J. Phys. Chem. 1994, 98, 12501- 12505
12501
A Classical Trajectory Study of IVR in Dimethyl Peroxide. 2. Dissociation Lifetimes and Comparison with Statistical Calculations F. E. Budenholzer,* M. Y. Chang,? and K. C. Huang’ Department of Chemistry, Fu Jen Catholic University, Hsinchuang 242, Taiwan, R.O.C. Received: June 9, 1994; In Final Form: September 9, 1994@
Average vibrational predissociation lifetimes have been calculated for the dissociation of CH300CH3. Classical trajectories were used to simulate overtone-induced dissociation over the three potential surfaces described in an earlier paper [J. Phys. Chem. 1991, 95, 42131. Comparisons are made with simple RRKM calculations, with parameters adjusted to fit the experimental bulb kinetic results. Qualitative agreement between the trajectory results and RRKM calculated lifetimes, as well as a general inspection of the trajectories, indicates the dissociation process may be described as statistical. A method to prevent the leakage of zero-point energy from the instantaneous normal modes of the molecule during the calculation of trajectories [J. Chem. Phys. 1989, 91, 28631 has been implemented, and the results are discussed.
I. Introduction In an earlier paper,’ subsequently referred to as paper I, we used the classical trajectory method to model intramolecular vibrational energy redistribution (IVR) and overtone-induced dissociation in dimethyl peroxide (DMP, CH300CH3). Trajectories were run over three potential surfaces: (I) a basic surface in which all stretches and bends (except the 0-0 stretch which was described by a Morse oscillator) were described as pure harmonic oscillators, (11) the basic surface with the addition of quadratic cross terms, and (111) the basic surface with all CH stretches replaced by Morse oscillators. Zero-point energy (ZPE) was systematically placed in the normal vibrational modes of the molecule, with the phases of the vibrations sampled randomly. Classical energy, corresponding to a local mode excitation, was placed in a single CH bond and the intramolecular flow of energy monitored. Because of computational limitations, only a few trajectories were run until the dissociation limit, on the order of tens of picoseconds. In this paper, we continue to study the DMP system. We have now been able to run ensembles of trajectories (up to 80 ps) over the three potential surfaces.26 We have also carried out an RRKM calculation of the dissociation lifetimes for comparison with the trajectory results. Finally, the method of Miller et aL2 has been used to prevent the vibrational energies in the instantaneous normal modes of the molecule from dropping below the zero-point energy. The comparison of ensemble averages of trajectories with statistical results allows a more refined understanding of IVR and dissociation processes in DMP. The comparison of the trajectory calculations over the three surfaces and the implementation of the algorithm to control zero-point leakage provide information related to two technical problems involved in classical trajectory calculations-the adequacy of different surfaces and the problem of how to deal with the quantum mechanical zero-point energy in a classical simulation. The detailed background and rationale for this research are given in the introduction of paper I and will not be repeated here. In section 11we describe the method of calculation, briefly reviewing the method of calculation described in paper I and + Present address: Department of Chemistry, National Chung-Cheng University, Chiayi, 621, Taiwan. Present address: Department of Computer Science and Information Engineering, National Chung-Cheng University, Chiayi, 621, Taiwan. Abstract published in Advance ACS Abstracts, November 1, 1994.
*
@
0022-3654/94/2098- 12501$04.5010
giving in more detail the new elements of the calculation. In section I11 we present our results and discuss their significance. 11. Method of Calculation
Potential Energy Surface. The the potential energy surface used in this calculation was based on the force field analysis of Butwill Bell and Laane? which in turn was based on the spectroscopic experiments of C h r i ~ t e . ~Surface I uses the harmonic bending and stretching terms of ref 4 to describe all but the 0-0 bond. The 0-0 bond was described by a Morse oscillator, using the experimentally determined activation energy of DMP? 37 kcal/mol, as the dissociation energy. Surface I1 is the same as surface I except that the quadratic cross terms coupling separate bonds and stretches were added. The constants are also taken from ref 4. Surface I11 is the same as the basic surface, surface I, except that all six harmonic CH stretches are replaced by Morse Oscillators. In all three surfaces the torsions were neglected, leading to three free internal rotors. For the detailed force constants and the equilibrium geometry of the molecule, see paper I and ref 4. Normal-mode frequencies and eigenvectors were calculated for all three surfaces. The small differences for the three surfaces were on the order of the differences between experimen@and the original force field analysis! Subsequent to the publication of paper I, we found an error in our computer code in the designation of the equilibrium configuration; however, the eigenvalues of the 21 active modes were not appreciably affected. (The eigenvalues for the rotational and translational modes were, however, much closer to their true zero values.) Trajectory Calculations. Classical trajectories were calculated in space-fixed Cartesian coordinates using a modified form of the program MERCURY by Hase.’ A zero-point energy of 49.39 kcal/mol was distributed in the molecule by using the “fixed normal-mode energy sampling” option of the program MERCURY. A classical energy E,, corresponding to the local mode overtone v , was then placed in a selected overtone stretch. The initial bond length and momentum, corresponding to E,, were selected randomly. Thus, Et0d = E, f Ezero point. Hamilton’s equations of motion were integrated numerically with a step size between 0.0125 x and 0.00625 x s, giving energy conservation to at least five significant figures. Calculation was continued either until dissociation (defined as 0 1994 American Chemical Society
Budenholzer et al.
12502 J. Phys. Chem., Vol. 98, No. 48, 1994
TABLE 1: Paramete@ for Standard RRKM Calculation of the Dissociation of Dimethyl Peroxide 34.999332b EO (kcdmol) nI
21 3
V
ref 16
nf
I' (amu Az) U
r200/
I (amu A 2 )
5
P
"f
+
n,' V+
tu
2
'
c- 1001
O h ' , ' 20
"
'
/
40 '
'
'
I
Time(p s )
60 '
" 80
Figure 1. Number of unreactive trajectories at time t, plotted as a function of t . A total of 500 trajectories were run over surface I at a total energy of 128 kcdmol. The solid line shows the computational results, with the individual points connected by a solid line. The dashed line is the best fit to eq 1. The predissociation lifetime is 25.5 ps. the 0-0 bond length exceeding 2.5 A) or until the trajectory time exceeded 80 ps for surfaces I and 111or 40 ps for surface 11. The dissociation times were least squares fit to an exponential decay curve
where N is the number of unreacted trajectories at time t and No is the total number of trajectories run. Figure 1 shows a typical fit to eq 1 for 500 trajectories run over surface I. The predissociation lifetime, z, is then defined as the time for the number of unreacted trajectories to fall to l/e of the initial value. There has been some discussion of late on the problem of how to deal with the so-called zero-point energy problem in a classical trajectory simulation. In back-to-back papers, Bowman et aL3 and Miller et d 2proposed a relatively simple way to guarantee that the energy in a given vibrational mode would not drop below a predetermined value, for example, the zeropoint energy. Simply put, the trajectories are integrated in phase space as usual. The energies in the vibrational modes are checked periodically, normally at each step in the numerical integration of the trajectory. If it is found that for a given trajectory the energy in one of the modes drops below the predetermined value, then the associated momentum is changed to its opposite sign while the associated spatial coordinate remains unchanged. Using this procedure, regions in phase space with mode energy below the zero-point energy can be excluded while the total energy of the trajectory is conserved and the trajectory of the spatial coordinates remains continuous. In our initial calculation, we assumed that the initial normal mode picture remained valid during the course of the reaction. (In the words of ref 2, we presumed the so-called "simple version".) At each step the initially calculated matrix of transformation was used to transform both the coordinates and momenta to the normal-mode representation. A check was then made to see whether the energies of any of the normal modes had dropped below the zero-point energy of that mode. If the energy had dropped below the zero-point energy, then pi(t>) was set to -pi@ and t< represent the time just instantaneously greater than and less than the time
2.7181527,2.7181527,8.8884105' 3.0,3.0, 1.0 119.4340, 110.1682, 15.78425 20 3 the same as v except (779 cm-') and COO bend freq decreased by 0.1803 2.9868875,2.9868875, 8.8884105' 3.0, 3.0, 1.0 195.31939, 178.99252,22.84530
I+' (amu AZ) d I+ (amu AZ) Definition of symbols: EO,thereshold energy; nf,the number of reactant molecular frequencies; n,, the number of free internal rotors; v, molecular frequencies; f,reduced moments of inertia of the intemal rotors of the molecule; u, symmetry number of internal rotors of the molecule; I, three overall moments of inertia of the molecule; n:, the number of frequencies of the complex; n,', the number of free intemal rotors of the complex; vf, frequencies of the complex; I+', reduced moments of inertia of the complex; a+,symmetry number of intemal rotors of the complex; I+, overall moments of inertia of the complex. * EO = Eb iZPEA+- ZPEA. E b is the barrier energy, 37.0 kcdmol. ZPEA+is the zero-point energy of the complex, 47.389613 kcdmol. ZPEA is the zero-point energy of the molecule, 49.390281 kcdmol. These are the frst approximation to the reduced moments of inertia, ref 23. of the check and pi and qi are the momentum and coordinate of the ith mode. We calculated a series of trajectories using this simple version. However, the calculation gave the counterintuitive result that constraining zero-point leakage decreased the dissociation lifetimes. Normally, one would expect the opposite, constraining the zero-point energy should make the rate constants smaller, that is, the lifetimes longer. Simply put, the zero-point energy constraint will decrease the amount of energy available for dissociation. Because of these anomalous results, we repeated the calculation using the "more general version" of Miller, Hase, and Darling.2 In this method the instantaneous normal modes are calculated at each point along the trajectory. This is done by diagonalizing the force constant matrix at the actual coordinate positions at time t, yielding eigenvalues {&*) and the corresponding eigenvectors. The modes are then checked to see whether they have dropped below the "instantaneous zero point energy". If for mode k the energy has dropped below the zeropoint value, Pk is replaced by -pk. When this more general version was used, there was a clear increase in the lifetime of the excited molecule, when compared with unconstrained trajectories. This is as would be expected. On reflection, it seems clear that the simple version would yield problematic results. The simple version presumes that the modes of the reacting system remain constant during the dissociation, a fact clearly not true of the C - 0 - 0 bending modes. RRKM Calculation.We also carried out an RRKM calculation on the system.8 We used an extended version of the RRKM program of Hase and Bunker.g The input for the program is given in Table 1. The program makes use of a semiclassical state counting algorithm as described in ref 8. The reaction coordinate is assumed to be identical with the 0-0 bond breaking. The RRKM rate constant depends sensitively on the normalmode frequencies of the reactant and transition state. The calculation of the normal-mode frequencies for the reactant is
Trajectory Study of N R in Dimethyl Peroxide
J. Phys. Chem., Vol. 98, No. 48, 1994 12503
CHART 1: Calculation of RRKM kud
k,(~*)(349.747~,*(~*))e-~*'~~ d~* Q,Q, F = E o - A G E , , 1 + ka(E*)/(k2[Ml)
k .=uN
S"
unit kum
-
rate constant of A product reaction vibrational partition function of A molecule internal rotation partition function of A molecule vibrational and internal rotational energy difference between A and A* threshold energy of the reaction energy constributed from external rotation rate constant of A*(E*,E*+dE*) -Af density of states in A*@*) ideal gas constant R = 1.987216 x temperature of the system rate constant of A* M A M reaction pressure of M
S-1
Q"
none none kcal/mol kcaVmol kcaVmol
Qr
E* Eo" AEm?
S-1
N,*(E*)' R T Kzd [MI
+
description
l/crn-' kcal/(mol K)
K
+
Torr-' s-I Torr
-
+
a EO = Eb ZPEA+- ZPEA. Eb is the barrier energy. ZPEA+is the zero-point energy of A+. ZPEA is the zero-point energy of A. AE,,,= -RT(1.5 - 0.5(c/Il $/I2 c/13)), where I1,12, and 1 3 are the principal moments of inertia of the molecule and and are the principal moments of the complex. This term is neglected in the calculation of Traj-k,d. k,(E*) and N,*(E*) are directly from the output of RRKM program, ref 9. The units are as above. kz is calculated using collision theory. The collision diameter is the weighted average of contributions from self-collisions of DMP (23 Torr) and collisions with methanol (400 Torr). For each molecule ad = (abc)lI3, where a, b, and c are the maximum differences of the equilibrium coordinates on the axes of the principal moments of inertia. For CHsOH, values for bond lengths and bond angles . . are taken from ref 22.
+
c,c,
+
TABLE 2: ComDarison of RRKM Results with Experiment temp (K) 427.8 431.2 436.0 437.0 439.7 440.0 446.7 450.2 453.0
exp k,,i @-')a 0.565 x 0.179 x 1.29 x 10-3 1.49 x 10-3 1.88 x 10-3 1.96 x 10-3 3.51 x 10-3 4.72 10-3 6.20 x 10-3
RRKM kuni(s-')* 0.615 x 0.852 x 1.34 x 10-3 1.47 x 10-3 1.88 x 10-3 1.93 x 10-3 3.53 x 10-3 4.80 x 10-3 6.12 x 10-3
"Experimental values are from ref 10. Total pressure 423 Torr (CH300CH3, 23 Torr; CH30H, 400 Torr). Calculated as described in Chart 1 with values of COO bending frequencies of the complex 0.1803 of the reactant values.
straightforward. However, for the transition state it is less clear. The problem of determining the transition state for a molecule dissociating into two free radicals has been discussed in the literature.24125Rather than carry out an extensive variational transition state calculation, we used a simple semiempirical approach to define the frequencies of the transition state. With the exception of the C - 0 - 0 bending modes, we presumed the frequencies of the transition state to be the same as those of the reactant. The transition state C - 0 - 0 frequencies will clearly go to zero in the dissociating DMP. Thus, we considered various fractions of the reactant C - 0 - 0 bending frequencies for the transition state. A 1954 paper of Takezaki and Takeuchi'O reported the unimolecular rate constants, kuni, for the dissociation of DMP at a total pressure of 423 Torr, at nine points between 428 and 453 K. In order to compare our computational results with experiment, we calculated kuni, as described in Chart 1. We found that, by using transition state bend frequencies 0.1803 of the reactant value (COO asymmetric bend, 55.723 cm-'; COO symmetric bend, 80.789 cm-') and a deactivating gas kinetic collision diameter of 2.16A, a good match to the experimental results was obtained as shown in Table 2. For comparison with the dissociation lifetimes, we considered only the rate constant k,(E*). This rate constant describes the transition of the molecule from the randomly energized state to the critical configuration for the reaction. E* is the nonfixed
TABLE 3: Dissociation Lifetimes of DMP after Local Mode Excitation excitation
(v)
total total reactive lifetime,* deviation' energy (kcaVmo1) trajectories trajectories t (ps) (ps)
10 12 14
112.0 120.0 128.0
Surfaced I (Time Limit 80 ps) 500 330 75.8 426 42.7 500 25.5 500 483
1.4 0.8 1.3
10 12 14
112.0 120.0 128.0
Surface I1 (Time Limit 40 ps) 94 11.6 100 100 100 6.6 100 4.2 100
1.5 0.4 0.6
10 12 14
112.0 120.0 128.0
Surface I11 (Time Limit 80 ps) 500 327 75.8 500 432 39.9 465 27.0 500
2.3 1.3 0.7
10 12 14
112.0 120.0 128.0
RRKM' 26.6 9.4 4.1
" Defined as the number of trajectories dissociating within the given time limit. Dissociation lifetime defined as the time for the number of unreacted trajectories to fall to l/e the original number. One standard deviation in t,in the fit to eq 1. Surface I, harmonic stretches and bends; surface II, same as surface I, with the addition of quadratic cross terms; surface ID,same as surface I, with C-H harmonic stretches replaced by Morse functions. e Input as in Table 1. energy in the degrees of freedom of the energized molecule (E* = Etod - Ezero If the dissociation process is statistical as defined by the RRKM assumptions, then the average lifetime of the energized molecule-defined as the time for l/e of the energized molecules to reach the critical configuration and dissociate-is given as k,(E*)-'.
III. Results and Discussion Table 3 gives the dissociation lifetimes for DMP at the total energies indicated, for both the trajectory calculations and the RRKM calculations. A number of clear trends emerge that were not clear in paper I where only individual trajectories could be examined. First, the predissociation lifetime clearly decreases with an increase in the total vibrational energy.
Budenholzer et al.
12504 .I Phys. . Chem., Vol. 98, No. 48, 1994
Second, the lifetimes for surfaces I and I11 are essentially identical. This is despite the fact that, as shown in paper I, the time constant for the decay of energy from the initially excited CH bond is up to 4 times faster for the harmonic CH stretch than for the Morse stretch. On further reflection, this is not surprising. First, the lifetimes for decay out of the initial bond are on the order of 0.05-0.24 ps. This is a relatively short period in the overall dissociation trajectory. Also, we argued in paper I that the longer decay of the initially activated CH Morse stretch could be explained by the less efficient coupling of the Morse oscillator with the neighboring harmonic bends and stretches. However, the lack of efficiency would also tend to channel energy away from energy sinks in the two methyl groups. Thus, the slowness of the initial decay may be compensated for by a similar slowness of the Morse oscillators absorbing energy within the excited DMP. A third obvious result is that the lifetimes calculated over surface I1 are considerably shorter than those for the other two surfaces. This indicates the importance of the quadratic coupling terms in the potential energy surface. It should be noted that all three surfaces give essentially the same normal-mode frequencies, indicating that the ability of a potential surface to reproduce the normal modes of a molecule is not a sufficient condition to guarantee the adequacy of the surface for dynamical calculations. Table 4 gives results analogous to those of Table 3, but the vibrational energy of the instantaneous normal modes has been constrained not to fall below the zero-point energy. It is clearly seen that the lifetimes calculated for trajectories constraining zero-point energy are 2-4 times larger than for nonconstrained trajectories. Figure 2 graphically represents the results of Tables 3 and 4. The solid lines are a fit to the RRK expression for ka(E) = l / t ,
TABLE 4: Dissociation Lifetimes of DMP after Local Mode Excitationa total excitation energy total reactive lifetime, deviation (v) (kcaymol) trajectories trajectories t (ps) (ps) Surface I (Time Limit 160 ps) 12 120.0 100 48 208.8 7.7 Surface I (Time Limit 80 ps) 14 128.0 100 52 107.8 3.2 Surface 11 (Time Limit 80 ps) 8 102.0 42 26 78.2 5.6 9 107.0 33 29 42.2 4.9 10 112.0 100 94 21.9 3.1 12 120.0 100 99 11.6 0.7 100 100 8.8 1.1 14 128.0 a
The zero-point energy was constrained as described in ref 2.
- k
t
in
SUR 111 A RRKbl (0.1803 reactant COO bend)
0 3 0 &
Q
100
a
ir
SUR I1 (hold ZPE)
110
120
130
Total energy (Kcal/mol)
where Eb is the barrier energy, 37 kcaVmo1, and E is the total energy available for reaction. For the unconstrained trajectories, E is just the total energy; for the constrained trajectories, E = Etod - ED. Table 5 gives the calculated values for s and A. In the simplest model, the constant A is just the frequency of the vibration along the reaction coordinate. For our system this would be the frequency of the 0-0 stretch, approximately 20 ps-'. If we include the three low-frequency torsions (free rotors in our calculation), DMP has 24 vibrational degrees of freedom. Thus, the results for the standard calculation (no constraints on zero-point energy) are in reasonable agreement for this simple model. In order to more directly compare our trajectory results with the gas phase bulb experiments, we used the RRK fit of ka(E) of the ZPE constrained trajectories (last entry of Table 5) to calculate k u ~ .For the nine experimental points, the ratio of experimental kuni to the trajectory kUdvaries from 6.54 to 7.27 with an average of 7.03. Because of the exponential Boltzmann term in the integral kuni, the calculation of the rate constant is very sensitive to the fit of ka(E), especially near the threshold. If Eb in eq 2 is reduced to 33.14 kcal/mol and the resulting best fit of eq 2 used to calculate kuni, the calculated results agree with experiment within about 18%. Such good agreement is obviously somewhat fortuitous given the good number of approximations in the calculation (the potential surface, the classical nature of the calculation, the value for k2, etc.). However, the fact that our trajectory calculations give values of kuni in agreement with the experimental values adds a note of realism to the overall calculation.
Figure 2. Dissociation lifetimes of DMP as a function of total vibrational energy. Trajectories were run over the surfaces as indicated. The RRKh4 values were calculated as explained in the text. The various symbols correspond to the actually calculated values. The solid lines are fits of the calculated values to eq 2. Note the log scale for the dissociation lifetimes. TABLE 5: RRK Parameters, Fit of k. = Yz to Eq 2 surface S A (vs-') deviation" (DS-') I 19.2 19.4 0.20 10-3 II 18.1 83.2 0.97 x 10-3 I11 18.3 14.2 1.03 x 10-3 IIb 4.92 1.45 4.54 x 10-3 One standard deviation in k,, in the fit to eq 2. Hold zero-point energy. A number of conclusions can be drawn from the above results. First, despite the fact that in the trajectory calculations energy is initially deposited in a single bond, the reaction is essentially statistical. Examination of both individual bond energies (Figures 2-5 of ref 1) and normal-mode energies as functions of time indicates that the energy tends to disperse throughout the molecule (on both sides of the 0-0 linkage) before dissociation takes place. In some cases, especially in the normal-mode picture, there is an indication of some energy buildup in the bending modes adjacent to the 0-0 stretch, prior to dissociation. However, the overall impression is one of energy being rapidly transferred between all the various modes of the molecule, with a sudden concentration in the 0-0 stretch, leading to dissociation. In paper I we characterized this sudden dissociation as "impulsive". Similar observations have been made for other systems.*l
Trajectory Study of IVR in Dimethyl Peroxide Second, for both the ZPE constrained and unconstrained calculations, dissociation lifetimes calculated using surface I1 with quadratic bending and stretching cross terms are in better agreement with RRKM results than those calculated using the simple harmonic surface I or the harmonic-Morse surface 111. This is not unexpected. RRKM theory presumes free exchange of energy between oscillators. Such exchange can only be enhanced by adding the cross quadratic terms. Most recent reports of classical trajectory simulations have noted the so-called “zero-point energy problem”. For example, in A BC type calculations, it is sometimes suggested to only count as reactive those trajectories where the (classical) vibrational energy of the product diatomic is greater than or equal to the product zero-point energy.” In studies of IVR in systems of comparable size to DMP, the most common way to deal with the “zero-point energy problem” has been to use varying fractions of the quantum mechanical zero-point energy as the initial c o n d i t i ~ n . ’ ~ -The ’ ~ consensus seems to be that given the other approximations involved in the quasiclassical simulation of IVR (approximate potential surfaces, classical mechanics, etc.) failure to deal with zero-point energy leakage will not effect the qualitative results. On the other hand, if we abstract from all the various approximations involved in classical trajectory simulations of IVR, and simply compare equivalent trajectory calculations with varying amounts of initial zero-point energy, there are obvious quantitative difference^.'^ For example, Lu and Hase15 used classical trajectories to estimate the line width of the C-H overtone bands in benzene. They find large differences depending on the fraction of initial zero-point energy deposited in the C-H local modes and on which normal modes were allowed to contain the zero-point energy. Using fractional initial zero-point energies obviously does not directly deal with the ongoing problem of zero-point energy leakage during the calculation of the trajectory. One solution that has been suggested is a combined semiclassical, quasiclassical method. This method has been applied to cluster systems where there exists a stiff diatomic bond, bound to a soft cluster bond.16 A second method is that developed by Miller and coworkers2and Bowman et aL3 As far as we know, the calculation reported here is the first time Miller’s method has been used in a full scale IVR simulation. In model calculations, Sewell and co-workers17used a power spectral analysis of the classical dynamics to show that the quasiperiodic trajectories in the free trajectory calculation become increasingly chaotic if the Miller-Bowman zero-point energy algorithm is used. Some have also wondered about the stability of calculations using the algorithm.l6 Our experience indicates that if the more general version of the zero-point energy constraining algorithm is used, there are no obvious numerical instabilities. Energy is well conserved throughout the trajectories. We did not, however, carry out a spectral analysis similar to that done by Sewell and co-workers,” and so, beyond the obvious increase in the lifetime of the activated molecule, we cannot comment on any detailed dynamical changes induced by the use of the algorithm. In our calculations the three torsional modes were treated as free rotors. A recent paper by Schranz and co-w~rkers’~ considered the effects of torsion on IVR simulations in HOOH and DMP (where the two methyl groups were modeled as united entities). Their study indicates that torsional modes, especially when in resonance with symmetric bending modes, can be significant channels of IVR. Presumably, if the torsion modes had been fully included, the predissociation lifetimes would have been somewhat shorter.
+
. I . Phys. Chem., Vol. 98, No. 48, 1994 12505
An interesting recent20 result with which to compare our work is the classical simulation of IVR leading to the dissociation of HOOH embedded in a cluster of 13 argon atoms. They find that the lifetime of the HOOH embedded in the Ar cluster decreases significantly as the excitation energy is increased. This is similar to our result and would be expected in any molecule with easily accessible energy sinks that would allow it to be described by an equation like eq 2. In conclusion, DMP dissociation can be described as “statistical” both from the examination of individual IVR trajectories and in the sense that there is a convergence between trajectory results, the RRKM results, and the experimental bulb kinetic values. However, as shown in paper I, this does not mean that the assumption of RRKM theory that all modes are equally accessible and all IVR channels equally important is literally true. Resonances between various modes will favor certain channels; however, the overall result is statistical. With reference to the “zero-point energy problem”, we have demonstrated that it is practical to use Miller’s2 method for a relatively large scale classical simulation, giving results in qualitative agreement with RRKM analysis and experimental rate constants.
Acknowledgment. This research was sponsored by the National Science Council of the Republic of China. Their assistance is gratefully acknowledged. References and Notes (1) Budenholzer, F. E.; Chen, C.; Huang, C. M.; Leong, K. C. J. Phys. Chem. 1991, 95, 4213. (2) Miller, W. H.; Hase, W. L.; Darling, C. L. J. Chem. Phys. 1989, 91, 2868. (3) Bowman, J. M.; Gazdy, B.; Sun, Q. J. Chem. Phys. 1989,91,2859. (4) Butwill Bell, M. E.; Laane, J. Spectrochim. Acta, Part A 1972.28, 2239. (5) Christe, K. 0. Spectrochim. Acta, Part A 1971, 27, 463. (6) Barker, J. R.; Benson, S. W.; Golden, D. M. Znt. J. Chem. Kine?. 1977, 9, 31. (7) Hase, W. L. MERCURY A General Monte Carlo Classical Trajectory Program. QCPE 1983, 3, 453. (8) Robinson, P. J.; Holbrook, K. A. Unimolecular Reactions; Wiley-Interscience: New York, 1972. (9) Hase, W. L.; Bunker, D. L. A General RRKM Program. QCPE, 1983, 11, 453. (10) Takezaki, Y.; Takeuchi, C. J. Chem. Phys. 1954, 22, 1527. (11) Varandas, A. J. C.; Brandao, J.; Pastrana, M. R. J. Chem. Phys. 1992, 96, 5137. (12) Cho, Y . J.; Vandelinde, S. R.; Zhu, L.; Hase, W. L. J. Chem. Phys. 1992,96, 8275. (13) Hase, W. L.; Darling, C. L.; Zhu, L. J. Chem. Phys. 1992.96, 8295. (14) Bintz, K. L.; Thompson, D. L. Chem. Phys. Lett. 1991, 187, 166. (15) Lu, D.; Hase, W. L. J. Chem. Phys. 1990, 91, 7490. (16) Alimi, R.; Garcia-Vela, A.; Gerber, R. B. J. Chem. Phys. 1992, 96, 2034. (17) Sewell, T. D.; Thompson, D. L.; Gezelter, J. D.; Miller, W. H. Chem. Phys. Lett. 1992, 193, 512. (18) Duchovic, R. J.; Hase, W. L. J. Chem. Phys. 1985, 82, 3599. (19) Schranz,H. W.; Collins, M. A. J. Chem. Phys. 1993, 98, 1132. (20) Finney, L. M.; Martens, C . C . J . Phys. Chem. 1992,96, 10626. (21) Yurtsever, E.; Gunay, H.; Uzer, T. J. Chem. Phys. 1993,99, 1135. (22) Lide, D. R., Ed. Handbook of Chemistry and Physics, 73rd ed.; CRC Press: Boca Raton, FL, 1992; p 9-34. (23) Pitzer, K. S. J. Chem. Phys. 1946, 14, 239. (24) Wardlaw, D. M.; Marcus, R. A. J. Chem. Phys. 1985, 83, 3462. (25) Smith, S. C. J. Phys. Chem. 1994, 98, 6496. (26) The original trajectories were run on a DEC VAX 8530. Acquisition of an IBM RISC/6000 workstation, Model 320, allowed the running of ensembles of trajectories.
