Tunneling in Hydrogen-Transfer Isomerization of n-Alkyl Radicals

Published: November 30, 2011

r 2011 American Chemical Society 319 dx.doi.org/10.1021/jp209360u | J. Phys. Chem. A 2012, 116, 319332

ARTICLE

pubs.acs.org/JPCA

Tunneling in Hydrogen-Transfer Isomerization of n-Alkyl RadicalsBaptiste 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

1. INTRODUCTION

Understanding the thermal decomposition reactions of alkylradicals is critical to the development of combustion models ofhydrocarbon fuels. In n-alkane oxidation at high temperatures,alkyl radicals are initially formed by CC or CH ssion in thefuel molecule or H-abstraction reaction by a free radical. Smallalkyl radicals (Ce3) decompose mainly by CC -scission, asthe 12 and 13 hydrogen shifts can occur only through three-or four-membered ring transition state structures with largeenergy barriers due to strain energies.1 For large alkyl radicals,the thermal decomposition is complicated by internal hydro-gen shifts.27 These isomerization processes occur throughcyclic transition state structures. Internal isomerization can occurthrough ve-, six-, or seven-member ring transition state struc-tures, which may be denoted as 14, 15, and 16 hydrogenshifts, respectively. Some of these transition state structures areassociated with ring strain energies, while others are practicallyunstrained. H-atom shifts are the predominant unimolecularreactions under lower temperature conditions. As the tempera-ture is increased, -scissions become increasingly important,but H-atom shift remains critical to the dissociation of alkylradicals and especially the product distributions resulting fromthe -scission processes. The combustion chemistry of olens

and smaller alkyl radicals, formed in the decomposition process,play a key role in the reactivity of the fuel and formation ofpollutants and polycyclic aromatic hydrocarbon (PAH) andsoot.812

The basic problemwith studying alkyl isomerization processesis that they involve large organic moieties (C5 or above). Exceptat the lowest temperatures, -scission will always make contribu-tions. Unless careful attention is paid to setting the reactionconditions to minimize mechanistic artifacts, it is usually dicultto derive truly quantitative kinetic parameters. In the case of singlepulse shock tube experiments, the isomerization rate constantsderived are all dependent on the initial -scission rate constants,which, in turn, are derived from literature values. Thus, theuncertainty in the -scission rate constants inevitably propagatesinto the isomerization rate constants. For this type of tightlycoupled unimolecular reaction involving competitive reactionpaths, theoretical reaction kinetics is often necessary to interpretthe experimental data.

Received: September 28, 2011Revised: November 22, 2011

ABSTRACT: The role of quantum tunneling in hydrogen shift inlinear heptyl radicals is explored using multidimensional, small-curvature tunneling method for the transmission coecients and apotential energy surface computed at theCBS-QB3 level of theory.Several one-dimensional approximations (Wigner, Skodje andTruhlar, and Eckart methods) were compared to the multidimen-sional results. The Eckart method was found to be sucientlyaccurate in comparison to the small-curvature tunneling results fora wide range of temperature, but this agreement is in fact fortuitousand caused by error cancellations. High-pressure limit rate con-stants were calculated using the transition state theory with treatment of hindered rotations and Eckart transmission coecients for allhydrogen-transfer isomerizations in n-pentyl to n-octyl radicals. Rate constants are found in good agreementwith experimental kinetic dataavailable for n-pentyl and n-hexyl radicals. In the case of n-heptyl and n-octyl, our calculated rates agree well with limited experimentallyderived 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 conrmed theoretically: older low-temperature experimental data, characterized by small pre-exponential factors and activation energies, can be reconciled with high-temperature data by taking into account tunneling; at lowtemperatures, transmission coecients are substantially larger for H-atom transfers through a ve-membered ring transition state thanthosewith six-membered rings; channelswith transition ring structures involving greater than8 atoms can be neglected because of entropiceects that inhibit such transitions. The set of computational kinetic rates were used to derive a general rate rule that explicitly accounts fortunneling. The rate rule is shown to reproduce closely the theoretical rate constants.

320 dx.doi.org/10.1021/jp209360u |J. Phys. Chem. A 2012, 116, 319332

The Journal of Physical Chemistry A ARTICLE

Reactions involving the transfer of a hydrogen atom are in-herently subject to quantum tunneling eect.13 The role of tunnel-ing in n-alkyl radical isomerizations was recently highlighted byTsang and co-workers.47 They showed that tunneling must beconsidered to reconcile older data obtained at low temperatureswith measurements made in their high-temperature single-pulseshock tube. Low-temperature kinetic data have been subject tocontroversies because of the unusually small pre-exponential factors(A) that were dicult to rationalize with the thermochemicalkinetic theory.14Directmeasurements of the isomerization reactionrates of alkyl radicals are dicult, and kinetic data were derivedusually from analyses of complex reaction mechanisms. Moreover,most experiments were performed in the fallo region and/orinvolved a chemically activated process. In such cases, the high-pressure limit rate constant must be extrapolated from the RiceRamspergerKasselMarcus (RRKM) theory and, for the lateststudies, from Master Equation (ME) analysis.

Among all n-alkyl radicals, the unimolecular reactions of1-pentyl and 1-hexyl have been studied most extensively in thepast. Figure 1 presents selected literature rate constants. End-renyi and Le Roy15 proposed the rst experimental rate constantfor the gas-phase 14 hydrogen migration in n-pentyl radicals:

1 C5H11 / 2 C5H11 R1They reported:

k1 s1 1:4 107e5440=T 1for temperatures 439 e Te 503 K at low pressures. Within theframework of transition state theory, they concluded that theunusual low pre-exponential factor can be explained only byconsidering a quantum tunneling eect. Watkins16 derived a rateexpression from experiments performed 297 e T e 435 K andpressures from 1 to 30 Torr:

k1 s1 3:3 108e7600=T 2Watkins also noted the unusually low A factor of rate constantand proposed that errors resulting from photochemical activationcould lead to an underestimation of the A factor. In a follow-uppaper,17 Watkins proposed a rate expression with a larger A factor

based on a newmechanistic interpretation of the data of Endrenyiand Le Roy and proposed:

k1 s1 5:0 1011e10600=T 3Similar low A factor values were observed from low-temperatureexperiments on n-hexyl radicals.Watkins andOstreko18 proposeda rate expression for the 15 hydrogen migration in n-hexylradicals:

1 C6H13 / 2 C6H13 R2for temperatures from 352 to 405 K and a pressure of 46 Torr:

k2 s1 2:0 107e4180=T 4The small A factor has been a subject of debate in theliterature.19,20 In the late 1980s, Dobe et al.21 reported an experi-mental investigation of k2 for 300 e T e 453 K and pressuresfrom 100 to 200 Torr and proposed a rate expression that againfeatures a small A factor:

k2 s1 3:16 107e5840=T 5Using the same experimental approach, these authors also pro-posed rate expressions for n-octyl radicals isomerization withsimilarA factors.Marshall22 examined the thermal decompositionof n-pentane in the temperature range of 737923 K and pres-sures below 200 Torr. The rate expression for the 14 hydrogenshift of reaction R1 was derived from the distribution of majorproducts as

k1 s1 9:1 1011e11800=T 6By considering earlier low-temperature measurements, he pro-posed that

k1 s1 1:2 1011e10100=T 7for 438 e T e 923 K. Imbert and Marshall23 followed a similarapproach to determine the high-pressure limit rate constant15 hydrogen transfer in n-hexyl radicals by n-hexane pyrolysis

Figure 1. Arrhenius plots of (a) 14 hydrogen shift in n-pentyl radical, and (b) 15 hydrogen shift in n-hexyl radical.



and proposed:

k2 s1 3:16 1010e8560=T 8for 723 e T e 823 K. Miyoshi and co-workers24 derived theexpressions for k1 and k2 from an analysis of shock-tube experi-ments for 900eTe 1400 K near the atmospheric pressure usinga complex reaction mechanism. They proposed expressions forthe high-pressure limit rate constants k from an RRKM analysisof their experimental results along with earlier, low temperaturedata. They included the tunneling eect via a Wigner correctionwith the imaginary frequencies taken fromHF/6-31G(d) calcula-tions. They proposed:

k1, s1 4:9 108T 0:846e9830=T 9and

k2, s1 6:7 107T 0:823e6570=T 10for 350 e T e 1300 K. The somewhat strong and positivetemperature exponents point to an apparent upward curvature ofthe Arrhenius form largely because of the tunneling eect beingconsidered in the rate calculation.

Miyoshi and co-workers25 reported the rst direct measure-ment of the rate coecients of 14 hydrogen shift in 1-pentylradical in the temperature range of 440520 K and pres-sures from 1 to 7 Torr. From an RRKM/ME analysis includingEckart tunneling, they evaluated the high-pressure limit rate con-stant to be

k1, s1 1:9 1010e8870=T 11for 440 e T e 520 K and

k1, s1 2:4 103T2:324e8180=T 9:1

105e5300=T 12300eTe 1300 K. The use of a bi-Arrhenius form is the result ofa large tunneling eect, leaving a high Arrhenius curvature.

More recently, Tsang and co-workers carried out a series ofsystematic studies on the thermal decomposition of n-pentyl,6

n-hexyl,7 n-heptyl,4 and n-octyl5 radicals in single-pulsed shocktube for temperatures from 850 to 1050 K and pressures between1.5 and 7 bar. From experimentally determined product branch-ing ratios, they deduced high-pressure limit rate expressions for14, 15, and 16 hydrogen transfers using an RRKM/MEanalysis. In the case of n-octyl, they also explored a hydrogenmigration channel through the eight-membered ring transitionstate structure, but found that such a transition is unimportantcompared to other reaction channels. Earlier, lower temperaturedata on n-pentyl and n-hexyl were included in their theoreticalanalysis considering Eckart tunneling. Notably, they proposedthe high-pressure limit rate constant:

k1, s1 1:0 1012e11300=T 13for 14 hydrogen shift in 1-pentyl from 850 to 1000 K, and

k2, s1 1:8 102T2:55e5550=T 14for 15 hydrogen migration in 1-hexyl for 500 e T e 1900 K.

Theoretically, Viskolcz et al.1 computed the energy barriers ofseveral isomerization reactions of linear alkyl radicals and thebarrier of the 14 hydrogen shift in 1-pentyl at the MP-SAC2//UHF/6-31G* level of theory. They discussed their results within

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 raterules for 12, 13, 14, 15, and 16 hydrogen shifts in alkylradicals. Hayes and Burgess27 calculated the energy barriers ofhydrogen transfer in alkyl, allyilc, and oxoallylic radicals usingseveral composite methods and showed that an EvansPolanyicorrelation can be developed in the case of linear alkyl radicals.Quantum tunneling eects were not considered in these studies.Truong and co-workers28 studied the 1,4-intramolecular hydro-gen migration in linear alkyl radicals using the class transitionstate theory. They calculated the rate constant for the referencereaction of n-C4H9 using canonical variational transition statetheory (CVTST) with the small curvature tunneling (SCT)correction. It is important to note here that SCT approximationallows for a multi-dimensional treatment of tunneling throughcomputationally expensive calculation of the Hessian for numer-ous points along the reaction path. In their study, the poten-tial energy surface (PES) was computed at the CCSD(T)/cc-pVDZ//BH&HLYP/cc-pVDZ level of theory. They found asignicant tunneling contribution at temperatures below 1000 K.Following their study on 14 hydrogen shift, they recentlyproposed high-pressure limit rate expressions for 13 to 16hydrogenmigrations in linear alkyl radicals.29 They described theisomerization reactions of n-propyl to n-hexyl radicals using thesame methodology. The rate constants for analogous reactionswere proposed on the basis of these prototype reactions withinthe framework of reaction class TST. Again, the tunneling eectwas found to be less prominent above 1000 K. Additionally, theycompared transmission coecients obtained from multidimen-sional SCT tunneling with those obtained from the one-dimen-sional Eckart function and showed that the Eckart tunnelingeect is more pronounced than SCT tunneling below 400 K.

Jitariu et al.30 carried out direct dynamic calculations withCVTST/SCT to study the decomposition and isomerizationpathways of n-pentyl radicals. They reported dual-level calcula-tions at the PUMP2-SAC/6-311G**///AM1 level of theory.Here, the triple slash (///) denotes interpolating optimizedcorrections (IOC) in the VTST calculations.31 In this method,the PES along the reaction path are corrected using higher-levelvalues at selected points. The Hessians, the potential energies,and moment of inertia along the reaction path were computedat the AM1 level of theory. UMP2/6-311G** geometries, vib-rational frequencies, moments of inertia, and PUMP-SAC2/6-311G** energies of the stationary points were then used toscale the low-level AM1 results. This rather elaborate approachled them to conclude that tunneling is pronounced at lowtemperatures in the 14 hydrogen shift of 1-pentyl. At 1000K, however, their proposed rate constant is about a factor 3 largerthan experimental values of Tsang et al.6,24 and Yamauchi et al.6,24

Zheng and Truhlar studied the 14H shift in the 1-pentyl radicaland the 14 and 15 hydrogen transfers in the 1-hexyl radicalusing CVTST/SCT theory.32 They calculated the PES withseveral levels of density functional theory (DFT) (e.g., M06-2X/MG3S andM08-HX/cc-pVTZ+) andmultilevel methods forthe stationary points. They used the interpolated variationaltransition state theory by mapping (IVTST-M) in all of theirdynamics calculations. This method gives PES data (energies,gradients, and Hessian) along the full reaction path based on alimited number of points calculated near the saddle point. Theycompared several one-dimensional approximations (Wigner, zerocurvature tunneling, and parabolic tunneling approximation) to



their multidimensional SCT results. They noted that at lowtemperatures the discrepancies among these methods are ratherlarge. An interesting point is that tunneling is more important inthe case of 14 hydrogen transfer than for 15 shift, in agree-ment with the work of Tsang and co-workers. Unfortunately, thewidely used Eckart method was excluded from their study, andits ability to reproduce results of higher dimensional tunnelingtreatments remains unclear.

In the present study, we intend to address several issuesregarding the role of tunneling in linear alkyl radical isomeriza-tions.We compute the transmission coecients using n-heptyl asthe model system employing a multi dimensional treatment. Theresults allow us to assess accuracy of the various one-dimensionalmethods for the calculation of transmission coecients. Beyondthe n-heptyl radical, we use the rigid rotor harmonic oscillator(RRHO) approximation with corrections for hindered internalrotation (HR), the TST theory with a selected 1-D tunnelingmethod, and we systematically determine the rate coecients for14 to 16 H-shifts in n-alkyl radicals ranging from n-pentyl ton-octyl. Note that as the size of the linear alkyl chain increases,most favorable isomerizations (ve-, six-, and seven-membertransition state structures) can involve hydrogen shift betweentwo secondary radicals (e.g., 2-heptyl to 3-heptyl). It is note-worthy to mention that there are almost no kinetic data availablefor these types of processes. Finally, we discuss the role oftunneling within the framework of correlations between struc-ture and reactivity, and its impact on the uncertainty of ratecoecients derived from theoretical calculations.

2. COMPUTATIONAL DETAILS

Electronic Structure Calculations. Potential energy andmolecular properties of stationary points were calculated usingGaussian 03 revision B.0333 and QChem version 3.1.34 For allstationary and critical structures, geometry optimizations wereperformed at the B3LYP/6-311G(2d,d,p) level of theory.35,36

Frequency calculations were performed at the same level oftheory for all optimized geometries to determine the nature ofstationary points. The composite CBS-QB3 method was appliedfor all stationary geometries and transition states.37 The CBS-QB3 model involves a five-step calculation starting with ageometry optimization and a frequency calculation (scaled by afactor 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 completebasis set extrapolation to correct the total energy. Dynamic calcu-lations along the reaction path were performed at the B3LYP/6-311G(2d,d,p) level of theory. We found that for reactionsexamined here, the energy barrier heights with zero-point energy

(ZPE) were quite close at the B3LYP/6-311G(2d,d,p) and CBS-QB3 levels of theory (Table 1).Hindered Internal Rotations. The description of isomeriza-

tion channels of large aliphatic chains requires an accurate treat-ment of low-frequency internal rotations. Using the harmonicoscillator (HO) approximation to describe these torsion modescan lead to large errors in the partition function. Vansteenkisteet al.38 calculated the thermodynamic properties of n-alkanesusing a quantum-mechanical treatment and showed that treatinghindered internal rotation is critical to obtain accurate entropiesand heat capacities values. As hydrogen transfers occur throughcyclic transition state structures, most of the internal rotations ofthe initial reacting radicals are locked in the ring, and the ratecoefficients will be strongly dependent on the entropy variation.As an example, Vansteenkiste et al. reported for n-heptane adifference of 9.3 cal/(mol K) at 1000 K in entropy betweencalculations using the assumption of harmonic oscillator (HO)and a more precise quantum mechanical treatment. The HOapproximation underestimates the entropy change. In the pre-sent work, hindered internal rotations were treated using thefollowing procedure. First, the potentials of each internal rotationof 1-pentyl, 2-pentyl, and 3-pentyl radicals were calculated at theB3LYP/6-311G(2d,d,p) level of theory using a relaxed energyscan. The energy barriers for these hindered internal rotors werethen calculated at the CBS-QB3 level of theory. The character-istics of the rotational potentials and the barriers of rotationobtained were used to correct the HO partition function usingPitzer andGwinn tabulations.39 Rotational potential functions andbarriers for each internal rotor in n-pentyl were used for similarinternal rotors found in largest alkyl radicals. Reduced moments ofinertia for internal rotations were calculated for each species usingB3LYP/6-311G(2d,d,p) geometries with the method of Pitzerimplemented in ChemRate.40,41 For cyclic transition states, thevibrational modes of the cyclic part of themolecular structure weredescribed within the HO approximation, and the lateral alkylgroup internal rotations were treated as hindered rotors (HR) viarelaxed scans (with cyclic bond lengths frozen) in the critical geo-metries for 1-pentyl to 2-pentyl and 1-hexyl to 3-hexyl. Parametersfor HR corrections for these two critical geometries were used forall other similar internal rotors found in the saddle point geome-tries of larger alkyl radicals. Hindrance potentials and barrierheights are given in the Supporting Information.Transmission Coefficient.Within a canonical TST or VTST

framework, quantum tunneling is taken into account by atemperature-dependent transmission coefficient k(T):

kT kTAeE=T 15Within the framework of the RRKM theory and to account fortunneling, Miller42 proposed one-dimensional tunneling prob-ability P(E,J) in the sum of states N(E,J) of the transition state:

NE, J nPE Eqn, J 16

where nq is the nth vibrational energy level and J is the angular

momentum. The standard microcanonical rate expression iscalculated using N(E,J):

kE, J NE, JhFE, J 17

where h is the Planck constant and F(E,J) is the density of energystates of the reactant. Note that k(T) can be calculated from P(E,J)

Table 1. Critical Energies (kcal/mol) Computed for H-AtomShifts in n-Heptyl Radicals at 0 K

B3LYP/

6-311G(2d,d,p)a CBS-QB3a G3MP2B327

7ps (1-heptyl/ 2-heptyl) 14.1 14.7 15.86ps (1-heptyl/ 3-heptyl) 14.6 15.0 16.15ps (1-heptyl/ 4-heptyl) 21.3 22.0 23.55ss (2-heptyl/ 3-heptyl) 23.2 22.8 23.9

aThis work.



by integrating over the Boltzmann-weighted energy distributionand that k(T) also includes nonclassical reflection effects.For a one-dimensional system, the transmission coecient

may be calculated by solving the Schrodinger equation with anappropriate potential function. In this approach, the potentialenergy surface is tted with a function for which the tunnelingprobabilities are known analytically. The simplest method is thatof Wigner43 who assumed a parabolic function for the PESleading to the transmission coecient:

kT 1 124

hImqkBT

" #218

where q is the imaginary frequency corresponding to thereaction coordinate, and kB is the Boltzmann constant. Skodjeand Truhlar44 proposed an improvement over the Wignermethod 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 ofthe one-dimensional PES. Moreover, the parameters of theEckart potential can be tted to reproduce the curvature at thesaddle point and the exothermicity of the reaction.45 This one-dimensional method is widely used in the theoretical combustionkinetics because they are rather inexpensive computationally yetaccurate enough for the temperatures of interest.More sophisticated methods to calculate the transmission co-

ecient have been proposed by Truhlar and co-workers. Parti-cularly, they developed semiclassical methods to calculate trans-mission coecients that take into account the multidimensionalnature of tunneling; that is, tunneling paths deviate drama-tically from the reaction path. Thesemethods are computationallyexpensive as they required detailed information (e.g., Hessians)along the reaction path. For reactions with a path curvature as-sumed to be small, the small curvature tunneling (SCT) methodwas proposed.46,47 It allows for corner-cutting approximately onthe concave side of the turning points of the vibrations transverseto theminimum energy path. For larger curvatures, complicationsarise as contributions from tunneling into excited states may haveto be considered. In that case, the large curvature tunneling (LCT)method can be used to calculate transmission coecients.48

For the n-heptyl radicals, we carried out calculations oftransmission coecients with the SCT method. The minimumenergy paths (MEP) were determined at the B3LYP/6-311G(2d,d,p) level of theory with direct dynamics calculations. MEPcalculations were performed in Cartesian coordinates with a stepsize of 0.01 using the Euler steepest-descents integrator. Thisstep size was found to be suciently small to converge thereaction path and the transmission probabilities in the range ofreaction coordinates ranging from 2.0 to +2.5 for 16 and15 hydrogen shifts, and2.8 to 3 and2.5 to +3 for 14and 25 hydrogen migrations, respectively. All of the SCTtransmission coecients reported here were calculated usingthe POLYRATE 9.749 and GAUSSRATE 9.750 codes. LCTcalculations were also performed to determine if a large curvaturemechanism could be important for the range of reaction co-ordinate studied here. We do not expect to obtain accuratetransmission coecients from the LCT method as the reactioncoordinate ranges may be too small for a proper description of alarge curvature path, and it may be necessary to include tunnelingin excited states to obtain accurate results.

One-dimensional transmission coecients were also com-puted using Wigner and Skodje and Truhlar approximationsdirectly from CBS-QB3 results on stationary points of the PES.The one-dimensional Eckart transmission coecients were calcu-lated using ChemRate software where the characteristic lengthof the Eckart function is obtained from forward and reversebarrier heights (E1 and E1) at 0 K along with the imaginaryfrequency (i) of the transition state using the equations reportedby Johnston and Heicklen.51 Note that in Chemrate, canonicaltransmission coecients are calculated by integrating microca-nonical transmission probabilities over the Boltzmann-weightedenergy distribution. Lennard-Jones parameters were taken fromthe JetSurF version 1.0 transport database.52,53 Calculated char-acteristic lengths, as well as parametersE1, E1, and i, are given inthe Supporting Information.

3. RESULTS AND DISCUSSION

Potential Energy Surfaces. The potential energy for hydro-gen shift in n-heptyl is presented in Figure 2. As the heptylradicals allow for secondary to secondary H shift, we will followthe notation proposed byHardwidge et al.19 that explicitly retainsthis information. Within this framework, a particular hydrogentransfer will be described as an iab process, where i is the ring sizeof the cyclic transition state structure, a refers to a primary orsecondary radical (noted p or s) for a reactant, and b is p or s for aproduct. Within this nomenclature, the 14 hydrogen transfer of1-pentyl radical would be referred to as a 5ps isomerization.It is known that transmission coecients are sensitive to the

barrier heights. Therefore, to be able to analyze the SCT trans-mission coecients obtained with molecular parameters ob-tained at the B3LYP/6-311G(2d,d,p) level of theory and theEckart k from CBS-QB3//B3LYP/6-311G(2d,d,p), the criticalenergies computed at these two level of energies were compared.From Table 1, it can be seen that the CBS-QB3 and DFT energybarriers computed for heptyl are very close to each other with thelargest deviation being 0.7 kcal/mol. Consequently, comparisonscan be made directly between the one-dimensional and SCTtransmission coecients.Comparisons are also made with the G3MP2B3 results of

Hayes and Burgess27 (Table 1). The CBS-QB3 critical energiesare 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. Thesedierences are perhaps within the limits of the uncertainty of the

Figure 2. Potential energy of H-atom shift in the n-heptyl radicalscomputed using the CBS-QB3 method at 0 K.



two methods, but we expect the present CBS-QB3 results to bemore reliable. It has also been reported that CBS-QB3 is able toaccurately predict thermochemical and kinetic data for hydro-carbon combustion.5458 In addition, Wang et al.59 studied thethermal decomposition and isomerization of 1-hexyl radical andshowed that the CBS-QB3 calculations were able to reproduceexperimental kinetic data.A comparison of the critical energies computed for H-shift in

dierent n-alkyl radicals is presented in Table 2. It can be seenthat for a given type of reaction, all critical energies lie within0.3 kcal/mol of each other, indicating that the reaction energeticsis primarily a function of the ring strain energy in the criticalgeometry, as will be discussed later.Transmission Coefficients. We shall start here using the

heptyl radical as the focal point of discussion. As shown inTable 3, transmission coefficients are calculated for 1-heptyl /4-heptyl hydrogen transfer considering four approximations. ForT g 800 K, the transmission coefficients are close to each otherfor all approximations considered. Below 800 K, however,discrepancies become progressively larger. For 300eTe 500 K,transmission coefficients calculated with the SkodjeTruhlar(S&T) approximation give the highest values, while Wignersmethod produces the lowest values; both are substantially dif-ferent from the higher-order SCT results (Figure 3). In com-parison, the Eckart method provides a reasonably good approx-imation of the SCT results above 300 K. The largest discrepancyis for the 7ps isomerization (1- to 2-heptyl) with the Eckartk value 2.8 times that of the SCT value at 300 K. This difference isgenerally smaller than that resulting from the various uncertain-ties in the electronic structure calculation itself. At 400 K, thedeviation becomes substantially smaller and is only about 20%.Above 500 K, SCT and Eckart transmission coefficients areessentially identical. It is noteworthy to mention that Ratkiewiczet al.29 compared Eckart and SCT transmission coefficient for ippisomerizations, with i = 4, 5, 6, and 7. They found somewhatlarger differences between the Eckart and SCT k(T) values at

300 and 400 K than the present results. In particular, their Eckarttransmission coefficients are larger than our values. This facthighlights the strong sensitivity of Eckart k(T) to the barrierheight 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 agree-ment is clear. The Eckart method inherently underestimates thetunneling probability by neglecting corner-cutting. In most cases,this underestimation is balanced by the fact that the Eckartfunction yields a potential energy curve narrower than the actualPES, as illustrated in Figure 4, thus leading to an overestimationof the tunneling probability. Hence, the good agreement betweenEckart and SCT transmission coecients is in many waysfortuitous because of error cancellation in the Eckart approxima-tion. For the same reason, the ability of the Eckart method inreproducing the SCT results should not be generalized to otherreaction 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 radicalusing multidimensional SCT calculations as well as the one-dimensional, Eckart, and Wigner transmission coecients. Theyreached a similar conclusion for their system. The Eckart methodworks well as compared to the SCT result, but they cautionedthat the agreement may not be generalized to other systems.Multidimensional tunneling calculations allow for the calcula-

tion of representative tunneling energies. This energy representsthe path with the greatest tunneling probability at a giventemperature. SCT representative tunneling energies for hydro-gen transfers in n-heptyl radicals system are presented in Table 4.Below 800 K, the representative tunneling energies are wellbelow the barrier top, and the tunneling paths can deviatedramatically from the reaction path. As an example, Figure 5presents the PES for 1-heptyl/ 4-heptyl and the representativetunneling energy at 300 K. Above 800 K, tunneling and reactionpaths are converging, and a one-dimensional approximation isgenerally adequate. For large critical ring structures, for example,6ps and 7ps, and at low temperatures, the representative tunnel-ing energy is closer to the energy of the saddle point than for the5ps or 5ss hydrogen shift. Hence, corner-cutting is enhanced forthe more strained transition state structures. It is for this reasonthat SCT calculations show that the transmission coecientincreases with a decrease in the critical ring size. For example,k(T) values are predicted to be 268 and 270 for 5ps and 5ssH-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 dierence remains signicant until 800 K, thetemperature above which the transmission coecients are closeto 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 thosefor 1-butyl 5pp and 1-pentyl 6pp of Ratkiewitcz et al.29 As shownin Table 5, k(300 K) values calculated for H-shift in the ve-membered ring transition structure is in reasonably good agree-ment with literature values. For six-membered ring structures,the k(300 K) value of Zheng and Truhlar is larger than ours byabout 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 alongthe MEP in both Ratkiewitcz et al. and our SCT/VTST calcula-tions. In particular, Ratkiewitcz et al. performed a considerablenumber of force constant calculations along the MEP to ensureconvergence of SCT calculations (150 points on each side of

Table 2. CBS-QB3 Critical Energies (kcal/mol, 0 K) Com-puted for ips and iss Hydrogen Shift

species 7ps 6ps 5ps 6ss 5ss

n-pentyl 22.3

n-hexyl 15.3 22.1

n-heptyl 14.7 15.0 22.0 22.8

n-octyl 14.5 15.0 22.0 15.7 22.6

Table 3. Transmission Coecients Computed for 5psH-Atom Shift in 1-Heptyl Radical Using Wigner, Skodje andTruhlar (S&T), Eckart, and SCT Approximations

transmission coecient, k(T)

T (K) Wigner S&T Eckart SCT

300 4.2 1.1 105 454 268400 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 1.3 1.4 1.4 1.4

1500 1.1 1.1 1.2 1.2



MEP, leading to a total of 300 points due to symmetry of thepotential function for ipp reactions, with i = 5, 6, and 7).Regardless, ve-membered ring critical geometries are shownto have a greater tunneling tendency than the six-membered ringtransition in all of the studies.

As was already mentioned above, large curvature calculationswere also performed in a somewhat qualitativemanner to examinewhether the small curvature approximation is adequate forH-shiftin n-heptyl radicals. The LCT approximation describes tunnelingpaths that are near and far from the minimum energy path, andallows for a large degree of corner-cutting. Calculations show thatanLCTtunnelingmechanismcanbe important over a small, selectedrange of energies along the reaction path, but for reactionsexamined here, the SCT approximation alone is suciently

Figure 3. Comparison of the transmission coecient 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.

Figure 4. Comparison of the potential energy () of 14 H shift in1-heptyl and an Eckart function t (- - -).

Table 4. Representative Tunneling Energies (Ert) as a Func-tion of Temperature Relative to the Vibrational Ground-StateEnergy of the Saddle Point (E0K

q) for Hydrogen Shift inn-Heptyl

E0Kq Ert (kcal/mol)

T (K) 5ps 6ps 7ps 5ss

300 8.1 2.3 2.3 4.0

400 5.1 1.3 1.7 2.3

500 1.8 0.7 1.5 2.0

600 1.0 0.4 1.0 1.7

800 0.3



accurate. LCT tunneling, even for a few energy levels, does leadto larger transmission coecients. For example, for 7ps H-atomshift in 1-heptyl, the transmission coecient is increased by 17%at 300 K and 41% at 250 K.Eckart transmission coecients are known to be sensitive to

the energy barrier height and the imaginary frequency at the

saddle point. Figure 6a presents the results of sensitivity analyseswith respect to the imaginary frequency and barrier height using1-heptyl/ 4-heptyl as an example. It is seen thatwithin(2 kcal/molvariation, k exhibits a small sensitivity with respect to the barrierheight. At 300 K, an increase or decrease of 2 kcal/mol in thebarrier height leads to 30% change in k. What impacts thetransmission coecient the most is the uncertainty in theimaginary frequency (i) of the critical geometry, as shown inFigure 6b. At 600 K, k calculated with 1.1 i is 20% larger thanthe reference value. At 300 K, however, deviations are as large as afactor of 4 due to a 10% change in i.High-Pressure Limit Rate Constants. Figure 7 presents high-

pressure limit rate constants computed for heptyl isomerizationusing the classical TST-HR approximation with Eckart tunneling.Molecular parameters were taken from electronic structurecalculations at the CBS-QB3 level of theory. As observed byZheng and Truhlar32 and Ratkiewicz et al.,29 almost no differencewas found between TST and CVT calculations, indicatingrecrossing to be infrequent (a maximum deviation of 0.3%).Figure 1 presents the computed k for reactions R1 and R2 alongwith literature data. The agreement is generally good, especiallyfor reaction R1 considering the fairly significant scatter in thedata. The high-pressure limit rate constants of reaction R2reported by Imbert and Marshall23 and Watkins and Ostreko18

are probably outside of the uncertainty bound of the currentcalculation and those of other studies.The high-pressure limit rate constants compare well with the

experimentally derived data of Tsang et al.,4,6,7 as illustrated inboth Figures 1 and 7. In general, the dierence is within a factorof 2 of each other. A very interesting point is that our calculationsconrm the conclusion of Tsang et al.4 regarding the dierence in6ps and 7ps hydrogen transfers. They proposed that the rate ofthe 6ps transition is about a factor of 2 greater than the 7pstransition and concluded that larger-ring transition structures willpossibly make a lesser contribution to isomerization because ofentropic constraints. The present results support this view. Fromthe PES displayed in Figure 2, it can be seen that the criticalenergies for 6ps and 7ps isomerizations are similar, yet thecomputed k(6ps)/k(7ps) is 5 over the temperature rangeof 5002000 K.

Figure 5. Vibrationally adiabatic ground-state potential energy as afunction of reaction coordinate for the 14 hydrogen shift in 1-heptyl.The dotted line denotes the representative tunneling energy at 300 K.

Table 5. Comparison of SCT Transmission Coecients at300 K

reaction k reference

5ps (1-pentyl/ 2-pentyl) 231 Zheng and Truhlar32

5ps (1-hexyl/ 3-hexyl) 245 Zheng and Truhlar32

5pp (1-butyl/ 1-butyl) 364 Ratkiewitcz et al.29

5ps (1-heptyl/ 4-heptyl) 268 this work6ps (1-hexyl/ 2-hexyl) 114 Zheng and Truhlar32

6pp (1-pentyl/ 1-pentyl) 38.5 Ratkiewitcz et al.29

6ps (1-heptyl/ 3-heptyl) 30.1 this work

Figure 6. Sensitivity of Eckart transmission coecient computed with respect to (a) critical energy (E0Kq) and (b) the imaginary frequency (i) for 14

H-shift in 1-heptyl with the base case E0Kq and vi computed at the CBS-QB3 level of theory. Symbols are computed values. Lines are drawn to guide

the eye.



Branching ratios of the four unimolecular channels of 1-heptylradical (5ps, 6ps, and 7ps isomerizations andCCbond-scission)are presented in Figure 8, based on the rate constants computedat the high-pressure limit. For CC -scission, the rate con-stant is calculated using the critical geometry of Sirjean et al.determined at the CBS-QB3 level of theory.55 For T



is estimated by adding the energy required to abstract an hydrogenatom of a given type (primary, secondary, or tertiary) to the ringstrain energy in the cyclic transition structure. The pre-exponential(A) factor can be calculated by estimating the entropy of activationby considering the loss or gain of internal rotor(s) and of opticalisomers. This semiempirical method relies on the group additivityprinciple, with values of ring strain energies taken from thosetabulated for cycloalkanes and entropies of activation estimatedfrom analogous cyclic and acyclic alkanes. For example, in an earlystudy,63 a4 cal/(mol K) change in the entropy was proposed foreach rotor locked into the cyclic transition structure. They derivedthis value on the basis of entropy difference between n-butaneand cyclobutane (12 cal/(mol K)) and the three internal rotorsin n-butane.63 Unfortunately, the corrections were derived fromexperimental data of hydrogen-transfer reactions that are inherently

subject to the tunneling effect. Additionally, the pre-exponentialfactors and activation energies derived from these methods alsoinclude the effects of tunneling. Here, we re-examine these raterules by decoupling the tunneling effect from the apparentactivation energies and A factors.We assume that for a given reaction class, transmission coe-

cients follow the same temperature dependency. This assump-tion is supported by the current theoretical results shownFigure 10. As seen, the transmission coecient is primarily afunction of the ring size in the transition state and not a functionof the reactant size. Within each reaction class, the maximumdeviation is smaller than 2% for T > 500 K. Below thistemperature, the maximum deviation is 4% for all reactionsexcept for ve-member ring structures where a 14% maximumdeviation is observed. Tunneling can therefore be taken into

Figure 9. Arrhenius plots for the high-pressure limit rate constants of hydrogen shifts in n-octyl radicals. The bottom right plot represents branchingratios computed for the thermal decomposition of 1-octyl radical.



account within the framework of a semiempirical correlation byusing a modied Arrhenius equation leading to the parameterpresented in Table 7. Here, small curvature transmission coe-cients determined for the n-heptyl system were used as referencedata. The temperature range for the t was chosen to minimizethe errors induced by the tting procedure. In all cases, the ttingerror is



values are consistently 2.5 kcal mol1 above those of cyclicalkanes. Hence, using the ring strain energy of cycloheptanewould lead to an overestimation of the activation energy.Contributions of the loss of internal rotation to the activation

entropy for each hydrogen-transfer reaction studied in this workare presented in Table 9. Only values at 298 K are reported here.Temperature was found to have a negligible eect on the entropyof activation. For ips hydrogen transfers, the loss of one internalrotor causes the entropy of activation to decrease by 3.61( 0.23cal/(mol K), while a mean value of 4.56 ( 0.23 cal/(mol K)per rotor was calculated for iss hydrogen shift. The dierence isnot surprising considering that ips involves CH2CH2 andCH2CH2 3 rotors, but iss involves CH2CH2 internalrotations only.The recommended hydrogen-transfer rate rule is summarized

in Table 7. Comparisons between the theoretical k values andthe rate-rule estimations are shown in Figure 11 for hydrogenshift in n-heptyl radicals. For the temperature range considered(5001500 K), the maximum deviation is about a factor of 2.5.

For most cases, the deviation is well within the estimateduncertainty of the theoretical calculation (around a factor of 2).

4. CONCLUSION

The role of quantum tunneling in hydrogen-transfer isomer-izations of linear alkyl radicals was studied in detail. Transmissioncoecients are shown to require a multidimensional treatmentbelow 800 K. Only above800 K is a one-dimensional treatmentappropriate. The Eckart method is shown to reproduce themultidimensional transmission coecients over the entire tem-perature range, but the agreement is due to a favorable errorcancellation. The inability of the Eckart approach to account forhigher-dimensional tunneling eect is compensated by therigidity of the Eckart function, leading to a potential energycurve narrower than the actual PES. Calculations show thatbelow 600 K the Eckart transmission coecient is highlysensitive to the value of the imaginary frequency, and hence issubject to the uncertainty in the electronic structure calculation.Regardless, high-pressure limit rate constants calculated usingthe classical transition state theory with treatment of internalrotation and the use of Eckart transmission coecients and a PESdetermined at the CBS-QB3 level of theory are in good agree-ment with literature data.

The present results are consistent with Tsang et al., whoconcluded that a large body of literature data at low temperatures(with small A factor and activation energy) can be reconciledwith high-temperature data by taking into account quantumtunneling. Our calculations also conrm their observation thattunneling is more pronounced with ve-membered ring transi-tion structures than with six-membered ring structures. Inaddition, hydrogen transfer through an eight-membered ringtransition structure is not competitive due to prohibitive entropyeects.

The systematic study of isomerization reactions from n-pentylto n-octyl led to a large set of theoretical kinetic data that can berationalized within a reaction class approach. It is shown that thetransmission coecient can be treated as an explicit structure/reactivity correlation parameter. This approach allows for a fastand suciently accurate estimation of the impact of tunneling on

Table 9. Contributions of Internal Rotor Losses to the Entropy of Activation in iab Hydrogen Shift

reaction channel

number of internal

rotors lost (n) S298Kq (cal/(mol K))

per rotor entropy loss ((Sq) = Sq/n,

cal/(mol K))

1-pentyl/ 2-pentyl (5ps) 3 10.00 3.331-hexyl/ 2-hexyl (6ps) 4 14.02 3.511-hexyl/ 3-hexyl (5ps) 3 10.23 3.411-heptyl/ 2-heptyl (7ps) 5 17.79 3.561-heptyl/ 3-heptyl (6ps) 4 14.57 3.641-heptyl/ 4-heptyl (5ps) 3 11.27 3.762-heptyl/ 3-heptyl (5ss) 3 13.08 4.361-octyl/ 2-octyl (8ps) 6 19.83 3.311-octyl/ 3-octyl (7ps) 5 18.80 3.761-octyl/ 4-octyl (6ps) 4 16.08 4.021-octyl/ 4-octyl (5ps) 3 11.57 3.862-octyl/ 3-octyl (6ss) 4 18.07 4.522-octyl/ 4-octyl (5ss) 3 14.41 4.80mean for ips 3.61 ( 0.23mean for iss 4.56 ( 0.22

Figure 11. Comparison of the high-pressure limit rate constants ()and rate-rule estimates ( 3 3 3 ) for hydrogen shift in n-heptyl radials.



the rate constant. It can be used also to extrapolate kinetic dataobtained in a limited temperature range. A rate rule was proposedfor hydrogen shift in linear alkyl radicals. In this rule, transmis-sion coecients are calculated using multidimensional tunnelingfor a representative reaction class. The rate constant withouttunneling may be estimated by considering the loss or gainof internal rotations in the transition state structure for pre-exponential factors. The apparent activation energy may be esti-mated from that of typical H-abstraction by an alkyl radicalcorrected for ring strain. The rate rule proposed here is shown toreproduce the theoretical high-pressure limit rate constant towithina factor of 2 for almost the entire range of temperatures considered.

ASSOCIATED CONTENT

bS Supporting Information. Optimized geometries at theB3LYP/6-311G(2d,d,p) level of theory for all species. Hindrancepotentials and barrier heights at the CBS-QB3 level of theory.Forward and reverse reaction barrier heights (E1 and E1) at 0 Kalong with imaginary frequencies (i) of the transition states andcalculated characteristic lengths. SCT transmission coecientsfrom 200 to 2000 K for all n-heptyl H-atom shifts. This material isavailable free of charge via the Internet at http://pubs.acs.org.

AUTHOR INFORMATION

Corresponding Author*Phone: (33) 383175202 (B.S.), (213) 740-0499 (H.W.). E-mail:[email protected] (B.S.), [email protected] (H.W.).

Present AddressesLaboratoire Reactions et Genie des Procedes, Nancy Universite,CNRS, BP 20451, 1 rue Grandville, 54001 Nancy, France.

ACKNOWLEDGMENT

This work was supported by the U.S. Air Force Oce ofScientic Research (AFOSR Grant Nos. FA9550-07-1-0168 andFA9550-08-1-0040) and by the Combustion Energy FrontierResearch Center (CEFRC), an Energy Frontier Research Centerfunded by the U.S. Department of Energy, Oce of Science, Oceof Basic Energy Sciences under Award No. DE-SC0001198. Partof this work was performed using HPC resources from GENCI-CINES (Grant 2011086686).

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Tunneling in Hydrogen-Transfer Isomerization of n-Alkyl Radicals

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