A theoretical ab initio and Monte Carlo simulation study of thereaction kinetics in the gas phase and in carbonpyridine + CCl

2tetrachloride solution using canonical Ñexible transition state theory

Josefredo R. Pliego, Jr. and Wagner B. De Almeida*

de Computacional e Modelagem Molecular (L QC-MM), Departamento deL aborato� rio Qu•�micaICEx, Universidade Federal de Minas Gerais (UFMG), Belo Horizonte, MG, 31270-901Qu•�mica,

Brazil

Received 9th November 1998, Accepted 4th January 1999

The potential energy surface for the reaction was studied at the ab initio MP4/6-311G(2df,p)pyridine] CCl2//MP2/6-31G(p) level of theory. The MP4/6-311G(2df,p) energies were evaluated by the additivityapproximation E[MP4/6-311G(2df,p)]B E[MP4/6-31G(p)]] E[MP2/6-311G(2df,p)][ E[MP2/6-31G(p)]. TheÐrst step proceeds by the addition of to pyridine forming a dipolar ylide structure without an activationCCl2barrier. Then this species rearranges to a more stable biradical like ylide on a picosecond time scale. Thegeneralized transition state for dipolar ylide formation occurs at a large center of mass distance between thespecies, and to calculate the reaction rate constant we have used canonical Ñexible transition state theory. TheconÐgurational integral was solved by Monte Carlo simulation and statistical perturbation theory, and thepotential of mean force in the gas phase was obtained. This procedure was extended to the liquid phase byincluding the solvent coordinates in the conÐgurational integral. The activation free energy in the gas phaseand in carbon tetrachloride solution was calculated as 1.44 and 2.62 kcal mol~1, respectively. Thecorresponding rate constants are 5.5 ] 1011 and 7.5] 1010 l mol~1 s~1. The last value is in reasonableagreement with the experimental result of 7] 109 l mol~1 s~1 determined in isooctane solution.

I IntroductionFirst principles ab initio quantum chemistry calculationscoupled with statistical theories of chemical kinetics hasbecome an important tool to study chemical reaction rate.1h8This approach involves a highly accurate computation of thepotential energy surface around a limited dynamically signiÐ-cative region, separating the reactants and products minima,and the reaction rate is calculated as the crossing rate of thesystem through this region. Transition state theory and itsvariations4,9,10 have been largely used to compute the cross-ing rate, and the general expression for the bimolecular reac-tion rate constant involving species A and B through ageneralized transition structure TS is :

k(T , s)\kbTh

QTS(s)QA QB

(1)

where is the (generalized) molecular partition function forQXunit volume, and s is the reaction coordinate, a variable indi-cating the location of the dividing surface (where the reactiveÑux is measured) on the reaction path. Alternatively, the rateconstant can be expressed in activation free energy terms :

k(T , s)\kbTh

e~*G(s)@kbT (2)

The nuclear motion at minima on the potential energysurface, i.e. the motion of the stable molecules, is generallytreated in the rigid rotorÈharmonic oscillator approximation.The partition function for the minima can then be factoredinto the translational, rotational, vibrational and electroniccontributions :

QX \ QX, TQX, RQX, V e~VX@kbT (3)

The reaction path and the respective reaction coordinate canhave several deÐnitions. The most dynamically appealing oneis the intrinsic reaction path (IRP), in which the sequence ofgeometries is ditched by the variation of the internal coordi-nates as a function of the intrinsic reaction coordinate(IRC).5,11,12 The IRP is originated by a small jump away ofthe saddle point in the direction of the transition vector, andposterior inÐnitely slow motion governed by the classicalmechanics equations of motion. The gradient vector in themass weight cartesian coordinates is at a tangent to the reac-tion path all the way along, and the potential energy surfacecan be separated into a contribution from the displacement onthe path and orthogonal to the path :13

V (s, q–j) \ V (s) ] ;

j/1, 3N~712u

j2(q

j[ q

j, s)2 (4)

The parameters are the harmonic frequencies obtained byujdiagonalizing the projected force constants matrix, and

is the displacement along the j normal mode orthog-qj[ q

j, sonal to the gradient vector in mass weight coordinates. TheHamiltonian for the generalized transition state is then sup-posed to be separable into rotational and vibrational contri-butions, and the harmonic frequencies orthogonal to thereaction path are used to build the vibrational partition func-tion. Thus, an expression similar that used for stable species isobtained for the generalized transition state partition functionQTS(s).Another deÐnition of the reaction path is to choose a dis-tinct internal variable, to maintain it frozen and to optimizethe remaining geometric coordinates.14 Several values of thisvariable are used, and the generated path is denominated thedistinguished reaction path (DRP). The frozen variable is thedistinguished reaction coordinate (DRC). To calculate thepartition function, the same procedure used for the IRP can

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be applied to the DRP. However, in this case the mass weightgradient vector is not tangential to the reaction path. Anadvantage of using a distinguished reaction path is thatgeometry optimization with one frozen variable is a relativelyeasy task. In contrast, to obtain the IRP, a sti† di†erentialeqn. has to be integrated, and in many situations the con-vergence is slow, requiring a very small integration pass and ahuge amount of computation.

In conventional transition state theory (TST), the dividingsurface is Ðxed at the saddle point on the potential energysurface.10 In this case projection of the hessian is not neces-sary, and the transitional motion is the imaginary frequencynormal mode. Recrossing on the saddle point can occur dueto a dynamical bottleneck away from the transition state. Forexample, normal modes can become more rigid across thetransition state, generating an e†ective barrier higher than thesaddle point one. To improve the reaction rate calculation, thedividing surface is varied until the reactive Ñux is a minimum.This point is then chosen as the generalized transition state,where the partition function is computed and used inQTS(s)eqn. (1). This approach is denominated the canonical varia-tional transition state theory (CVTST).4,5,9,14h17

Harmonic approximation is generally used to account forthe modes orthogonal to the gradient vector. However, insome situations the harmonic modes can degenerate to hin-dered rotation. This occurs in dissociation or association reac-tions involving two fragments A and B, for which thegeneralized transition state is a very Ñoppy structure locatedat a large AÈB separation. Reactions that do not present acti-vation barriers are in this class. These large amplitude vibra-tional degrees of freedom are transformed in free rotation ofeach A and B fragment as the AÈB distance increases. An ade-quate approach to this problem is a Ñexible transition statetheory.18h24 The idea is to separate the small amplitude inter-nal vibrations of each fragment from the almost free rotationof A and B frameworks. The small amplitude motions areconsidered harmonic modes, while the rotation of A and Bspecies are treated using classical mechanics for two rigidbodies interacting between them with a Ðxed center of massdistance, which is taken as the reaction coordinate. In canon-ical Ñexible transition state theory (CFTST), the partitionfunction for the transition state can be written as :

QTS(s)\ QAB, T QAB, PD(s)QA, RQB, RQA, VQB, V

]/ e~U(s, usA, us B)@kT du– A du– B

(8n2)2(5)

The terms account for translation of AB, pseudo-diatomic ABrotation, rotation of the A and the B framework, internalsmall amplitude vibration of A and B, and the intermolecularinteraction between A and B species. The u– vector is in termsof Eulerian angles and s is the center of mass distance. A morecomplete version of the CFTST also considers the deforma-tion of A and B fragments and variations of the internal vibra-tions along the reaction path. But if the generalized transitionstate, also determined by a variational procedure, leads to alarge separation of A and B fragments, these contributions canbe ignored.

Modeling chemical reactions in solution from Ðrst prin-ciples is a challenge for theoretical chemistry. The solvent canact slightly on the reaction rate or play a very important rolein the reaction mechanism. Transition state theory can beextended to the liquid phase, where the potential energysurface is substituted by the free energy surface or potential ofmean force.25h27 This approach considers the classical parti-tion function for the whole system, and separates the motioninto internal, involving the fragments, and external, involvingthe solvent molecules. The full partition function for thesystem is then integrated in the external variables, generatingan e†ective partition function for the reactants motion. In an

analogous form for the calculation of the reaction rate in thegas phase, the reaction in the liquid phase can be calculatedusing expression (1), but now the structure and vibrational fre-quencies for species A and B are determined by the potentialof mean force. This approach is impractical, as the integrationon the solvent coordinates is needed for each conÐguration ofA and B species. Fortunately, an important approximationcan be introduced : the reaction path in solution, as well as thestructure of the stable species, is assumed to be the same as inthe gas phase.26h29 In addition, the alteration of the vibra-tional frequencies is neglected, with the result that the solventjust a†ects the potential energy along the reaction path. Withthese assumptions, the rate constant will be given by :

ksolution(T , s) \ kgas(T , s)e~*W(s)@kT (6)

*W (s) being the di†erence in solvation free energy betweenthe generalized transition state at the value s of the reactioncoordinate and the free A and B molecules. This treatmentonly considers equilibrium solvent e†ects. Dynamical e†ectssuch as frictional forces against the reacting motion are nottaken into account. Developments in this Ðeld have beenattained by the e†orts of several authors.30h33

Carbenes are an important class of reactive intermediates.Many reactions involving these species do not have activationbarriers, and the generalized transition state can occur at largeAÈB distance. In particular, pyridine reacts with carbenes inan almost di†usion controlled regime, forming pyridiniumylides which are very useful to the experimentalist in the studyof carbenes kinetics with relation to several organic functionalgroups.34h39 Thus, Ñexible transition state theory can be aspecially attractive approach for some reactions involvingcarbenes. In the present article we report a theoretical study ofthe reaction kinetics in the gas phase and inpyridine] CCl2solution using canonical Ñexible transition state theory. TheconÐguration integral involving the A and B molecules orien-tation in the gas phase were computed using the Monte Carlomethod and statistical perturbation theory. The sameapproach was used in the solution phase, now includingexplicitly the solvent molecules, and determining the potentialof mean force as a function of the center of mass distance. Thegeneralized transition state and the respective rate constantwere determined by the variational procedure. Dynamicsolvent e†ects were not included. In the next sections we willpresent the details of the calculations and the results.

II Ab initio quantum chemistry calculation ofaddition to pyridineCCl

2In previous work we have studied the interaction of withCCl2water,40h43 formaldehyde2 and methylimine.44 The inter-action of this carbene with heteroatoms having double bondsleads to the formation of an ylide, accomplished by the break-ing of the p bond and formation of a biradical like species. Inthe case of addition to methylimine, dipolar and biradicalylides can be formed. An adequate treatment of these biradicallike species requires the inclusion of electron correlation, andsimilar trends are expected for the addition to pyridine.CCl2Thus, to study this system we have used the MP2 method toperform the geometry optimizations employing the 6-31G*basis set for chlorine and 6-31G for the remaining atoms. Thisbasis set will be denominated 6-31G(p). Harmonic frequencyanalysis was also performed at this level of theory. To obtainhigh level electronic energies, single point calculations at theMP4/6-31G(p) and MP2/6-311G(2df,p) levels were performedin conjunction with an additivity approximation of the corre-lation energy resulting in an e†ective MP4/6-311G(2df,p) cal-culation. The GAUSSIAN 94 program was used for all abinitio computations.45

Similarly to the interaction of with methylimine, itsCCl2interaction with pyridine also leads to the formation of both

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Fig. 1 Minimum energy structures (MS1 and MS2) and the transition state structure (TS2) for the MS1 ] MS2 isomerization located on thepotential energy surface for the reaction. Geometry optimizations were performed at the ab initio MP2/6-31G(p) level.pyridine ] CCl2

dipolar and biradical ylide species, which are depicted in Fig.1. The dipolar form (MS1) is less stable, and a rotation aroundthe CwN bond leads to a transition state TS2, which falls tothe biradical isomer (MS2). Fig. 2 shows the potential energysurface proÐle for the reaction. A search for apyridine] CCl2transition state connecting the free species to the dipolar ylidewas unsuccessful, indicating that this reaction step proceedswithout a barrier. The enthalpy variation for this step is[24.53 kcal mol~1, while the free energy variation is [12.53kcal mol~1. The isomerization transition state (TS2) is just0.90 kcal mol~1 (at the MP2/6-31G(p) level) above MS1. Atthe MP4/6-311G(2df,p) level, this barrier decreases to 0.02kcal mol~1. These results indicate that MS1 could not be aminimum at a higher level of geometry optimization, and if itsurvives as a minimum, its lifetime should be of the order of

Fig. 2 Potential energy surface proÐle for the reac-pyridine ] CCl2tion. Thermodynamics data were obtained using the ab initio MP4/6-311G(2df,p)//MP2/6-31G(p) results.

Fig. 3 Potential energy surface proÐle for the reac-pyridine ] CCl2tion. Parameter R is the CÈN distance, and the remaining variableswere fully optimized at the MP2/6-31G(p) level. The zero of energy wastaken as R\ O. A, MP2/6-31G(p) ; B, MP4/6-311G(2df,p) ; C, *GE.

picoseconds. The biradical ylide (MS2) has a *H¡ \ [35.51kcal mol~1 and *G¡ \ [23.15 kcal mol~1 from the freespecies. Thus this molecule is formed irreversibly, and certain-ly is responsible for the intense absorption observed in theproduct of addition to pyridine.CCl2In order to calculate the reaction rate constant for this reac-tion system, we need to determine the reaction path andlocate the maximum free energy point. We have chosen theN(pyridine)wC(carbene) bond length as the distinguishedreaction coordinate, and full geometry optimizations were per-formed at several frozen NÈC distances, resulting in a curve ofenergy vs. R(NÈC). High level single point calculations, energygradients and a force constant matrix were also determinedalong with the reaction path. The hessian matrices were pro-jected in a space orthogonal to the gradient in mass weightcoordinates and to rotational and translational motions, usingthe technique of Miller et al.13 The diagonalization of the pro-jected hessian was carried out to obtain the harmonic fre-quencies along the reaction path. These data were utilized tocalculate the activation free energy at selected points on thepath. The results are given in Fig. 3.

Dichlorocarbene presents a very accentuated interactionwith pyridine. At R\ 3.0 the interaction energy is [5.4Ókcal mol~1 and decreases to [26.7 kcal mol~1 whenR\ 1.617 corresponding to the dipolar ylide minimum.Ó,This strong stabilization energy and the long CÈN distance,which result in large amplitude motion of the fragments ofpyridine and dichlorocarbene, lead to a very low activationfree energy, as can be seen in Fig. 3. The highest value of *GE

in this graph is 0.2 kcal mol~1, occurring at 2.8 (we haveÓnamed this structure TS1-28). The consequence is a very highrate constant of 4 ] 1012 l mol~1 s~1. However, it is probablethat the true transition state is located at higher

distance, where the interaction energy is verypyridineÈCCl2low and the reaction rate is close to the collision rate of struc-tureless particles. Using this model and supposing that thecenter of mass separation for contact is 7 the collision rateÓ,constant will be about 4 ] 1011 l mol~1 s~1. Thus, we shouldsearch for a generalized transition state in this region. Never-theless, by the fact that the intermolecular interaction is veryweak at this separation, the relative vibration between thepyridine and moieties will become an almost two rigidCCl2bodies free rotation. As a consequence an adequate treatmentof this problem will require the use of the canonical Ñexibletransition state theory, which will be applied to this system inthe next sections.

III Reaction rate constant by canonical Ñexibletransition state theoryThe generalized transition state for the reac-pyridine] CCl2tion is probably located at a large separation of the fragments,thus the center of mass can be taken as an adequate reactioncoordinate. In addition, each molecule can be considered a

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Table 1 Parameters for the intermolecular potentials

Molecule Site q/e e/kcal mol~1 p/Ó

CCl2 C 1.627 0.105 3.750Cl [0.079 0.300 3.400Lp [1.469 0.0 0.0

Pyridine N [0.578 0.170 3.250C2 0.376 0.110 3.750C3 [0.169 0.110 3.750C4 0.164 0.110 3.750

rigid body, rotating upon its center of mass and interacting byintermolecular forces. The internal vibration of each fragmentis supposed to be separable from the rigid body rotation. Withthese considerations, the partition function along the reactionpath is represented by eqn. (5). Substituting (3) and (5) into (1),we obtain :

k(T , R)\kbTh

QTS,T QTS, PD(R)

QA,TQB,Te~*W(R)@kT (7)

where :

e~*W(R)@kT\/ e~U(R, usA, us B)@kT du– A du– B

(8n2)2(8)

In eqn. (8) we are considering that for R] O, U(R, u– A ,Thus, *W (R) is the potential of mean force in the gasu– B)] 0.

phase as a function of the center of mass distance (R). Eqn. (8)can be extended for reaction in the liquid phase. It is onlynecessary to include all solvent coordinates in the conÐgu-q–rational integral, and the result will be :

e~*Wsolution(R)@kT\/ e~U(R, usA, us B, qs)@kT du– A du– B dq–

(8p2)2(8p2V )Ns(9)

It should be noted that we are considering the solvent mol-Nsecules as polyatomic non-linear rigid bodies. Eqn. (9) can besubstituted into eqn. (7) to obtain the reaction rate constant insolution (excluding dynamic solvent e†ects) by the variationalprocedure, i.e. minimizing the rate constant as a function of R.

Another form in which to write the rate constant is toexpress the partition function in the free energy form. Eqn. (7)can be written as :

k(T , R)\kbTh

e~*Gj(R)@kbT (10)

and

*Gj(R)\ *GPDj (R)] *W (R) (11)

Here the Ðrst term on the right side is the gas phase activationfree energy for a pseudo-diatom reaction at center of mass

Fig. 4 Interaction energy between and as a function ofC5H5N CCl2the CÈN distance, and the remaining geometric parameters frozen inthe TS1-28 geometry. A, Ab initio MP2/6-31G* level. B, empiricalforce Ðeld level, using the ChelpG charges derived from a MP2/6-31G* wavefunction with dipole restriction.

distance R. This expression can be applied in the gas phaseand in solution, and by the variational procedure, the activa-tion free energy to be used in the computation of the rateconstant is that which maximizes the activation free energy.

IV Evaluation of the conÐgurational integralFor reactions in the gas phase, the conÐgurational integralcan be solved numerically by direct integration. However, forreactions in the liquid phase, the solvent coordinates have tobe integrated too, with the result that a statistical type ofapproach is needed. The evaluation of the potential of meanforce by a combination of Monte Carlo simulation and sta-tistical perturbation theory has been applied by Jorgensen etal. to several systems.26,27,46h49 This method can also be usedfor evaluation of the integrals (8) and (9) in the gas phase or insolution. In the present problem, the perturbations are smalldisplacements along R, and the variation of the potential ofmean force will be given by :

*W (Rn`1) [ *W (R

n) \ [kbT ln

C Pe~dU@kT

] o(R, u6 1, u6 2 , q– ) du6 1 du6 2 dq–D

(12)

o being the density of probability for the conÐguration of thesystem at and dU the variation of the potential due to theR

nnew geometry of the reactant molecules. The above integral isthen evaluated using Monte Carlo simulation :

*W (Rn`1) [ *W (R

n) \ [kbT lnSe~dU@kTT (13)

When R] O, *W (R) ] 0. Thus, we begin the integration at alarge R value, and perform the perturbation in a way to gener-ate the *W (R) curve. To evaluate the asymptotic contribution,the following annihilations can be performed :

C5H5NÉ É ÉCCl2(Ra) ] C5H5N *G1 (14)

CCl2 ] nothing *G2 (15)

and the potential of mean force at can be obtained by theRarelation :

*W (Ra) [ *W (O) \ *G2 [ *G1 (16)

V Intermolecular potentialsPolyatomic molecules in general present a high anisotropy inthe intermolecular potentials, such that a unique centerexpansion of the potential leads to a slow convergence of theenergy. The interaction sites model, with expansions centeredin each atom is more adequate. This approach has beenlargely applied to liquid simulations, and in particular theLennard-Jones plus point charges model. We have used thispotential to model the interaction in the gaspyridine] CCl2phase and in solution. In the last case, interaction with thesolvent molecules are included too. The analytic form of thispotential is :

Uab\ ;iEj

qiqj

rij

]4e

ijpij12

rij12

[4e

ijpij6

rij6

(17)

where i and j are the sites on the interacting A and B mol-ecules, q is the point charge and e and p are the Lennard-Jones parameters. The point charges were obtained by theChelpG procedure of Ðtting charges to the electrostatic poten-tial. In the present case, we have used the MP2/6-31G* wave-function with dipole restriction to derive the point charges.The values of e and p were taken from OPLS parameters inconjunction with the combination rules for crossing inter-

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actions :

eij\ Je

iiejj

pij\ Jp

iipjj

(18)

Table 1 presents the parameters used to represent thepotentials of pyridine and The hydrogens of pyridineCCl2 .were united to the respective carbon to represent just one site,and a fourth site was introduced into the dichlorocarbene asalready carried out in previous studies.40,41 The quality of thispotential can be evaluated by studying Fig. 4. It presents theinteraction energy as a function of CÈN distance calculated atthe ab initio MP2/6-31G* level and using the empirical poten-tial. The geometry was taken from free fragments and theintermolecular geometric parameters were taken from theTS1-28 structure. The results show that at distances greaterthan 4 the empirical potential reproduces very well the abÓ,initio potential. This NÈC distance translates to about 5.5 Ófrom the center of mass distance.

VI Monte Carlo simulationsThe calculation of the relative values of the conÐgurationalintegrals were performed by Monte Carlo simulations coupledwith statistical perturbation theory using double-wide sam-pling.26,27,49 Steps of 0.25 in R were used, and for R\ 10Ó Óannihilations (14) and (15) were performed to evaluate theassymptotic contribution to the potential of mean force [eqn.(16)]. For simulations in solution, we have used theisothermalÈisobaric ensemble (NPT) and a rectangular boxwith 128 molecules. The cut-o† was set at 12 forCCl4 ÓsoluteÈsolvent and solventÈsolvent interactions. The soluteswere admitted to keep the distance between them Ðxed, androtations of each fragment over their center of mass weresampled. The equilibration was attained with 9] 105 conÐgu-rations, and the average was calculated using 2 ] 106 conÐgu-rations. Translations and rotations were adjusted to lead to anacceptation rate of about 50%, the environmental conditionsbeing Ðxed at 1 atm¤ and 25 ¡C. All simulations were per-formed using the BOSS program.50

VII Results and discussionThe potential of mean force in the gas phase presents amonotonic decrease from inÐnite separation to 5 andÓ,begins to increase for lower R as can be seen from Fig. 5. Thisfeature at low R is due to the inadequacy of the potential atthis region, where chemical forces begin to actuate. Indeed,Fig. 4 shows an accentuated deviation of the empirical poten-

Fig. 5 Potential of mean force for the interaction inpyridine ] CCl2the gas phase (A) and in carbon tetrachloride solution (B). R is thecenter of mass distance.pyridineÈCCl2

¤ 1 atm \ 101 325 Pa.

tial from the ab initio data below an NÈC distance of 4 Ó,which corresponds to a center of mass distance of about 5.5 Ó.In carbon tetrachloride solution, the potential of mean forceincreases from R\ 10 to R\ 6.5 which corresponds to aÓ Ó,maximum, and falls to a minimum at 5.5 Below this pointÓ.the potential increases again, but analogous to the gas phaseresults this region is not adequately described by the empiricalpotential. Thus, this increase in both cases is artiÐcial. Thedi†erent behavior of the potential of mean force in the gasphase and in solution is notable. In the gas phase thereCCl4is an intermolecular attraction, and the rotations of the pyri-dine and moieties lead to a small increase of the freeCCl2energy with relation to the energy on the minimum energypath. In the liquid phase, another e†ect is taking place. Theapproximation of pyridine and dichlorocarbene species expelsthe solvent molecules between them, increasing the occupiedcavity in the solvent. The consequence is the increase in thefree energy of the system, and the appearance of a barrier withthe top occurring at 6.5 This e†ect is contrary to thatÓ.observed for compact transition states.51 In the last case, thereis a decrease in the cavity occupied by the solutes, and the freeenergy decreases. As a consequence the reaction in solution isgreater than in the gas phase. In the present case, the inverseis occurring. So, we can note that the rate constants ofbimolecular reactions involving neutral species in low polaritysolvents can be increased or decreased in relation to the gasphase. The property that determines this behavior is the loca-tion of the transition state. Loose transition states with largeinterfragment separation will lead to smaller reaction rates insolution than in the gas phase. Tight transition states with acompact structure will lead to a greater reaction rate in thesolution than in the gas phase.

The variation of the activation free energy in the gas phaseand in solution are in Fig. 6. In the gas phase the maximumoccurs at R\ 7.5 where kcal mol~1 andÓ, *GE\ 1.44results in a reaction rate constant of 5.5 ] 1011 l mol~1 s~1.In carbon tetrachloride solution, the maximum occurs atR\ 6.5 The respective free energy and rate constants areÓ.

kcal mol~1 and k(298.15)\ 7.5] 1010 l mol~1*GE\ 2.62s~1. There are no experimental kinetics data for this reactionin the gas phase, but in isooctane28 the rate constant wasdetermined as being 7] 109 l mol~1 s~1, which traduces anactivation free energy of 4 kcal mol~1. It is probable that asimilar value would be found for the reaction in solu-CCl4tion, so the di†erence between the experimental and theoreti-cal e†ective value is about 1.4 kcal mol~1. As the*GE

intermolecular potential seems accurate, the origin of the dif-ference is probably due to the di†erence between the solventsand solvent dynamics e†ects, which can lead to recrossing onthe dividing surface, and a decrease in the rate constants. Wethink that this e†ect is important in this system because thetransition state is very Ñoppy, occurring at a large separationof the species and the potential of mean force is very Ñat in

Fig. 6 Activation free energy for the reaction in thepyridine ] CCl2gas phase (A) and in carbon tetrachloride solution (B) as a function ofthe center of mass distance (R).pyridineÈCCl2

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this region, facilitating the recrossing. We intend to addresssolvent dynamics e†ects in future work.

Acknowledgementswould like to thank the Conselho Nacional de Desenvol-We

vimento Cient•� Ðco e Tecnolo� gico (CNPq), the deFundacÓ a8 oAmparo a Pesquisa do Estado de Minas Gerais (FAPEMIG)and the Programa de Apoio ao Desenvolvimento Cient•� Ðco eTecnolo� gico (PADCT II) for supporting this research project.We also thank the Centro Nacional de Processamento de AltoDesempenho de Minas Gerais e Regia8 o Centro-Oeste(CENAPAD-MG/CO) for computational resources.

References1 S. S. Kumaran, M-C. Su, K. P. Lim, J. V. Michael, S. J. Klip-

penstein, J. DiFelice, P. S. Mudipalli, J. H. Kiefer, D. A. Dixonand K. A. Peterson, J. Phys. Chem. A, 1997, 101, 8653.

2 J. R. Pliego, Jr. and W. B. De Almeida, J. Chem. Phys., 1997, 106,3582.

3 A. M. Mebel, E. W. G. Diau, M. C. Lin and K. Morokuma, J.Phys. Chem, 1996, 100, 7517.

4 D. G. Truhlar, B. C. Garrett and S. J. Klippenstein, J. Phys.Chem., 1996, 100, 12771.

5 M. A. Collins, Adv. Chem. Phys., 1996, 93, 389.6 M. T. Nguyen, D. Sengupta, G. Raspoet and L. G. Vanquicken-

borne, J. Phys. Chem., 1995, 99, 11883.7 X. Duan and M. Page, J. Chem. Phys., 1995, 102, 6121.8 T. H. Dunning, Jr., L. B. Harding, A. F. Wagner, G. C. Schatz

and J. M. Bowman, Science, 1988, 240, 453.9 J. I. Steinfeld, J. S. Francisco and W. L. Hase, Chemical Kinetics

and Dynamics, Prentice Hall, New Jersey, 1989.10 S. Glasstone, K. J. Laidler and H. Eyring, T heory of Rate

Process, McGraw-Hill, New York, 1941.11 K. Fukui, Acc. Chem. Res., 1980, 14, 363.12 C. Gonzales and H. B. Schlegel, J. Phys. Chem., 1990, 94, 5523.13 W. H. Miller, N. C. Handy and J. E. Adams, J. Chem. Phys.,

1980, 72, 99.14 J. Villa and D. G. Truhlar, T heor. Chem. Acc., 1997, 97, 317.15 D. G. Truhlar, W. L. Hase and J. T. Hynes, J. Phys. Chem., 1983,

87, 2664.16 D. G. Truhlar and B. C. Garrett, Acc. Chem. Res., 1980, 13, 440.17 B. C. Garrett and D. G. Truhlar, J. Phys. Chem., 1979, 83, 1052.18 S. H. Robertson, A. F. Wagner and D. M. Wardlaw, J. Chem.

Phys., 1995, 103, 2917.19 S. H. Robertson, A. F. Wagner and D. M. Wardlaw, Faraday

Discuss., 1995, 102, 65.20 S. C. Smith, J. Phys. Chem., 1993, 97, 7034.21 S. C. Smith, J. Chem. Phys., 1991, 95, 3404.22 S. J. Klippenstein and R. A. Marcus, J. Chem. Phys., 1987, 87,

3410.23 D. M. Wardlaw and R. A. Marcus, J. Chem. Phys., 1985, 83, 3462.24 D. M. Wardlaw and R. A. Marcus, Chem. Phys. L ett., 1984, 110,

230.

25 T. L. Hill, Statistical Mechanics, Dover Publications, New York,1987.

26 W. L. Jorgensen, Acc. Chem. Res., 1989, 22, 184.27 W. L. Jorgensen, Adv. Chem. Phys., Part II, 1987, 70, 469.28 W. L. Jorgensen, J. F. Blake, D. Lim and D. L. Severance, J.

Chem. Soc., Faraday T rans., 1994, 90, 1727.29 K. M. Merz, Jr., J. Am. Chem. Soc., 1990, 112, 7973.30 G. A. Voth and R. M. Hochstrasser, J. Phys. Chem., 1996, 100,

13034.31 R. F. Grote and J. T. Hynes, J. Chem. Phys., 1981, 74, 4465.32 R. F. Grote and J. T. Hynes, J. Chem. Phys., 1980, 73, 2715.33 S. H. Northrup and J. T. Hynes, J. Chem. Phys., 1980, 73, 2700.34 M. Robert, I. Likhotvorik, M. S. Platz, S. C. Abbot, M. M.

Kirchho† and R. Johnson, J. Phys. Chem. A, 1998, 102, 1507.35 A. Admasu, A. D. Gudmundsdottir, M. S. Platz, D. S. Watt, S.

Kwiatkowski and P. J. Crocker, J. Chem. Soc., Perkin T rans. 2,1998, 1093.

36 A. Admasu, A. D. Gudmundsdottir and M. S. Platz, J. Phys.Chem. A, 1997, 101, 3832.

37 M. Robert, J. P. Toscano, M. S. Platz, S. C. Abbot, M. M. Kirch-ho† and R. P. Johnson, J. Phys. Chem., 1996, 100, 18426.

38 J. E. Chateauneuf, R. P. Johnson and M. M. Kirchho†, J. Am.Chem. Soc., 1990, 112, 3217.

39 J. E. Jackson, N. Soundararajan and M. S. Platz, J. Am. Chem.Soc., 1988, 110, 5595.

40 J. R. Pliego, Jr. and W. B. De Almeida, J. Phys. Chem. A, 1999, inpress.

41 J. R. Pliego, Jr., M. A. and W. B. De Almeida, Chem.FrancÓ aPhys. L ett., 1998, 285, 121.

42 J. R. Pliego, Jr. and W. B. De Almeida, J. Phys. Chem., 1996, 100,12410.

43 J. R. Pliego, Jr. and W. B. De Almeida, Chem. Phys. L ett., 1996,249, 136.

44 J. R. Pliego Jr., and W. B. De Almeida, J. Chem. Soc., PerkinT rans. 2, 1997, 2365.

45 GAUSSIAN 94, Revision D.2, M. J. Frisch, G. W. Trucks, H. B.Schlegel, P. M. W. Gill, B. G. Johnson, M. A. Robb, J. R. Cheese-man, T. Keith, G. A. Petersson, J. A. Montgomery, K. Raghava-chari, M. A. Al-Laham, V. G. Zakrzewski, J. V. Ortiz, J. B.Foresman, J. Cioslowski, B. B. Stefanov, A. Nanayakkara, M.Challacombe, C. Y. Peng, P. Y. Ayala, W. Chen, M. W. Wong,J. L. Andres, E. S. Replogle, R. Gomperts, R. L. Martin, D. J.Fox, J. S. Binkley, D. J. Defrees, J. Baker, J. P. Stewart, M. Head-Gordon, C. Gonzalez and J. A. Pople, Gaussian, Inc., Pittsburgh,PA, 1995.

46 W. L. Jorgensen and D. L. Severance, J. Am. Chem. Soc., 1990,112, 4768.

47 W. L. Jorgensen, J. Am. Chem. Soc., 1989, 111, 3770.48 J. K. Buckner and W. L. Jorgensen, J. Am. Chem. Soc., 1989, 111,

2507.49 W. L. Jorgensen and C. Ravimohan, J. Chem. Phys., 1985, 83,

3050.50 W. L. Jorgensen, BOSS version 3.5, Yale University, New Haven,

CT, USA, 1995.51 B. M. Ladanyi and J. T. Hynes, J. Am. Chem. Soc., 1986, 108, 585.

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