional approximation is generally adequate. For large critical ring structures, for example, 6ps and 7ps, and at low temperatures, the representative tunneling energy is closer to the energy of the saddle point than for the 5ps or 5ss hydrogen shift. Hence, corner-cutting is enhanced for the more strained transition state structures. It is for this reason that SCT calculations show that the transmission coefficient increases with a decrease in the critical ring size. For example, k(T) values are predicted to be 268 and 270 for 5ps and 5ss H-atom shifts at 300 K, respectively. For 6ps and 7ps transitions, k(T) values are notably smaller and equal to 30.8 and 14.3, respectively. The difference remains significant until 800 K, the temperature above which the transmission coefficients are close to unity. We compare our SCT results to those of Zheng and Truhlar32 for 1-pentyl 5ps and 1-hexyl 5ps and 6ps H-atom shifts and those for 1-butyl 5pp and 1-pentyl 6pp of Ratkiewitcz et al.29 As shown in Table 5, k(300 K) values calculated for H-shift in the fivemembered ring transition structure is in reasonably good agreement with literature values. For six-membered ring structures, the k(300 K) value of Zheng and Truhlar is larger than ours by about a factor of 3 and that of Ratkiewitcz et al. by a factor of 4. Note that the Hessians were computed for all of the points along the MEP in both Ratkiewitcz et al. and our SCT/VTST calculations. In particular, Ratkiewitcz et al. performed a considerable number of force constant calculations along the MEP to ensure convergence of SCT calculations (150 points on each side of
Table 2. CBS-QB3 Critical Energies (kcal/mol, 0 K) Computed for ips and iss Hydrogen Shift species
7ps
6ps
5ps
n-hexyl n-heptyl
14.7
15.3 15.0
22.1 22.0
n-octyl
14.5
15.0
22.0
n-pentyl
6ss
5ss
22.3 22.8 15.7
22.6
Table 3. Transmission Coefficients Computed for 5ps H-Atom Shift in 1-Heptyl Radical Using Wigner, Skodje and Truhlar (S&T), Eckart, and SCT Approximations transmission coefficient, k(T) T (K)
Wigner
S&T
Eckart
SCT
300
4.2
1.1 105
454
268
400
2.8
55.5
13.0
15.6
500
2.2
5.4
4.2
4.9
600
1.8
2.7
2.6
2.8
800
1.5
1.7
1.7
1.7
1000 1500
1.3 1.1
1.4 1.1
1.4 1.2
1.4 1.2
two methods, but we expect the present CBS-QB3 results to be more reliable. It has also been reported that CBS-QB3 is able to accurately predict thermochemical and kinetic data for hydrocarbon combustion.5458 In addition, Wang et al.59 studied the thermal decomposition and isomerization of 1-hexyl radical and showed that the CBS-QB3 calculations were able to reproduce experimental kinetic data. A comparison of the critical energies computed for H-shift in different n-alkyl radicals is presented in Table 2. It can be seen that for a given type of reaction, all critical energies lie within 0.3 kcal/mol of each other, indicating that the reaction energetics is primarily a function of the ring strain energy in the critical geometry, as will be discussed later. Transmission Coefficients. We shall start here using the heptyl radical as the focal point of discussion. As shown in Table 3, transmission coefficients are calculated for 1-heptyl / 4-heptyl hydrogen transfer considering four approximations. For T g 800 K, the transmission coefficients are close to each other for all approximations considered. Below 800 K, however, discrepancies become progressively larger. For 300 e T e 500 K, transmission coefficients calculated with the SkodjeTruhlar (S&T) approximation give the highest values, while Wigner’s method produces the lowest values; both are substantially different from the higher-order SCT results (Figure 3). In comparison, the Eckart method provides a reasonably good approximation of the SCT results above 300 K. The largest discrepancy is for the 7ps isomerization (1- to 2-heptyl) with the Eckart k value 2.8 times that of the SCT value at 300 K. This difference is generally smaller than that resulting from the various uncertainties in the electronic structure calculation itself. At 400 K, the deviation becomes substantially smaller and is only about 20%. Above 500 K, SCT and Eckart transmission coefficients are essentially identical. It is noteworthy to mention that Ratkiewicz et al.29 compared Eckart and SCT transmission coefficient for ipp isomerizations, with i = 4, 5, 6, and 7. They found somewhat larger differences between the Eckart and SCT k(T) values at 324
dx.doi.org/10.1021/jp209360u |J. Phys. Chem. A 2012, 116, 319–332
The Journal of Physical Chemistry A
ARTICLE
Figure 3. Comparison of the transmission coefficient k(T) for 1- and 2-heptyl radicals calculated using the Wigner, Skodje and Truhlar (S&T), Eckart, and SCT approximations. Symbols are computed values. Lines are drawn to guide the eye.
Table 4. Representative Tunneling Energies (Ert) as a Function of Temperature Relative to the Vibrational Ground-State Energy of the Saddle Point (E0Kq) for Hydrogen Shift in n-Heptyl E0Kq Ert (kcal/mol) T (K)
5ps
6ps
7ps
5ss
300 400
8.1 5.1
2.3 1.3
2.3 1.7
4.0 2.3
500
1.8
0.7
1.5
2.0
600
1.0
0.4
1.0
1.7
800
0.3
