Variational transition state theory calculation of proton transfer

Mar 1, 1993 - Steve Scheiner, Tapas Kar, and Martin Čuma. The Journal of ... Donald G. Truhlar , Bruce C. Garrett , Stephen J. Klippenstein. The Jour...
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1765

J. Phys. Chem. 1993,97, 1765-1769

Variational Transition State Theory Calculation of Proton Transfer Dynamics in (H3CH-CH3)Alan D. Isaamn' Department of Chemistry, Miami University, Oxford, Ohio 45056

Lan Wang and Steve Scheiner Department of Chemistry & Biochemistry, Southern Illinois University, Carbondale, Illinois 62901 Received: October 2. 1992

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The energetics of proton transfer in the (H3CH-CHs)- complex, which is one stage in the gas-phase reaction CH4 CH3- CH3- CH4, have been investigated with a b initio calculations employing a 4-31G basis. The minimum-energy reaction path was determined by a steepest-descent technique in mass-weighted coordinates from the symmetric saddle point (1 -35kcal/mol above separated CH, CH3-) to the reactant/product association well (9.25 kcal/mol below separated CH4 + CH3-). Motion along this path is found to occur in two relatively distinct phases: motion of the proton between two fixed carbons followed by separation of the two hydrocarbon fragments. Rateconstants computed by variational transition state theory, including an adiabatic approximation for the vibrational modes transverse to the reaction coordinate, demonstrate the importance of including the contributions from tunneling at energies below the top of the 7.88 kcal/mol vibrationally adiabatic barrier for low to moderate temperatures. In addition, incorporating the effects of reaction-path curvature in the tunneling calculation is found to be important, especially at low temperature. However, due to the dual nature of the reaction path, various model barriers fit to the saddle point information yield rate constants that are much too large at low temperatures, thus underscoring the importance of computing the reaction path explicitly.

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1. Introduction

Advances in the first-principles calculation of Born-Oppenheimer electronic energies, gradients, and higher derivatives of potential energy surfaces' together with the development of reliable approaches to computing dynamical i n f o r m a t i ~ nhave ~~~ made possible a proper theoreticaltreatment of reactions involving more than three or four atoms. In addition, recent developments4 have made "direct dynamics" calculationspractical for polyatomic reactions. Rather than requiring information on the entire potential energy surface for a reacting system, this approach describes a chemical process using electronicstructureinformation only in the region of configuration space along a reaction path. Due to its simplicity, the proton transfer reaction is an ideal candidate for the prediction of kinetic data by such theoretical means. Information about these reactions is important also because of the widespread Occurrence of proton transfers in reactions of chemical and biological relevance. One of the least complex systems in which proton transfer dynamics have been computed is (H3C-H-.CH3)-, where the proton is transferred between two simple anions.s*6Following the identification of the equilibrium geometry, (H3C-H-CH3)-, and the transition state for the proton transfer, (H~C*QH-CH~)-, rate constants at various temperatures were computed previouslys using microcanonical transition state theory. Tunnelingwas incorporated at each energy below the adiabatic barrier by replacing the density of states with a tunneling transmission coefficient. The latter was evaluated analytically after first approximating the shape of the adiabatic barrier along the minimum energy path by a function of Eckart form. The results indicated a strong dependence of the kinetics upon tunneling below 400 K, as evidenced by the fact that the rate constants level off sharply at lower temperatures. The present communication concerns itself with a calculation of the kinetics of proton transfer in the same system, but using an entirely different method, the direct dynamicsapproach. Unlike the previous study,5 which was based on information only at the twostationary points on the potential energy surface, this method follows as closely as possible the minimum energy path leading 0022-3654/93/2097-1765S04,00/0

between the equilibrium structure and the saddle point on the surface, extracting information at each point along this path. To summarize, the various stages comprising the full reaction CH, CH3- CH3- CH4are shown schematically in Figure 1. Association of the CH4and CH3-speciesto form the H-bonded complexisfmt,followed bythe proton transfer within thecomplex. Subsequent to the completion of the transfer, the CH3- and CH4 speciesseparate from one another. It is theunimolecular transfer within the complex which is the focus of this work. In section 2, the methods used in this study are discussed, and the results are presented in section 3. Section 4 summarizesour conclusions and discusses possible avenues of future research.

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2. Details of tbe Calculrtions

The required features of the potential energy surface of (H3CH-CH3)- were computed using the ab initio Gaussian-88 suite of programs? at the Hartree-Fock level with a 4-31G basis set.* While this level of theory is not adequate for quantitative reproduction of the true surface, previous results with related systems offer evidence that many features of the SCF/4-3 1G surface are in near coincidence with those computed with correlated methods employing larger, more flexible basis sets.g Indeed, as detailed below, the proton transfer barrier computed here is quite similar to those obtained with more sophisticated methods. Moreover, the aim of this work is not so much an evaluation of the actual rate constant in this particular system, but morea test of the dynamical methods themselves,based upon a given potential energy surface. The SCF/4-3 1G surface should serve in this capacity quite well. The exploration of the minimum energy path (MEP) of the (H3C-H-CH3)- system began with the characterization of the saddle point and the equilibrium structure, whose geometries are illustrated in Figure 2. At the saddle point, the central proton, H,, is located at the midpoint of the C--C axis and the two methyl groups are staggered with respect to one another, as shown. The intercarbon distance at the saddle point is 2.914 A, as compared to 3.598 A in the equilibrium structure, (H$H--CH3)-, wherein the central hydrogen is 1.099 A from the nearer C atom. The Q 1993 American Chemical Society

Isaacson et al.

1766 The Journal of Physical Chemistry, Vol. 97, No. 9, 1993 (a,c.. 'aa. . c a d I \

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PROTON W S r E R

COWLLXATION

SWARATION

Figure 1. Schematic diagram of the SCF energetics (roughly drawn to scale for the basis set used in the present work) for the three stages that comprise the proton transfer reaction CH, + CHpCH3- + CHI.

-.

e: = V,,(4

saddle point

equilibrium structure Figure 2. Geometries of the equilibrium structure and saddle point.

TABLE I: Harmonic Vibrational Fr

Uencies (cm-I) Computed for Equilibrium aod Saddk?oint CeOmeMes Saddle Point

30.9 513.0 1627.8 3019.7 1425.2i

298.5 1244.0 1627.8 3060.6

289.5 1251.6 1793.4 3060.6

494.5 1611.9 1793.4 3068.4

5 13.0 1611.9 3011.0 3068.4

151.9 1576.5 1746.6 3 174.0

343.6 1576.5 2880.2 3213.5

Equilibrium Structure 10.1

343.6 1636.2 2909.4 3213.3

140.4 1095.0 1636.2 2909.4

151.9 1454.8 1746.6 2972.6

were taken along the negative of the energy gradient (the vector of the first derivatives of the energy with respect to the massweighted coordinates). A gradient step size 6s of 0.005 A was used for following the MEP, so that the system geometry, energy [which is denoted as VMEp(S)], and gradient were computed for every 0.005 A ins. In addition, the matrix of second derivativeswas computed for every 0.025 A (the Hessian step size, As) in s. Using the projection operator technique of Miller, Handy, and Adams,13generalized vibrational frequencies and eigenvectors were obtained at each Hessian point along the reaction path. Within the harmonic approximation, this information allows for thecalculationof both the vibrational partition function along the MEP as well as the ground-state vibrationally adiabatic potential curve,

latter structure is a true minimum in the PES, as demonstrated by the fact that all of its harmonic vibrational frequencies are positive; these are listed in the lower part of Table I. The single imaginary frequency in the upper part of this table characterizes (H3C-H-CH3)- as a true saddle point. After placing the origin at the center of mass of the saddle point geometry, the MEP from the reactant equilibrium structure to the product one was found by following the path of steepest descent from the saddle point in a mass-weighted coordinate system, in which the same reduced mass of 1 amu is associated with motion in each degree of freedom. In this way, we obtain the "intrinsic" reaction coordinate1'J-I2rather than an arbitrary reference path. (Due to the symmetry of the problem, only onehalf of the MEP, e.g., the product side, needed to be computed here.) The reaction coordinate (s)is defined as thesigned distance along the MEP from the saddle point, with s > 0 referring to the product side. In determining the MEP, the first step from the saddle point was taken along the normal mode eigenvector corresponding to the negative force constant (eigenvalue of the Hessian or second derivative matrix) there, and subsequent steps

+ &(s)

(1)

where ez,(s) is the total zero-point energy for the bound vibrational degrees of freedom orthogonal to the MEP at s. As discussed below, the latter provides an effective barrier for reaction-path tunneling. Although the accuracy of the reactionpathinformationand therateconstantsderivedfromit (seebelow) obtained with the above step sizes has not been tested, we note that the values of 6s and As used in the present study are smaller than those reported to be necessary for obtaining rate constants of other reactions that are converged to within 15%,.14 Unimolecular canonical variational transition state theory (CVT) rate constants2JJs and semiclassical transmission coefficients16J7for the proton transfer process were calculated using the POLYRATE program.18J9 In this work, bound vibrational and rotational motions were assumed to be separable, and the vibrational partition functions were computed quantum mechanically within the harmonic approximation. For roughly threefigure convergence in the rate constants, four-point Lagrange interpolation of the reaction-path data was used to obtain properties (geometry, energy, generalized normal mode frequencies, and reaction-path curvature components) at every 0.02 bohr along the path from the saddle point to the equilibrium structure. The generalized free energy of activation was then calculated at each of these points, and both the location of the variational transition state and the CVT rate constant kcvTwere determined from interpolating to the maximum of this function. As discussed below, this maximum occurred at s = 0 (i.e., at the saddle point) for all temperatures considered in this study. Quantum mechanical tunneling effects were included by multiplying kcvTby a transmission coefficient K . I ~ Two approximations to the transmission coefficienthave been employed here, namely, the minimum energy path semiclassicaladiabatic groundstate approximation KCVT/MEBAG 16 and the curvature-dominant smallcurvature approximation f l / c - A G . 1 7 J 9 The KCW1M-G factor is calculated efficiently as the Boltzmann average of the semiclassical probability, P(E),of tunneling through the groundstate vibrationally adiabatic potential curve, @(s), at energy E; the methods are described elsewhere.20 The K ~ factor additionally includes the effect of reaction-path curvature, which can lead to a "corner-cutting" that shortens the tunneling path, thereby predicting a larger tunneling contribution. Since the heavy-light-heavy mass combination in the present proton transfer process results in a large degree of reaction-path curvature, the CD-SCSAG method, which is reliable for systems in which tunneling occurs in regions of small to moderate reaction-path curvature, is expected to provide only a qualitative estimate of the tunneling factor. Methods for large curvature cases have been devel0ped,3*~I but they require moreglobal information about the potential energy surface of the system than was determined in the present study. In addition to theabove K ~ factor based ~ on tunneling ~ through the true ground-state vibrationally adiabatic potential curve, CVT/MEPSAG transmission coefficients were

ei(s),

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Proton Transfer Dynamics

The Journal of Physical Chemistry, Vol. 97, No.9, 1993 1767

TABLE II: SCF Ewrgies (kcal/mol) of Equilibrium Structure and Saddle Point

,ZEP

equilibrium structure

saddle point

-9.25 49.87 0 0

1.35 57.75 10.60 7.88

also obtained based on tunneling through various model barriers that are fit to the position and second derivative of the true curve at its maximum, which occurs at the saddle point in the present calculations. One widely-used form, especially for proton transfer reactions,ss22 is the Eckart potential, which for a symmetric case is given by

e(s)

VE(s) = 4EoA/( 1

+ A)'

(2)

e

where Eo is the adiabatic barrier height (Le., the value of (s = 0) measured relative to the ground vibrational level of the reactant equilibrium structure) and A = exp(as), where a = (2F/Eo)l/2 and F is the magnitude of the second derivative of q ( s ) at s = 0. [Following the usual procedure, F is closely approximated by the second derivativeof VM&) at s = 0;in the present work, the imaginary frequency at the saddle point (1425.25icm-1) leads toavalueof 172.264 kcal/(mol A2) for F.] However,as shown below, this Eckart potential is much narrower in the "tail" region than the true adiabatic potential, leading to transmissioncoefficientsthat are much larger than those obtained with the true adiabatic potential at low temperatures. In order to mimic the potential more accurately, several additional model potentials were considered. Since a parabola and a Gaussian function yield barriers that are even narrower than the Eckart potential, they were not considered further. Forms (i) and (ii) arise from considering the functionsx exp(-ax) and x2exp(-ax), respectively, in the regions beyond their peaks. They are given by

V(s) = E,(alsl+ 1) exp(-a)sl)

(3)

Vi,(s)= E,(b(s(/Z + 1)' exp(-bls))

(4)

wherea = (F/Eo)l/2and b = (2F/E0)Il2. Finally, forms (iii) and (iv) are the square r w t and fourth root of the Eckart potential, respectively:

Viii(s)= 2E0/ [exp(as) Vi"($)= (2)'/'EO/[exp(bs)

+ exp(-as)] + e~p(-bs)]'/~

(5)

(6)

where a = (F/Eo)I/2 and b = ( ~ F / E o ) ' / ~As . discussed below, while forms (i)-(iv) have broader "tails" than the Eckart potential, they are still narrower than the true adiabatic potential, leading to much larger transmission coefficients at low temperatures. This underscores the importance of utilizing information on the entire reaction path, rather than just on the stationary points of a reaction, for obtaining rate constants, especially at low temperature.

3. Results md Discussion The geometries of the reactant equilibrium structure and the saddle point optimized at the SCF level with a 4-31G basis are displayed in Figure 2. Correspondingenergies, both without and with the zero-point vibrational energy (ZPE) contribution, are given in Table 11. Here the classical energy VMEPis the BornOppenheimer electronic energy and the vibrationally adiabatic energy is the electronic energy plus the ZPE contribution. Both VMEPand are measured from the electronic energy of the reactants (Le., separated CHI + CHI- at their equilibrium

e

e

i

\ 0 0.0

I

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0.8

0.4 S

1.2

b"S-weighted

1.6

2.0

A)

Figure 3. Classical potential energy, VMEP(lower curve and left scale), and ground-state vibrationally adiabatic potential energy, (upper solid curve and right scale), as functions of the reaction coordinates. The long-dashed portions are extrapolations (see text). The short-dashed curve is the Eckart potential fit to the adiabatic barrier.

geometries), while AVMEPis measured from the value of VMEP for the equilibrium structure and AJf is measured from the for the equilibrium structure. value of These energetic features of the surface can be compared with results computed earlier using larger basis sets and including correlation. Theclassical barrier of 10.6kcal/mol is only slightly smaller than values in the 12-14 kcal/mol range obtained at the correlated MP2 level with a fully polarized 6-31+G** basis set.6 On the other hand, the energy of separated CH, and C H j relative to the equilibrium structures is considerably higher in the SCF/ 4-3 1G case. The larger dissociationenergy at this level is probably due in large measure to basis set superpositionerror. Fortunately, this error is expected to be nearly constant along the minimum energy path between equilibrium geometry and saddle point.23 Since it is this path which is followed to extract the reaction kinetics, and since the SCF/4-31G barrier height is reasonably accurate, the calculated rate constants are probably not very different from those that would be obtained from a better correlated surface. Figure 3 shows the classical potential energy curve ( VMEP)and the ground-state vibrationally adiabatic potential energy curve along the reaction path (solid curves). Due to the relatively large step size 6s used for following the gradient, the steepestdescent method breaks down when the potential energy becomes sufficiently flat. That is, for s > 1.4 A, the steepest-descent path predicted with the present value of 6s "zig-zags" about the true MEP. Thus, we extrapolated the reaction-path properties from s = 1.4 A to the equilibriumstructure, which on energetic grounds was chosen to occur at 1.93 A. (On geometric grounds, a better choice for the position of the equilibrium structure is 2.20 A; however, the rate constant results presented below are stable to at least three figures with respect to the choice for this position.) The sudden flattening of VMEP(S)and around s = 0.3 Acoincides with the region of the MEP with the highest reaction path curvature. To understand this, we consider the important geometric changes along the MEP, which are plotted in Figure 4. (Other than separating, no significant internal or relative changes occur in the two CH3 groups as the system moves along the MEP from the saddle point to the product equilibrium structure.) Near the saddle point, positive reaction coordinate motion corresponds to the motion of the central hydrogen (HE) toward Cbwith a nearly constant C-C distance, while for s > 0.35 A, the reaction coordinate motion corresponds to an increase in the C-C distance with a constant Hc-Cb distance. This twophase nature of the reaction path, Le., proton transfer followed by separation of the two hydrocarbon fragments, thus leads to

(e)

e(s)

Isaacson et al.

1768 The Journal of Physical Chemistry, Vol. 97, No.9, 1993 T

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1.0 0.0

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0.8

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1.2

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-20

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2.0 0

s (mass-weighted A)

Figure 4. Carbon+xrbon and carbon-hydrogen distances as functions of the reaction coordinate s. (See Figure 2 for definitions of these distances.) a relatively steep change in the energy along the MEP near the saddle point and a relatively broad tail, while the coupling of the carbon and hydrogen motions along the reaction path around s = 0.3 A yields a high degree of reaction path curvature. Such behavior has been proposed to explain the proton transfer in the hydrogen abstraction from methane by a methyl radical,22.24and was also observed in the concerted hydrogen atom tunneling in the water trimer.25 In addition, we note that the Eckart potential fit to the trueadiabatic potential (the shortdashedcurvein Figure 3) does not have a broad tail, and thus differs substantially from the true adiabatic potential at lower energies, leading to a much larger prediction for the degree of reaction coordinate tunneling at low temperatures (see below). Another obvious feature in Figure 3 is the local maximum in around s = 0.35 A. This peak arises from a sudden increase in the frequency of the US vibrational mode, which can be best described as an "asymmetric umbrella" motion, Le., one CH3"closes" as C moves toward H, while the other CH3 "opens" as C moves away from H,. Perhaps due to the formation of the H,-Cb bond around s = 0.3 A,the frequency of this mode increases from 1450cm-Iats= 0.26Ato2200cm-lats=0.36A,resulting in the observed increase in In addition, it is important to note that between s = 0.32 and s = 0.58 A,the 4-31G frequencies for the lowest doublydegenerate bending vibrationsare imaginary. This suggests that the 4-3 1G basis set predicts that the vibrational potential along these modes ("wagging" vibrations that involve the motion of C,, H,, and c b in one direction as the two H3 groups swing around in the other) is double-welled over this region of the MEP. Under the harmonic approximation used in these calculations,the POLYRATE program ignores the contributions of imaginary-frequency modes to the ZPE and to the reaction path curvature, and, to avoid placing the variational transition state in regions where such behavior occurs, sets the vibrational partition functions of these modes to 1 X 10'0, which results in a large, negativevalue for the generalized free energy of activation there. Rate constants for the proton transfer process calculated with various levels of approximation are shown as functions of temperature in Figure 5 . At all temperatures considered in this study, the variational transition state (Le., the highest maximum in the generalized free energy of activation curve) occurs at the saddle point, so that the CVT rate constants are identical to those computed withconventional transition state theory. [The relative maximum in *(s) indicates that, at high temperature, the generalized free energy of activation curve would exhibit another maximum that is higher than the one at the saddle point (and, hence, yield a CVT rate constant that is lower than the

c(s)

e(s).

"\

I4

8

12

\-J

16

20

1000/T(K)

Figure 5. Arrhenius plot of calculated canonical rate constants for proton transfer in (HICH.-CH,)-.

TABLE IIk Tnnsm&sion Coefficients K Variow Potential Barriers. T,K true &kart (i) 40 60 80 100 150 200 250 300 400 500

600 800

loo0

9.25(27) 1.40(15) 4.03(9) 5.48(6) 2.91(3) 1.12(2) 2.07(1) 7.95(0) 3.09(0) 2.02(0) 1.62(0) 1.31(0) 1.18(0)

6.14(33) 7.00(19) 1.13(13) 1.32(9) 1.89(4) 1.85(2) 2.19(1) 7.47(0) 2.86(0) 1.91(0) 1.56(0) 1.28(0) 1.17(0)

7.12(29) 2.70(16) 1.37(10) 4.58(6) 6.11(2) 2.73(1) 7.40(0) 3.84(0) 2.08(0) 1.59(0) 1.38(0) 1.20(0) 1.12(0)

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for

/

(ii)

(iii)

(iv)

2.55(31) 5.15(17) 1.49(11) 3.07(7) 1.65(3) 4.46(1) 9.66(0) 4.52(0) 2.26(0) 1.67(0) 1.42(0) 1.22(0) 1.13(0)

3.61(32) 6.73(18) 1.66(12) 2.78(8) 7.85(3) 1.18(2) 1.74(1) 6.58(0) 2.72(0) 1.86(0) 1.53(0) 1.27(0) 1.16(0)

3.74(30) 1.72(17) 8.93(10) 2.76(7) 2.31(3) 6.34(1) 1.26(1) 5.48(0) 2.51(0) 1.78(0) 1.49(0) 1.25(0) 1.15(0)

a Number in parenthats is the power of 10 by which the preceding number should bc multiplied.

conventional one) if the imaginary frequencies were not present.] From the curvature of the three upper Arrhenius curves in Figure 5, it is clear that the present results predict tunneling to be very important for this proton transfer process for temperatures below 500 K. For example, the CVT/MEPSAG rate constants (which include tunneling along the MEP) are larger than the CVT results (which ignore tunneling) by factors of 7.9, 110, and 5.5 X 106 at 300,200, and 100 K, respectively. In addition, incorporating the curvature of the reaction path in the calculation of the tunneling correction factor (CVT/CD-SCSAG) further increases the results significantly for temperatures below 300 K. For example, the CVT/CD-SCSAG results are larger than the CVT results by factors of 11,250, and 8.1 X lo7 at 300,200, and 100 K, respectively. The major contribution to the enhancement of the CVT/CD-SCSAG results over the CVT/MEPSAG results arises from the region of the MEP around s = 0.3 A, where the character of the reaction-path motion changesfrom proton transfer to hydrocarbon fragment separation. Finally, in Table I11 we compare the CVT/MEPSAG transmission coefficients obtained with the various model potential barriers described in section 2 with those obtained with the true ground-state vibrationally adiabatic potential curve. Since all of the model barriers are narrower than the true adiabatic potential, they all overestimate the transmission coefficient at low temperatures. The effect on the predicted CVT/MEPSAG rate constant is shown for the case of the Eckart potential by the uppermost curve in Figure 5. In addition, although forms (i)(iv) have broader tails than the Eckart potential, they are also much broader than the true adiabatic potential in the saddle point region, leading to significant underestimates of the trans-

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Proton Transfer Dynamics

The Journal of Physical Chemistry, Vol. 97, No. 9, 1993 1769

mission coefficient at moderate to high temperatures. Thus, the two-phase nature of the reaction path prevents a single functional form for the adiabatic barrier from leading to reliable estimates of the CVT/MEPSAG transmission coefficient at both higher and lower temperatures. Of the models tested, the fourth root of the Eckart potential [form (iv)] yields transmissioncoefficients that on the whole appear to be the closest to those obtained from the true potential, but even this choice gives results that are in error by at least an order of magnitude at temperatures below 100 K.

Acknowledgment. The authors are grateful to Bruce Garrett and Donald Truhlar for a portion of the code used to calculate the semiclassical transmission coefficients, and to the latter for helpful suggestions. Some of the calculations reported here were carried out on the Miami University V A X computer, and the computer time is appreciated. Financial support was provided by the National Institutes of Health (GM29391).

4. summary

References and Notes

We have presented here a model study of the energetics and dynamics of the proton transfer process in (HJCH.-CH~)-based on an ab initio potential energy surface. Rate constants were computed by variational transition state theory with semiclassical vibrationally adiabatic ground-state transmission coefficients using a limited set of information about the potential energy surface along the reaction path. This information was obtained directly from ab initio electronic structure calculations of the energy and its first and second derivativeswith respect to Cartesian coordinates. The character of the reaction path was observed to change from proton transfer near the saddle point to hydrocarbon fragment separation as the product equilibrium structure is formed, leading to a broadening of the energy as a function of the reaction coordinate. Below 500 K, reaction-path tunneling was found to be important in this system. In fact, the results indicate that in order to obtain reliable rate constants, one must compute the transmission coefficient using the true vibrationally adiabatic potential along the reaction path and include the effect of reaction-path curvature. One disadvantage of obtaining reaction path information directly from ab initio calculations is that the reaction path must be recomputed for each isotopiccomposition. Thus, the prediction of kinetic isotope effects in the (H3CH4H3)- system would require recalculating the MEP and the properties along it. The use of a smaller gradient step size, which is important for ensuring that the MEP is followed accurately, would also require a recalculation of the entire MEP. In addition, better ab initio calculations (i.e., the use of a bigger basis set and the inclusion of electron correlation) would allow one to make more reliable predictions about the true (H3CH-CH3)- system than can be made from the present model study employinga 4-3 1G basis. For example, in the present calculations the reactant and product equilibrium structures correspond to deep local wells in the vibrationally adiabatic potential separated by an adiabatic barrier. Thus, the calculation of the transmission coefficient, which assumes a Boltzmann distribution of translational states rather than a discrete set of bound levels in the adiabatic wells, is only approximately correct in the present case. However, adiabatic energies obtained with a considerably larger basis6 predict the adiabatic wells of the reactant and product equilibriumstructures

to be much shallower, making the assumptions in the transmission coefficient calculation more correct.

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