Reaction class transition state theory: Hydrogen abstraction reactions by hydrogen atoms as test cases

Reaction class transition state theory: Hydrogen abstraction reactions byhydrogen atoms as test casesThanh N. Truong Citation: J. Chem. Phys. 113, 4957 (2000); doi: 10.1063/1.1287839 View online: http://dx.doi.org/10.1063/1.1287839 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v113/i12 Published by the AIP Publishing LLC. Additional information on J. Chem. Phys.Journal Homepage: http://jcp.aip.org/ Journal Information: http://jcp.aip.org/about/about_the_journal Top downloads: http://jcp.aip.org/features/most_downloaded Information for Authors: http://jcp.aip.org/authors

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JOURNAL OF CHEMICAL PHYSICS VOLUME 113, NUMBER 12 22 SEPTEMBER 2000

Reaction class transition state theory: Hydrogen abstraction reactionsby hydrogen atoms as test cases

Thanh N. TruongHenry Eyring Center for Theoretical Chemistry, Department of Chemistry, University of Utah,315 South 1400 E, room 2020, Salt Lake City, Utah 84112

~Received 7 April 2000; accepted 8 June 2000!

We present a new method called Reaction Class Transition State Theory~RC-TST! for estimatingthermal rate constants of a large number of reactions in a class. This method is based on thetransition state theory framework within the reaction class approach. Thermal rate constants of agiven reaction in a class relative to those of its principal reaction can be efficiently predicted fromonly its differential barrier height and reaction energy. Such requirements are much less than whatis needed by the conventional TST method. Furthermore, we have shown that the differentialenergetic information can be calculated at a relatively low level of theory. No frequency calculationbeyond those of the principal reaction is required for this theory. The new theory was applied to anumber of hydrogen abstraction reactions. Excellent agreement with experimental data shows thatthe RC-TST method can be very useful in design of fundamental kinetic models of complexreactions. ©2000 American Institute of Physics.@S0021-9606~00!01133-8#

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I. INTRODUCTION

One of the goals of computational science is to predobservables where experiments have not yet been donhave difficulties in carrying out. In the area of chemical knetic theory, there has been much progress in develodirect ab initio dynamics methods based on the transitstate theory framework for calculating rate coefficients frofirst principles.1–10 These methods have achieved a ratexcellent level of accuracy even for large biologicsystems.11–13 It should be noted that applications of sumethods are done for one chemical system at a time. Amthe existing methods, the simplest and most cost effecone is the well-known transition state theory14 ~TST!, whichrequires only structural, energetic and vibrational frequeinformation at the reactant and transition state. For mcombustion systems, kinetic models often consist of theder of thousand of elementary reactions. There are a lanumber of these reactions whose kinetic parameters areknown. It is still impractical to carry out calculations of themal rate constants for every one of such reactions even uthe TST method at a sufficiently accurate level of electrostructure theory. Furthermore, in practice only a musmaller set of reactions from the detailed kinetic modeimportant to the combustion system under a given operacondition. To determine such a set, one only needs a gestimate of the unknown kinetic parameters to perform ssitivity analysis on the mechanism. Thus, it is much betteperform accurate rate calculations for reactions after thatbeen determined to be important to the mechanism.

The central task is to have a good estimate of kineparameters for a large number of reactions. A simplecommon practice is to assign the unknown kinetic paraeters by those of a similar reaction. A better approach isemploy the Evan–Polanyi linear free-energy relationshiptween the activation energies and bond dissociation ener

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or heats of reaction of similar reactions to estimate theknown activation energy. Both approaches are empiricalhave large uncertainty in the estimated rate constants. Thave been some efforts to determine whether rate consare additive so that rate constants can be estimated fseparate components. However, it was proven that ratestants are not. In the case where some limited rate infortion is available, for instance a single rate measurement,can use a procedure called the thermochemical kinetics~TK-TST! method developed by Benson and co-worker15

and later refined by Cohen and co-workers16–19 to extrapo-late to other temperatures.

Recently, we have introduced the concept of reactclass into both electronic structure as well as dynamicalculations. This concept recognizes that all reactions ingiven class have the same reactive moiety, thus they shhave certain similarities on their potential surfaces alongreaction coordinate. We have shown that coupling this idwith the ONIOM methodology20–22 leads to an efficientcomputational strategy for accurate determination of barheights and reaction energies of reactions in a class.23 Inaddition, by exploring similarities on the potential surfacof reactions in a class, we found that several featuresconserved and thus can be transferred among reactions iclass. This led to the development of two new tunnelimodels9,10 as approximations to the multidimensional semclassical small curvature tunneling~SCT! ~Ref. 24! method.These tunneling models have shown to be quite accuratethey require substantially less computational demand cpared to that of the SCT method. In this study, again usthe reaction class concept we present a new computatimethodology for calculating thermal rate coefficients forlarge number of elementary reactions efficiently. Thmethod is based on the transition state theory frameworkdetermining relative rate constants of reactions in the cla

7 © 2000 American Institute of Physics

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4958 J. Chem. Phys., Vol. 113, No. 12, 22 September 2000 Thanh N. Truong

The focus of this study is to illustrate the theory and to tesby performing actual applications to an important classcombustion reactions, namely, the hydrogen abstractionactions by hydrogen atoms.

II. REACTION CLASS TRANSITION STATE THEORY„RC-TST…

From the transition state theory framework,14 thermalrate coefficient can be expressed as

k~T!5k~T!skBT

h

Q‡~T!

FR~T!e$2DV‡/kBT%, ~1!

where k is the transmission coefficient accounting for tquantum mechanical tunneling effects;s is the reaction sym-metry number;Q‡ and FR are the total partition functions~per unit volume! of the transition state and reactant, resptively; DV‡ is the classical barrier height;T is the tempera-ture; kB and h are the Boltzmann and Planck constants,spectively.

Let us consider a reaction class where all elemenreactions in the class have the same reactive moiety, witk1

the rate coefficient of the principal reaction R1, the smallreaction in the class, andk2 the rate coefficient of some othereaction R2 in the same class. Due to the small size,k1 canbe calculated from first principles using an accurate dynacal theory with potential energy information computed froa sufficiently high level of electronic structure theory. Thuk1 can be readily available. The task is to evaluate thecoefficientk2 .

The principle idea of the RC-TST method is to factor tratio of the two rate constantsk2 andk1 into different com-ponents, namely, tunneling, reaction symmetry, partitfunction, and potential energy, as defined below:

k2

k15 f k f s f Qf V . ~2!

By using the reaction class approach, these factors caefficiently evaluated so that a large number of reactionsthe class can be estimated fromk1 and these factors. Wediscuss how to approximate these factors separately bel

A. Ratio of the transmission coefficient, f k

f k~T!5k2~T!

k1~T!. ~3!

The transmission coefficientk1 of the principal reaction canbe calculated from the simple Eckart model25 or from a moreaccurate method such as the SCT~Ref. 24! approach. Totake advantage of the possibility for cancellation of errobothk1 andk2 in Eq. ~3! should be calculated from the samtunneling model. For simplicity, we propose to use the Eart model. However, in this study, we modified the Ecktunneling model25 proposed earlier to fit with the reactioclass methodology. In the previous model, the Eckart fution representing the zero-point energy corrected potencurve for tunneling is assumed to have the same width athe classical potential curve. This potential width depennot only on the imaginary frequency but also on the classbarrier height and reaction energy. Since the barrier he


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and reaction energy vary for different reactions in a class,potential width is not a conserved quantity. However, froanalyzing transition state properties of different reactions iclass discussed below, we found the imaginary frequencymore conserved quantity. Thus, in the RC-TST method,assume that reactions in the class have the same imagfrequency and zero-point energy corrections to the barand reaction energy as of the principal reaction, and theyused to calculate the potential for tunneling for differentactions in the class. This change introduces only a smdifference in the calculated transmission coefficients at vlow temperatures compared to those from the original Eckmodel. Thus, for each reaction we only need the differenbarrier height and reaction energy from those of the princireaction to determine the potential curve for tunneling.shown in our previous study, the differential reaction eneand barrier height can be calculated accurately at a relatilow level of theory. Note it does not require any frequencalculation for the reactants or the transition state of theaction R2.

B. Ratio of the reaction symmetry number, f s

f s5s2

s1. ~4!

Since the reaction symmetry numbers for reactions R1R2 are known, this factor is determined exactly.

C. Ratio of the partition function f Q

f Q~T!5S Q2‡

F2RD Y S Q1

‡

F1RD 5S Q2

‡

Q1‡D Y S F2

R

F1RD . ~5!

Note that each partition function is a product of translationrotational, and vibrational partition functions. It can bshown that the translational and rotational components ya constant multiplicative factor inf Q . The temperature dependent component off Q solely comes from the vibrationacomponent. By rearranging the ratio of the partition funtions of reactions R1 and R2 as shown in the second ratiEq. ~5! above, an important point regarding reaction clacan be made. To illustrate this point, we use a class ofdrogen abstraction reactions where the principal reactionthis class is the H1CH4 reaction and different substituenR1, R2, and R3, as shown in Fig. 1, yield different reactionin this class. The ratio of the vibrational partition functionsthe transition states of the reactions R2 and R1@the numera-tor of the second ratio in Eq.~5!# mainly comes from thesubstituents and the substituent effects on the frequenciethe reactive moiety. The same is true for the denominathe ratio of the vibrational partition functions of the reactanof the reactions R2 and R1. The overall ratio further remothe principle components of the substituents. Thus, the rf Q results mainly from the differences in the coupling‘‘cross’’ terms of the force constants of the two reactionsthe transition states and reactants. If there is no coupbetween the substituents and the reactive moiety thenvibrational component off Q is unity. In other words, thevibrational component off Q is the ratio of the differences inthe effects of the substituents on vibrational frequencies


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4959J. Chem. Phys., Vol. 113, No. 12, 22 September 2000 Reaction class transition state theory

the reactive moiety and in the effects of the reactive moion the frequencies of the substituents at the transition sand at the reactants. Since these effects should have thetrends at the reactants and at the transition state, the temture dependent component is significantly lessened in tratio. Consequently, we expect thatf Q does not dependstrongly on the temperature. As shown in Fig. 2, these raare nearly constant for a rather larger number of reactionthis class. Only a small temperature dependence inf Q isobserved for temperatures below 300 K for these reactioFor simplicity, we can make an approximation thatf Q is aconstant and has the high temperature limit value ofH1C2H6 reaction. This would make only about 60% errorthe rate coefficients for temperatures above 300 K forreaction class. This is certainly an acceptable level of ac

FIG. 1. Pictorial illustration of the factor of partition functionf Q . Thereactive moiety consists of atoms in the box. Different substituents R1, R2,and R3 yield different reactions in the class.

FIG. 2. Plot of lnfQ vs T for a number of hydrogen abstraction reactionsthe H atom.~p! denotes abstraction of primary hydrogen;~s! of secondaryhydrogen;~t! of tertiary hydrogen;~e! of equatorial hydrogen;~a! of axialhydrogen.


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racy in kinetic modeling. Thus,f Q can be predetermined foeach class of reaction from frequency calculations foradditional small reaction in the class, so that vibrational fquency calculations for other reactions in the class arerequired.

D. Ratio of the potential energy, f V

f V5expH 2DDV‡

kBT J 5expH 2~DV2

‡2DV1‡!

kBT J , ~6!

whereDDV‡, the difference in the classical barrier heighof reactions R2 and R1, respectively, can be rewritten as

DDV‡5$E2‡2E1

‡%2$E2R2E1

R%, ~7!

where E is the total potential energy. Thus,DDV‡ is thedifference between the substituent effects in the potentialergy at the transition state and at the reactants. Fromexpression in Eq.~7!, interactions in the reactive moiety thaare critical for accurate evaluation of the barrier height infirst term and reaction energy in the second term are remoby taking the difference from those of the principal reactioAs shown in previous studies,26,27 DDV‡ can be accuratelypredicted from a relatively low level of theory.

In summary, the reaction class transition state theoryquires the following information:

~1! Accurate potential energy information of the principreaction, namely, structures, energies, and frequencieleast at the reactants, transition state, and products. Sthe principal reaction is small, this condition can easbe met. Thermal rate constants for the principal reactcan be calculated from the above potential informatusing the TST method with Eckart tunneling correctionHowever, they can also be calculated from a more acrate directab initio dynamics method, though this woulrequire much more potential energy information alothe reaction path. Note that experimental measuremcan also be used.

~2! The ratio of the partition function,f Q . As shown above,to a good approximation it can be assumed to be a cserved function of a given reaction class and can betermined from additional frequency calculations for aother small reaction in the class.

Thermal rate constants for any other reactions in the ccan be estimated from knowing only the differential reactienergy and barrier height. No frequency calculation, hoever, is required.

III. RESULTS AND DISCUSSION

We have chosen the class of hydrogen abstraction rtions by hydrogen atoms to test the RC-TST method. Pticularly, several reactions were selected so as to provsevere tests of the theory and they are

H1CH4→H21CH3, ~R1!

H1C2H6→H21C2H5, ~R2!

H1H–CH2CH2CH3→H21CH2CH2CH3, ~R3!


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H1H–CH~CH3!2→H21HC~CH3!2, ~R4!

H1C3H8→H21products, ~R5!

H1c-C3H6→H21c-C3H5, ~R6!

H1H–CH2F→H21CH2F, ~R7!

H1H–CHF2→H21CHF2, ~R8!

H1H–CF3→H21CF3. ~R9!

For this reaction class, R1 is the principal reaction. R2used to provide thef Q factor. R3–R9 reactions are usedtest the theory. For the principal reaction, accurate therrate calculations using canonical variational TST theory pSCT tunneling correction were done previously;7,8 we do notneed to discuss further here. However, since the RC-Tmethod has its strength in its practicality, we envision thaactual applications of the theory, simple TST calculation aEckart tunneling corrections for the principal reaction woube sufficient. The accuracy of the TST/Eckart methodbeen discussed in our previous studies.9,25,28 In this case,geometries and frequencies of the reactants, transitionand products were calculated at the BH&HLYP/cc-pvlevel of theory. Single point energy calculations at tPMP4/cc-pvtz level were done to improve the accuracythe calculated barrier height and reaction energy of the pcipal reaction. Our previous studies8,29 have shown that thecombination of BH&HLYP and PMP4 calculations providsufficiently accurate potential energy information for thermrate determination. For the R2 reaction, we only needoptimize the geometry and to calculate the frequencies ofstationary points at the BH&HLYP/cc-pvdz level. FoR3–R9 reactions, only geometry optimizations of the stionary points were performed at the BH&HLYP/cc-pvdlevel. Note that R3–R9 reactions were selected to provdifferent variations of reactions in this class that can sevetest the new theory. In particular, reaction R3 is a primhydrogen abstract reaction, whereas reaction R4 is a secary abstract and R5 is the overall reaction. The ring cstraint of cyclo-propane in R6 would severely test the retion class concept in using potential energy informationthe principal reaction H1CH4. Electronegative fluorine substituents in R7–R9 reactions distort the geometry of trantion states far from that of the principal reaction. In additiothese R3–R9 reactions provide a wide variation in the barheight from 9.3 to 15.5 kcal/mol and in the reaction enefrom 22.6 to 4.6 kcal/mol. The calculated differential barriheights and reaction energies are listed in Table I. To calate the barrier height or reaction energy for a particularaction, the respective differential barrier height or reactenergy is added to that of the principal reaction. Note thatdifferential barrier heights and reaction energies calculatethe BH&HLYP compare well with the PMP4 results. Eletronic structure calculations were done using theG98

program.30

A. H¿C2H6

Arrhenius plot of the calculated rate constants for treaction along with experimental data is shown in Fig. 3. O


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calculated rate constants are in excellent agreement withperimental data18,31–33for a wide range of temperatures. Rcall that no PMP4 single-point calculations were donethis reaction. The ratio of partition function,f Q for this re-action class is determined from this and the principal retions. f Q is set equal to its high temperature limit of 0.09and is used in estimating rate constants for reactions R3–

B. H¿C3H8

Arrhenius plots for the primary, secondary, and total hdrogen abstraction of C3H8 are shown in Figs. 4~a!–4~c!,

TABLE I. Calculated differential reaction energies and barrier heig~kcal/mol!.a

Reactions

DDE5DE2DE(R1) DDV‡5DV‡2DV‡(R1)

PMP4 BHLYP PMP4 BHLYP

H1H–CH3~R1! ~2.90!b ~1.40! ~14.58! ~12.60!H1H–CH2CH3 23.01 22.52 22.83 22.90H1H–CH2CH2CH3 22.73 22.27 22.64 22.57H1H–CH~CH3!2 25.45 25.79 25.26 25.21H1c-H–CHC2H4 4.51 4.38 0.22 0.13H1H–C~CH3!3 25.80 28.47 26.38 27.11H1H–CH2F 23.77 23.11 21.96 22.31H1H–CHF2 23.38 23.04 21.92 22.32H1H–CF3 1.71 2.36 0.92 0.56

aPMP4 denotes PMP4/cc-pVTZ//BH&HLYP/cc-pVDZ; BHLYP denoteBH&HLYP/cc-pVDZ.

bValues in parentheses are the classical barrier height and reaction enethe principal reaction.

FIG. 3. Arrhenius plot of the H1C2H6→H21C2H5 reaction. Dashed line isfrom the TK-TST extrapolation of experimental data from Ref. 18. Fillcircles are from Ref. 31; filled diamonds from Ref. 32; open squares frRef. 33.


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FIG. 4. Arrhenius plots of the H1C3H8 reaction.~a! For the primary hy-drogen abstract reaction H1C3H8→H21CH2CH2CH3 ~dashed line is fromRef. 34 and dashed–dotted line is from Ref. 18!; ~b! for the secondaryhydrogen abstract reaction H1C3H8→H21CH~CH3!2 ~dashed line is fromRef. 18, open squares from Ref. 34, open circles from Ref. 35, filled cirfrom Ref. 36, and filled triangles from Ref. 37!; ~c! for the overall reactionH1C3H8→H21products~dashed line is from Ref. 18, open squares froRef. 32, open circles from Ref. 38, filled circles from Ref. 39, filled tangles from Ref. 40, crossed squares from Ref. 41, and filled squaresRef. 42!.

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respectively. Note that the experimental rate constants18,34–37

for the primary and secondary hydrogen abstract arefrom direct measurements but are derived either from fittto a complex reaction mechanism or from the detailed bance condition. Thus, it is not informative to compare ocalculated rate constants with the experimentally deridata. However, for the overall reaction where direct meas

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ments can be done, our calculated overall rate constshown in Fig. 4~c! are in excellent agreement with the avaable experimental data.32,38–42

C. H¿c -C3H6

There were only two experimental studies43,44 done byMarshall and co-workers for the hydrogen abstraction fr


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cyclopropane. Our calculated results as shown in Fig. 5 amuch better with the earlier experiment,43 whereas Cohen’sTST interpolation results18 fit better with the later one.44 Inany case, our calculated rate constants are within the exmental uncertainty.

D. H¿CH3F, H¿CH2F2, H¿CHF3

Despite the fact that experimental data for these retions are limited, our calculated rate constants for theseactions shown in Figs. 6–8, respectively, are in good agment with the available experimental measurements.25,45–47

Our results are slightly larger than those from accurate Tcalculations by Berryet al.48 though both are within the experimental uncertainty.

In summary, our calculated results for the R3–R9 retions using the RC-TST method described above showcellent agreement with experimental data. This is verycouraging since limited electronic structure calculationamely, only geometry optimizations of the stationary poiat the BH&HLYP/cc-pvdz level of theory, were performefor these reactions. These results show that the RC-Tmethod can be a powerful tool for estimating kinetic paraeters of a large number of elementary reactions in desigkinetic models for complex combustion systems, thoumore work needs to be done. Several issues have notdiscussed in this study but are important to the developmof this theory, namely~1! efficient methodology for obtaining the differential barrier height and reaction energy;~2!applications to other reaction classes such as those involaromatic molecules;~3! treating reactions without barrierThese issues will be systematically addressed in future sies.

FIG. 5. Arrhenius plot for the H1c-C3H6→H21c-C3H5 reaction. Dashedline is the TK-TST extrapolation of experimental data from Ref. 18, fillcircles are experimental data from Ref. 44, and filled diamonds from R43.


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IV. CONCLUSION

We have presented a new formalism called ReactClass Transition State Theory~RC-TST! for efficient calcu-lations of thermal rate constants of reactions in a class frfirst principles. The RC-TST method is based on the trantion state theory framework and properties of the react

f.

FIG. 6. Arrhenius plot for the H1CH3F→H21CH2F reaction. Dashed lineis the TST results from Ref. 48, filled triangles from Ref. 45, and fillcircles from Ref. 46.

FIG. 7. Arrhenius plot for the H1CH2F2→H21CHF2 reaction. Dashed lineis from Ref. 48 and filled circles from Ref. 45.


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class in deriving the expression for relative rate constaWith the readily available potential energy information fthe principal reaction, thermal rate constants of any otreaction in the class can be efficiently estimated from otwo energetic properties, namely, the differential barrheight and reaction energy. We have shown that these diential energetic properties can be calculated at a relatilow level of electronic structure theory and thus requmuch less computational demand than calculations ofactual barrier height and reaction energy. Consequently,potential energy information required for the RC-TSmethod is much less than what is needed by the simpleventional TST method. We have applied the RC-Tmethod to several hydrogen abstraction reactions. The calated rate constants are in excellent agreement with availexperimental data in all test cases. This is particularly vencouraging since the practical implication of the RC-Tmethod in design of kinetic models for complex combustsystems is in fact enormous.

ACKNOWLEDGMENTS

This work is supported by the University of Utah Centfor the Simulation of Accidental Fires and Explosionfunded by the Department of Energy, Lawrence LivermoNational Laboratory, under subcontract No. B341493 angenerous gift from the Dow Chemical Company. The autwould like to thank Shao-Wen Zhang and Thanh-ThaiTruong for assistance with the calculations. We also ththe Utah Center for High Performance Computing for coputer time support.

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FIG. 8. Arrhenius plot for the H1CHF3→H21CF3 reaction. Dashed line isfrom Ref. 48, filled triangles from Ref. 45, and filled circles from Ref. 4


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