Computational and Theoretical Chemistry 963 (2011) 357–364

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Computational and Theoretical Chemistry

journal homepage: www.elsevier .com/locate /comptc

Rate coefficients of the reactions of isopentane with H and CH3 radicals: Quantummechanical approach

Xin Hong, Hongyan Sun ⇑, Chung K. LawDepartment of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, United States

a r t i c l e i n f o a b s t r a c t

Article history:Received 17 August 2010Received in revised form 27 October 2010Accepted 27 October 2010Available online 3 November 2010

Keywords:IsopentaneH-abstractionDFT and ab initioSmall-curvature tunneling correctionRate coefficients

2210-271X/$ - see front matter � 2010 Elsevier B.V.doi:10.1016/j.comptc.2010.10.047

⇑ Corresponding author.E-mail address: hsun@princeton.edu (H. Sun).

The rate coefficients of the H-abstraction reactions of isopentane by H and CH3 radicals were determinedby transition state theory (TST) and canonical variational transition state theory (CVT) with the potentialenergy surfaces calculated at the CBS-QB3 and G3B3 levels, and with the quantum mechanical tunnelingeffect corrected using the small-curvature tunneling (SCT) and Eckart methods. It was found that theG3B3 method predicts good barrier heights for the reactions of isopentane + H radical, and that theCBS-QB3 method predicts good barrier heights for the reactions of isopentane + CH3 radical. Furthermore,the energy barriers for the abstraction of primary, secondary, and tertiary hydrogen of isopentane werefound to be consistent with the corresponding H-bond dissociation energies derived by the enthalpy dif-ference between the C5H11 radical + H and isopentane. By using isodesmic reaction analysis, the enthal-pies of formation (DHo

f 298) of the four initial H-abstraction radical products of isopentaneC�H2CH2CH(CH3)CH3, CH3CH2CH(CH3)C�H2, CH3C�HCH(CH3)CH3, and CH3CH2C�(CH3)CH3 were determinedto be 12.0, 11.9, 9.4, and 7.0 kcal mol�1, respectively. To account for quantum mechanical tunnelingeffects, the rate coefficients corrected by the SCT method were found to be closer to the experimentaldata than those by the Eckart method. Thermal rate coefficients for the abstraction of primary, secondary,and tertiary hydrogen atom within isopentane by H and CH3 radicals in the temperature range of 300–1500 K were determined and also compared with available experimental data and theoretical results.

� 2010 Elsevier B.V. All rights reserved.

1. Introduction

Reactions of H and CH3 radicals with alkanes play a significantrole in the combustion process of hydrocarbons. At high tempera-tures, the b-scission of large alkyl radicals rapidly generate H andCH3 radicals, which can be the primary chain carriers in the ther-mal decomposition of hydrocarbons. Despite their importance,the rate coefficients for these types of reactions with many hydro-carbons are not well characterized, especially for the branchingratios of the primary reaction channels of large alkanes. A majordifficulty in the experimental study of such reactions is that sec-ondary reactions involving the intermediate alkyl radicals aremuch faster than the primary ones, and as such theory can playan important role in providing the necessary kinetic information.

Experimentally, Baldwin and Walker [1] studied the rate constantsof H atom and OH radical with C2–C5 alkanes and then derived therelationship of the rate parameters as functions of primary, secondaryand tertiary C–H bonds. Specifically, the rate constants for anunknown alkane reaction with the H atom can be estimated bythe relationship k = 2.2 � 1010 np exp(�4715/T) + 5.1 � 1010 ns

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exp(�3030/ T) + 4.9 � 1010 nt exp(�4005/T), where np, ns, and nt

are the numbers of primary, secondary and tertiary C–H bonds,respectively. Theoretically, Truong et al. [2,3] studied the thermalrate constants of the H-abstraction reactions of alkane by the Hand CH3 radicals with the reaction class transition state theory(RC-TST). They derived first-principle parameters from the RC-TSTtheory with the linear energy relationship (LER) method, so thatthe rate constants for any reaction in the class can be calculated byusing only the reaction energy computed either from the densityfunctional BH&HLYP/cc-pVDZ method, or the semi-empiricalmolecular orbital AM1 method. Their approach is based on the con-cept that all reactions in a given class have the same reactive moiety,and thus they are expected to have similar features on their potentialsurfaces along the specific reaction coordinate. The RC-TST methodis used to explore such similarities so that they can be transferredfrom one reaction to another without having to explicitly calculatethem, and as such it is quite cost effective and has reasonableaccuracy.

Isopentane (i-C5H12), being the smallest alkane with primary,secondary, and tertiary hydrogens, is a component in practicalfuels and is also used as a blowing agent and solvent. Furthermore,it is the smallest alkane with rotational conformers. It is thereforeof interest to explicitly characterize its primary, secondary, and

358 X. Hong et al. / Computational and Theoretical Chemistry 963 (2011) 357–364

tertiary H-abstraction reactions by active radicals. Because of itsmoderate molecular size, it is suitable to characterize the primary,secondary, and tertiary H-abstraction reactions within the mole-cule by using rigorous first-principle methods.

In view of the above considerations, we have performed densityfunctional and ab initio quantum calculations for the equilibriumgeometry of isopentane and its four radicals derived from the H-atom loss, as well as the variational transition states for reactionswith the H and CH3 radicals, and determined the rate coefficientsof the primary, secondary, and tertiary H-abstraction reactionsexplicitly with the application of conventional transition-state the-ory and variational transition-state theory.

Fig. 1. The rotational potential energies as a function of the torsion angle along theCH3CH2–CH(CH3)2 bond calculated at the B3LYP/6-311+G(d,p) level.

2. Calculation methods

All the electronic structure calculations were performed usingthe Gaussian 03 program [4]. The geometries and harmonic vibra-tional frequencies of all the stationary points (the reactants, transi-tion state, and products) were calculated using the hybrid B3LYPdensity functional theory [5] with the split valence basis set, aug-mented with diffuse and polarization functions on carbon atoms,6-311+G(d,p). It was reported that the experimental rate coeffi-cients of the H-abstraction of hydrocarbon by the CH3 radical weremost accurately predicted with the CBS-QB3 method [6], whichcombines results of several electronic structure calculations andempirical terms to predict molecular energies to an accuracyaround 4 kJ mol�1 [7]. Hence the composite CBS-QB3 [8–9] andG3B3 [10] methods were selected to refine the enthalpy values ofthe participating species in the target reactions.

The rate coefficients of the reactions studied were first deter-mined using the canonical transition state theory with correctionof quantum mechanical tunneling effects,

kðTÞ ¼ jðTÞr kBTh

Q –ðTÞQ RðTÞ

e �DV–=kBTf g ð1Þ

where j is the transmission coefficient accounting for quantummechanical tunneling effects, r the reaction symmetry number,Q– and QR the total partition function of the transition state, DV–

the classical barrier height, T the temperature, and kB and h theBoltzmann and Planck constants, respectively. The partition func-tions were calculated by statistical mechanics using B3LYP/6-311+G(d,p) moments of inertia and frequencies scaled by a factorof 0.9806 as recommended by Scott and Radom [11]. All the vibra-tions were treated as harmonic oscillators except the torsional mo-tion of the methyl groups and the central CAC bond of isopentane(include corresponding TS and radical products), which were trea-ted as hindered internal rotors using the polynomial expressionproposed by Ayala and Schlegel [12] to compute the hindered rotorpartition function QHR.

The tunneling factor, r, was evaluated using two methods,namely the Eckart tunneling method and the small curvature tun-neling (SCT) method [13]. The Eckart method approximates thepotential by a one-dimensional function that is fitted to reproducethe zero-point energy corrected barrier, the enthalpy of reaction at0 K, and the curvature of the potential curve at the transition state[14,15]. It also depends on the reaction coordinate corresponding tothe barrier maximum, as well as a parameter characterizing thebarrier width which is related to the imaginary frequency of thetransition state. Depending on the curvature of the reaction path,the SCT tunneling method is within the framework of variationaltransition state theory to correct the quantum mechanical effecton the motion along the reaction coordinate, and as such it is a moresophisticated method for an accurate evaluation of the tempera-ture-dependent transmission coefficient jðTÞ. In this method,jðTÞ was computed as the ratio of the Boltzmann-weighted

multidimensional semiclassical transmission probability to theBoltzmann-weighted classical transmission probability.

Since the calculation of tunneling factors using the SCT methodrequires information on the curvature of the reaction path for mul-tidimensional estimates of the tunneling probabilities, the ratecoefficients of the H-abstraction reactions of isopentane by H andCH3 radicals were also calculated within the framework of canon-ical variational transition state theory (CVT/SCT). Mathematically,the forward rate constant is obtained by minimizing the general-ized transition state theory (GT) rate constant with respect to s:

kCVTf ðTÞ ¼ min

skGT

f ðT; sÞ ¼ kGTf T; sCVT

� ðTÞ� �

ð2Þ

where

kGTðT; sÞ ¼ kðTÞr kBTh

Q GTðT; sÞQ RðTÞ

e�DV–ðsÞ=kBT ð3Þ

and the reaction coordinate s is the distance along the minimum en-ergy path (MEP) from the saddle point. For each s a generalized tran-sition state is defined perpendicular to the MEP and intersecting it atthat s, where QGTðT; sÞ is the partition function of the generalizedtransition state at the reaction coordinate s, and DV–ðsÞ is the classi-cal potential energy along the minimum energy reaction path withthe zero of energy at the reactants. The thermal rate coefficientsfor the reactions involved in this study were calculated using theRate program [16] in the temperature range of 300–1500 K.

3. Results and discussion

3.1. Geometries

The conformational analysis of isopentane was performed byscanning the rotation on the CH3CH2–CH(CH3)2 bond of isopentane.The potential energy as a function of the torsion angle was deter-mined by scanning this torsion angle from 0� to 360� at 15� incre-ments and allowing the remaining molecular structuralparameters to be optimized at the B3LYP/6-311+G(d,p) level. Twoenergy minima were found in a three-fold torsional potential withan energy difference of 0.89 kcal mol�1, as shown in Fig. 1. Further-more, the energy barrier between the two energy minima rotationalisomerism was found to be 5.85 kcal mol�1. The two energy minimaof the torsional potential were then fully optimized at the B3LYP/6-311+G(d,p) level, and the two conformers of isopentane were foundwith C1 and Cs symmetry respectively, as shown in Fig. 2. It is notedthat the conformer with Cs symmetry has one Cs plane bisecting the

X. Hong et al. / Computational and Theoretical Chemistry 963 (2011) 357–364 359

three backbone carbon atoms and the two branched methyl groups.Furthermore, the conformer with C1 symmetry was found to be theglobal energy minimum. This is ascribed to the fact that the termi-nal methyl group in the straight chain adjacently interacts with oneof two branched methyl groups in the C1 conformer, but it interactswith two branched methyl groups in the Cs conformer, as shown bythe Fischer projections in Fig. 1. The higher energy of the Cs con-former than that of the C1 conformer was also reported in a fewexperimental [17,18] and theoretical studies [19]. The geometryoptimization of the four isopentane radicals derived from the H-atom loss from the parent molecule was based on the electronicstructure of isopentane with C1 symmetry.

The transition states of H-abstraction reaction of isopentane by Hradical were labeled as TS1, TS2, TS3, and TS4 to designate the forma-tion of C�H2CH2CH(CH3)CH3, CH3CH2CH(CH3)C�H2, CH3C�HCH(CH3)CH3, and CH3CH2C�(CH3)CH3 radicals respectively. The transitionstates of the H-abstraction reaction by the CH3 radical were labeledas TS5, TS6, TS7, and TS8, respectively. The transition state structureswere verified to have only one imaginary frequency associated withthe vibrational motion along the reaction coordinates. The opti-mized geometries of the two isopentane isomers, four radical prod-ucts, and eight transition states with selected geometric parametersat the B3LYP/6-311+G(d,p) level are presented in Fig. 1. The corre-

Fig. 2. Optimized geometries of isopentane isomers, radicals, and transition states

sponding vibrational frequencies and moments of inertia for theevaluation of the partition functions are provided in Tables S1 andS2 of the Supplementary materials.

For the reactions of isopentane with the H radical via TS1, TS2,TS3, and TS4, the cleaving C–H bond length was found to be 1.369,1.377, 1.330, and 1.268 Å respectively, while the forming H–Hbond lengths are 0.926, 0.923, 0.962, 1.024 Å, respectively. Forthe reactions by the CH3 radical, the cleaving C–H bond lengthvia TS5, TS6, TS7, and TS8 was found to be 1.326, 1.331, 1.312,1.291 Å, respectively, while the forming C–H bond lengths are1.384, 1.382, 1.410, 1.438 Å, respectively. From these bond lengthdata, it is seen that the cleaving C–H bond length is reduced, andthe forming H–H (C–H) bond length increased, in the order of pri-mary, secondary, and tertiary H-abstraction. Therefore it is con-cluded that the transition state occurs earlier in the order oftertiary, secondary, and primary H-abstraction, and results in in-creased energy barriers corresponding to tertiary, secondary, andprimary H-abstraction reactions.

3.2. Energies

The energy barriers and reaction exothermicities at 0 K and298 K calculated by using the CBS-QB3 and G3B3 methods for the

at the B3LYP/6-311+G(d,p) level, bond length in Å, and bond angle in degree.

Fig. 3. Energy barriers for the primary, secondary, and tertiary H-abstraction as afunction of reaction exothermicity. Solid lines: from CBS-QB3 calculations; dashlines: from G3B3 calculations.

360 X. Hong et al. / Computational and Theoretical Chemistry 963 (2011) 357–364

eight H-abstraction reactions of isopentane by H and methyl radicalare given in Table 1. It was found that the G3B3 energy barriers ofthe H-abstraction reaction are consistently higher than those fromthe CBS-QB3 calculation; specifically, 0.7–0.8 kcal mol�1 higher forthe reactions of isopentane + H, and 1.5–1.7 kcal mol�1 higher forthe reactions of isopentane + CH3, as shown in Table 1. Since the en-ergy barriers are sensitive to the abstraction rate, it was found thatby using the G3B3 energy barriers the calculated rate coefficientsagree well with the experimental data of isopentane + H, and itwas also found that by using the CBS-QB3 energy barriers the calcu-lated rate coefficients agree with the theoretical experimental dataof isopentane + CH3. This finding will be further discussed in thesection of rate coefficient calculations. Here, we focus on the trendof energy barriers with the corresponding reaction exothermicitiesfor the abstraction of the primary, secondary, and tertiary hydro-gens within isopentane.

For reactions with the H radical, the energy barriers obtained byusing the CBS-QB3 method at 298 K for the abstraction of the twoprimary H are 9.00 and 8.90 kcal mol�1, and are 6.47 and4.16 kcal mol�1 for the abstraction of the secondary and tertiaryhydrogens. For reactions with the CH3 radical, the CBS-QB3 energybarriers for the abstraction of the two primary H atoms at 298 Kare 13.24 and 13.03 kcal mol�1, and 10.80 and 8.61 kcal mol�1 forthe abstraction of the secondary and tertiary H atoms. The trendof calculated energy barriers for the abstraction of primary, sec-ondary, and tertiary hydrogens are consistent with that of geomet-ric parameters for cleaving the C–H bond and the forming H–H (C–H) bond length in the corresponding transition states, as discussedabove. Furthermore, it was found that the energy barriers for theprimary, secondary, and tertiary H-abstraction have a linear rela-tionship with the corresponding reaction exothermicities, asshown in Fig. 3. However, the good linear relationship for abstrac-tion of the primary hydrogen is limited to those bonded in the car-bon of straight chain rather than those in branched methyl groups(see Fig. 3).

The trend for the energy barriers identified above is also foundto be consistent with their corresponding bond dissociation ener-gies (BDEs) derived by the enthalpy difference between the C5H11

radical + H and isopentane. Because most of the DHof 298 of the

C5H11 radicals were unknown, they were determined by usingisodesmic reaction analysis and ab initio CBS-QB3 and G3B3 meth-ods in this work. The isodesmic reactions used for each C5H11 rad-ical are listed in Table 2. It is shown in Table 2 that the DHo

f 298

values of the C�H2CH2CH(CH3)CH3, CH3CH2CH(CH3)C�H2,CH3C�HCH(CH3)CH3, and CH3CH2C�(CH3)CH3 radicals determinedby the two ab initio composite methods are in excellent agreement.It is noted that only the DHo

f 298 of one of the C5H11 radicals, the ter-tiary radical CH3CH2C�(CH3)CH3, was listed in the NIST chemistrydatabase, being 6.7 kcal mol�1 [20]. Our calculated data for theDHo

f 298 of CH3CH2C�(CH3)CH3, 7.0 kcal mol�1, is in good agreementwith this value. According to these values of DHo

f 298 of the C5H11

Table 1Calculated energy barriers and reaction exothermicities at 298 Ka.

Reaction DH– (CBS-QB3)

DH– (G3B3) DHof rxn (CBS-

QB3)DHo

f rxn

(G3B3)

TS1 8.90 (9.89) 9.61 (10.60) �3.86 �3.80TS2 9.00 (10.02) 9.73 (10.75) �3.13 �3.06TS3 6.47 (7.43) 7.28 (8.22) �6.14 �5.67TS4 4.16 (5.08) 4.98 (5.89) �7.57 �7.33TS5 13.24 (14.04) 14.76 (15.57) �3.92 �3.46TS6 13.03 (13.82) 14.57 (15.39) �3.19 �2.72TS7 10.80 (11.47) 12.44 (13.12) �6.19 �5.33TS8 8.61 (9.16) 10.30 (10.88) �7.61 �6.99

a Unit: kcal mol�1. The data in the parentheses are the energy barriers at 0 K.

radicals, the H-bond dissociation energies can be derived by theenthalpy difference between the products C5H11 + H and reactantC5H12. Table 2 also shows that the BDEs derived from the two abinitio methods are in excellent agreement: 100.8 kcal mol�1 forthe primary H, 98.2 for the secondary H, and 95.8 for the tertiaryH bond dissociation energies, respectively. This supports the factthat the energy barrier heights for the primary, secondary, and ter-tiary H-abstractions decrease with the corresponding decreasedbond dissociation energies.

3.3. Minimum energy paths of H-abstraction

Intrinsic reaction coordinate calculation (IRC) was performed atthe B3LYP/6-311+G(d,p) level to obtain the minimum energy paths(MEP). The calculation was carried out by starting from the fullyoptimized saddle-point geometry, and then moving downhill alongthe reactant and product channels, in mass-weighted Cartesiancoordinates. About one hundred points were calculated in eachdirection at a gradient step size of 0.01 bohr. The reaction coordi-nate s is defined as the signed distance from the saddle point, withs < 0 referring to the reactants side and s > 0 to the products side.Force constants, harmonic vibrational frequencies and normal-mode vectors for the (3N � 7) degrees of freedom that areorthogonal to the reaction path were computed at the B3LYP/6-311+G(d,p) level, at selected points along the IRC, to constructthe minimum energy paths for the H-abstraction reactions. To ob-tain an accurate potential energy curve, more points in the saddlepoint zone �1.0 < s < 1.0 were selected for the calculation of theHessian and energy gradients. It was found that the single-pointenergy of the saddle point for each channel of isopentane + H cal-culated at the G3B3 and CBS-QB3 levels is consistently 0.29 and0.63 kcal mol�1 higher than that of the B3LYP/6-311+G(d,p) level,respectively. Consequently, the B3LYP/6-311+G(d,p) energies alongeach minimum energy path were corrected by the single-point en-ergy difference between the two levels of theory at the saddlepoint, respectively, such that the energy maximum correspondingto the saddle point is not shifted from the original calculations inorder to avoid the problem of numerical defect that is mistakenfor any variational effect [21].

Fig. 4a presents the classical potential energies (VMEP) correctedby the single-point energy difference between the DFT and G3B3levels at the saddle point, the local zero-point energies (Eint), andthe ground state vibrationally adiabatic potential energies (VG

a )

Table 2Calculated DHo

f 298 values and bond dissociation energiesa.

Isodesmic reaction DHof 298 (CBS-QB3) DHo

f 298 (G3B3)

C�H2CH2CH(CH3)CH3 + CH3CH2CH2CH3 ? CH3CH2CH(CH3)CH3 + C�H2CH2CH2CH3 12.04 11.26CH3CH2CH(CH3)C�H2 + CH3(CH3)CHCH3 ? CH3CH2CH(CH3)CH3 + C�H2(CH3)CHCH3 11.91 11.86CH3C�HCH(CH3)CH3 + CH3CH2CH2CH3 ? CH3CH2CH(CH3)CH3 + CH3C�HCH2CH3 9.36 9.41CH3CH2C�(CH3)CH3 + CH3(CH3)CHCH3 ? CH3CH2CH(CH3)CH3 + CH3(CH3)C�CH3 6.96 6.33

BDE (CBS-QB3) BDE (G3B3)CH3CH2CH(CH3)CH3 ? C�H2CH2CH(CH3)CH3 + H 100.87 100.09CH3CH2CH(CH3)CH3 ? CH3CH2C(CH3)C�H2 + H 100.74 100.69CH3CH2CH(CH3)CH3 ? CH3C�HCH(CH3)CH3 + H 98.19 98.24CH3CH2CH(CH3)CH3 ? CH3CH2C�(CH3)CH3 + H 95.79 95.16

a Units in kcal mol�1. The DHof 298 of following species is from the NIST database (kcal/mol): The DHo

f 298 of following species is from the NISTdatabase (kcal/mol): �36.73 for isopentane, �30.03 for butane, 11.0 for tert-butyl radical, 17.0 for isobutyl radical, 16.0 for 2-butyl radical, and18.74 for 1-butyl radical (this value is from J.W. Bozzelli’s thermo database).

X. Hong et al. / Computational and Theoretical Chemistry 963 (2011) 357–364 361

along the reaction coordinate for the four channels in the reactionsof isopentane + H. It is noted that the scale of Eint is expressed inthe right axis of the ordinates to show the facile change of the Eint

curve for each reaction channel. It was found that the maximum ofthe potential energy curves of VMEP and VG

a along the reaction coor-dinates are close to the reaction coordinate s = �0.05; while the Eint

curves show a drop near the saddle point, with the minimum of the

Fig. 4. Classical potential energies (VMEP), the local zero-point energies (Eint), andthe ground state adiabatic potential energies (VG

a ) along the reaction coordinates forthe H-abstraction by (a) H radical, (b) CH3 radical.

Eint curves occurring at s = �0.4, �0.25, and 0.1 for the primary,secondary, and tertiary H-abstraction, respectively. Furthermore,each Eint curve is unique and the variational transition states werefound to be located in the reaction coordinates before the maxi-mum of the potential energy curves of VMEP and VG

a .Fig. 4b presents the classical potential energies (VMEP) corrected

by the single-point energy difference between the DFT and CBS-QB3 levels at the saddle point, Eint, and VG

a curves along the reactioncoordinate for the H-abstraction of isopentane by the CH3 radical;the scale of Eint is also expressed in the right axis of the ordinates toexhibit the facile change of the Eint curve for each reaction channel.It was found that the maximum of the potential energy curves ofVMEP and VG

a along the reaction coordinate are close to the reactioncoordinate s = 0.04 and each Eint exhibits a unique curvature. Forthe local zero-point energy Eint curves, they all show a substantialdrop between the s range of �0.5 < s < 0.5. Furthermore, the varia-tional transition states were found to be located near the maxi-mum of the potential energy curves of VMEP and VG

a .

3.4. Rate coefficients

The Cartesian coordinates of reactants, transition-state andproducts, the vibrational frequencies calculated at the B3LYP/6-311+G(d,p) level, the energies at 0 K determined by the CBS-QB3or G3B3 method, the force constant matrix (Hessian) and the en-ergy gradient along the minimum energy path, forward and re-verse reaction symmetry numbers in different H-abstractionpaths were used to calculate the rate coefficients by both the TST

Fig. 5. Comparison of rate coefficients of isopentane + H with the tunneling effectcorrected by Eckart and SCT methods.

Table 3Calculated transmission coefficients by SCT and Eckard methods for the reactions of isopentane with H radical.

T (K) C�H2CH2CH(CH3)CH3 CH3CH2C(CH3)C�H2 CH3C�HCH(CH3)CH3 CH3CH2C�(CH3)CH3

SCT Eckart SCT Eckart SCT Eckart SCT Eckart

300 18.91 4.84 18.32 5.55 6.88 5.17 1.85 3.61400 5.84 2.47 5.66 2.63 3.27 2.60 1.55 2.20500 3.23 1.87 3.16 1.94 2.23 1.94 1.40 1.76600 2.31 1.61 2.27 1.65 1.78 1.66 1.31 1.55700 1.86 1.47 1.84 1.50 1.54 1.50 1.26 1.43800 1.62 1.38 1.61 1.40 1.40 1.40 1.22 1.35900 1.47 1.32 1.46 1.34 1.31 1.34 1.19 1.29

1000 1.37 1.27 1.36 1.29 1.25 1.29 1.17 1.261100 1.29 1.24 1.29 1.25 1.20 1.26 1.15 1.221200 1.24 1.21 1.24 1.22 1.17 1.23 1.14 1.201300 1.20 1.19 1.20 1.20 1.14 1.20 1.13 1.181400 1.17 1.17 1.18 1.18 1.12 1.19 1.12 1.171500 1.15 1.16 1.15 1.17 1.11 1.17 1.11 1.15

362 X. Hong et al. / Computational and Theoretical Chemistry 963 (2011) 357–364

and CVT methods via Eqs. (1) and (2). Specifically, the forwardreaction path degeneracies for abstraction of primary, branchedprimary, secondary, and tertiary hydrogen of isopentane are 3, 6,2, and 1 respectively. It was found that the calculated rate coeffi-cients of the reactions of isopentane + H radical using the CBS-QB3 classical barrier heights are higher than the experimental dataof Baldwin and Walker [1], but they agree well with the experi-mental data when using the higher classical barrier heights deter-mined by G3B3 method. While the calculated rate coefficients ofthe reactions of isopentane + CH3 radical using the CBS-QB3 classi-cal barrier heights in general agree with those predicted by usingthe analytical rate expressions of Kungwan and Troung [3]. Regard-ing the tunneling effect, the rate coefficients of isopentane + H withtunneling effect corrected by the Eckart and SCT methods using theCBS-QB3 energies are shown in Fig. 5. It is seen that the rate coef-ficients at low temperatures are affected significantly by the twodifferent tunneling methods. The transmission coefficients as afunction of temperature for reactions with the H radical predictedby the two different methods are listed in Table 3. It is seen that at300 K, the SCT transmission coefficients are a factor of four larger,1.7 larger, and 1.6 smaller than the Eckart transmission coefficientsfor the primary, secondary, and tertiary H-abstraction, respec-tively. However, the transmission coefficients as a function of tem-perature for reactions with the CH3 radical predicted by the twodifferent tunneling methods are similar. It is noted that since the

Fig. 6. Arrhenius plots of rate coefficients of isopentan

SCT transmission coefficients depend on the curvature of the reac-tion path for multidimensional estimates of the tunneling proba-bilities, it should be more accurate than those from the Eckartmethod that approximates the potential by a one-dimensionalfunction. In fact, the TST rate coefficients corrected by the SCTmethod were found to be closer to the experimental data forabstraction reactions by the H radical. Therefore, the SCT transmis-sion coefficients were adopted to correct the quantum mechanicaltunneling effect for the calculated TST rate coefficients.

The forward rate coefficients k1–k4 for the H-abstraction reac-tions of isopentane by the H radical, and k5–k8 for the H-abstractionreactions of isopentane by the CH3 radical, in the temperature rangeof 300–1500 K are listed in the three parameter Arrhenius expres-sion as follows, with k in units of cm3 molecule�1 s�1. We note thatthe subscript of the rate k for each reaction channel is consistentwith the sequence of numbering the transition states.

k1 ¼ 4:80� 10�22T3:409 expð�2125:1=TÞ

k2 ¼ 2:54� 10�21T3:326 expð�2287:3=TÞ

k3 ¼ 4:55� 10�20T2:845 expð�1613:9=TÞ

k4 ¼ 8:23� 10�18T2:127 expð�1323:0=TÞ

e + H with comparison of available literature data.

Fig. 7. Arrhenius plots of rate coefficients of isopentane + CH3 radical with comparison of available literature data.

X. Hong et al. / Computational and Theoretical Chemistry 963 (2011) 357–364 363

k5 ¼ 2:34� 10�27T4:443 expð�3732:5=TÞ

k6 ¼ 2:30� 10�28T4:844 expð�3346:0=TÞ

k7 ¼ 1:29� 10�25T3:884 expð�3041:0=TÞ

k8 ¼ 1:26� 10�24T3:609 expð�2134:0=TÞ

The Arrhenius plots of the temperature dependence of the cal-culated rate coefficients for H and CH3 abstraction reactions arepresented in Figs. 6 and 7, respectively, together with the com-parison of available literature data. It is seen that our calculatedrate coefficients k1, k2, k3, and k4 for primary, secondary, and ter-tiary H-abstractions agree well with the experimental values ofiso-pentane + H by Baldwin and Walker [1]. The rate coefficientsfrom the analytical rate expressions of Zhang and Troung [2] arealso plotted in Fig. 6; it is seen that our calculated k1, k2, k3 agreeclosely with those of Zhang and Troung [2], but it shows somediscrepancies for k4 of the tertiary H-abstraction of isopen-tane + H between the two different approaches. Our theoreticalk4 agrees more closely to the experimental data than that ofZhang and Troung [2], indicating that it is essential to use rigor-ous first-principle methods for obtaining accurate coefficients.For the H-abstraction reactions by the CH3 radical, it was foundthat our calculated k5 agrees well with that predicted by usingthe analytical rate expressions of Kungwan and Troung [3] withC2H6 + CH3 being a reference reaction, and our rate coefficientsk6, k7, k8 show resonable agreement with those of Ref. [3]. Fur-thermore, it was found that our theoretical k8 agrees with theexperimental data of the tertiary H-abstraction reaction of isobu-tane + CH3 in the temperatures of 500–855 K [22], as shown inFig. 7. It is noted that although it is better to have more experi-mental data of this system at the intermediate temperatures(700–1500 K) to validate these theoretical abstraction rates, therate coefficients obtained in this work should be useful to futureefforts at modeling C5 hydrocarbon pyrolysis and oxidation inthe combustion processes.

4. Concluding remarks

The kinetics of the H-abstraction reactions of isopentane by Hand methyl radicals was studied by using density functional andab initio G3B3 and CBS-QB3 calculations with application of transi-tion state theory (TST) and canonical variational transition statetheory (CVT). It was found that the G3B3 and CBS-QB3 methodsare respectively suitable to determine the barrier heights for thereactions of isopentane + H radical and of isopentane + CH3 radical.Furthermore, the energy barriers for the abstraction of the primary,secondary, and tertiary hydrogen of isopentane were found to beconsistent with the corresponding H-bond dissociation energiesderived by the calculated enthalpy difference between the C5H11

radical + H and isopentane. To account for quantum mechanicaltunneling effects, the rate coefficients corrected by the SCT methodwere found to be closer to the experimental data than by the Eckartmethod. Thermal rate coefficients for the abstraction of the pri-mary, secondary, and tertiary hydrogen by H and CH3 radicals inthe temperature range of 300–1500 K were determined and alsocompared with available experimental data and theoretical results.

Acknowledgement

This work was supported by the US Army Research Office underthe technical monitoring of Dr. Ralph A. Anthenien.

Appendix A. Supplementary material

Supplementary data on the vibration frequencies and momentsof inertia of isopentane, isopentane radicals, and transition statescan be found, in the online version, at doi:10.1016/j.comptc.2010.10.047.

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