10626
J . Phys. Chem. 1992, 96, 10626-10635
(4) Cleveland, C. B.; Jursich, G. M.; Trolier, M.; Wiesenfeld, J. R. J . Chem. Phys. 1987,86, 3253. ( 5 ) Buss, R. J.; Casavecchia, P.; Hirooka, T.; Sibener, S.J.; Lee, Y. T. Chem. Phys. Lett. 1981,82, 386. (6) Fitzcharles, M. S.;Schatz, G. C. J . Phys. Chem. 1986, 90, 3634. ( 7 ) Kuntz, P. J.; Niefer, B. I.; Sloan, J. J. J . Chem. Phys. 1988,88, 3629. (8) Dunne, L. J. Chem. Phys. Lett. 1989, 158, 535. (9) Badenhoop, J. K.; Koizumi, H.; Schatz, G. C. J . Chem. Phys. 1989, 91, 142. (10) Rynefors, K.; Elofson, P. A.; Holmlid, L. Chem. Phys. 1985,100, 53. (1 1) Tsukivama. K.: Katz. B.: Bersohn. R. J . Chem. Phws. 1985.83.2889. (12j T o n o h a , K.;'Matsumi; Y.; Kawasaki, M.; Kasatani, K:J. Chem. Phys. 1991, 95, 5065. (13) Matsumi, Y.; Shafer, N.; Tonokura, K.; Kawasaki, M.; Kim, H. L. J . Chem. Phys. 1991, 95, 4972. (14) Hilber, G.; Largo, A,; Wallenstein, R. J . Opt. SOC.Am. 1987, 8 4 , 1753.
(15) Marinero, E. E.; Rettner, C. T.; &re, R. N.J. Chem. Phys. 1984. 80,4142. (16) Felder, P.; Haas, B. M.; Huber, J. R. Chem. Phys. Lett. 1991,186, 177. (17) Sparks, R. K.; Carlson, L. R.; Shobatake, K.; Kowalnyk, M. L.; Lee, Y. T. J. Chem. Phys. 1980, 72, 1401. (18) Satyapal, S.;Park, J.; Bersohn, R.; Katz, B. J. Chem. Phys. 1989,91, 6873. (19) Shafer, N.; Tonokura, K.; Matsumi, Y.; Tasaki, S.;Kawasaki, M. J . Chem. Phys. 1991, 95, 6218. (20) Park, C. R.; Wicscnfeld, J. R. Chem. Phys. L r t . 1989, 163, 230. (21) Johnston, G. W.; Kornweitz, H.; Schechter, 1.; Persky, A.; Katz, B.; Bersohn, R.;Levine, R. D. J . Chem. Phys. 1991, 94, 2749. (22) Whitlock, P. A.; Muckerman, J. T.; Kroger, P. M. In Potential Energy Surfaces and Dynamic Calculationsfor Chemical Reactions and Molecular Energy Transfer; Truhlar, D. G., Ed.; Plenum: New York, 1981, p 551.
Overtone-Induced Unimoiecuiar Decomposition of Poiyatomic Molecules in Rare Gas Clusters: A Classical Trajectory Study of H202-Ar,3 Lisa M. Finney and Craig C. Martens* Department of Chemistry, University of California, Imine, Imine, California 9271 7 (Received: July 16, 1992; In Final Form: September 24, 1992)
The effects of intermolecular interactions on the dynamics of intramolecular energy transfer and unimolecular dissociation are studied by considering the overtone-induced unimolecular decomposition of a polyatomic molecule embedded in a rare gas cluster. The system studied is H202-Ar13.Classical trajectory calculations are performed on both the isolated molecule and the molecule-cluster complex. The rates and mechanisms of intramolecular energy transfer and molecular decomposition of the complex are investigated and compared with the behavior of isolated H202 Three main mechanisms leading to pronounced differences in the intramolecular dynamics and unimolecular decay rates are identified: vibrational deactivation of the excited molecule, modification of intramolecular vibrational energy redistribution (IVR)pathways by molecule-cluster interactions, and recombination of the nascent OH fragments induced by binding to the cluster and subsequent diffusion on its surface.
I. Introduction The dynamics of intramolecular energy redistribution and unimolecular dissociation of highly excited molecules are currently the subjects of great interest and research activity. Significant progress has been made in recent years in understanding the detailed dynamics of chemical processes, particularly for isolated polyatomic molecules in the gas phase.14 Here, modern experimental methods such as supersonic molecular beam technology and time- and frequency-resolved laser spectroscopy have allowed detailed measurements to be made on molecular systems in the absence of external perturbations. In the simplest cases, the small number of coupled degrees of freedom and the level of detail supplied by experiments allow a close comparison to be made with first-principles theory . Although the study of gas-phase chemical dynamics has led to important advances, much chemistry of practical significance occurs in condensed phases. Here, the chemical dynamics of polyatomic molecules can be strongly influenced by the perturbing effects of the solvent, and the outcome of a chemical reaction in solution is dictated by energy-transfer processes involving both intramolecular and intermolecular coupling. Thermal activation of the reactant species, energy redistribution within reactant molecules, crossing (and recrossing) of the transition state in the presence of solvent perturbations, and energy relaxation from excited product species into the bath are all key steps in the overall chemical process, and all involve energy transfer between the relevant molecular and solvent degrees of freedom.5 In this paper, we investigate the effects of solvent interactions on the dynamics of energy transfer and unimolecular reaction of Author to whom correspondence should be addressed.
polyatomic molecules. We consider the unimolecular dissociation dynamics of hydrogen peroxide resulting from l o c a l i i overtone excitation of an OH bond. The "solvent" is represented by an associated rare gas cluster. Finite clusters provide an intermediate case between isolated molecules and true c o n d d - p h a s e systems. The method of classical trajectory integration is employed to model the unimolecular decomposition of H202embedded in an Ar13 cluster, and, for comparison, the dynamics of the isolated molecule on the same intramolecular potential surface and with the same molecular initial conditions. Ensembles of trajectories are integrated and the unimolecular lifetimes arc determined as a function of the initial OH excitation energy. Individual trajectories are also examined, in order to reveal the dynamical mechanisms at work in determining the overall rates of reaction and the role played by the cluster in modifying the behavior of the isolated hydrogen peroxide molecule. Overtone-induced proctsses in hydrogen peroxide have been the subject of extensive experimental2vWand theoretical'*ls investigation. Crim and co-workers measured the overtone spectra of hydrogen peroxide at room temperature using laser-induced fluorescence and compared the unimolecular rate implied by the overtone line width and productstate distributionswith statistical models.2-6 Butler et al. measured the overtone spectra in a molecular beam, yielding a lower limit of 3.5 ps for the unimolecular lifetime of the v = 6 overtone level.' Scherer et al. used timeresolved photofragment spa%mcopy to study the avertoneinduced decomposition of hydrogen peroxide! Rizzo and *workers have applied double resonance techniques to investigate the intramolecular dynamics and unimolecular decomposition of H2OP9 A number of groups have employed the method of classical trajectory simulation and concepts from nonlinear dynamid6 to investigate the detailed mechanisms of intramolecular energy
0022-3654/92/2096-10626%03.00/0 @ 1992 American Chemical Society
Classical Trajectory Study of H2O2-ArI3 transfer and decomposition of hydrogen peroxide. Sumpter and Thompson studied intramolecular energy transfer in several four-atom systems, including hydrogen peroxide.I0 Thty observed no dissociations resulting from u = 6 excitation on the time scale of their trajectory integrations, but attributed this to their initial condition selection, which did not include zero-point energy. Uzer, Hynes, and Reinhardt investigated the overtone-induced dissociation of H202and HOOD resulting from excitation of the u = 6 OH overtone (including m p o i n t energy in the other Vibrati~nal modes) and identified the important roles played by anharmonicity, kinetic coupling, and nonlinear resonance in the energy-transfer processes.ll They found a unimolecular lifetime on the order of picoseconds. Uzer et al. studied mode specificity in this system by determining the dependence of unimolecular lifetime on the location of the initial excitation.12 Sumpter and Thompson have investigated the influence of overall molecular rotation on the intramolecular dynamics and decomposition rate of H202,and found that adding rotational energy to the system enhanced the rate of both energy transfer and dissociati~n.'~Getino et al. have employed classical trajectory methods to evaluate the sensitivity of intramolecular energy redistribution and unimolecular decay lifetimes on the details of the potential energy surface.14 The lifetime for the u = 6 level was found to be in the range of 20-25 ps for their system, considerably longer than previous estimates. Getino et al. performed a detailed analysis on the effects of the form of the potential energy surface, rotation, and initial condition selection methods on the unimolecular dissociation of this molecule.15 They identified a number of mechanisms governing the dynamics; the coupling of the OH stretch and OOH bend degrees of freedom and the anharmonicities induced by force constant switching functions were found to play particularly important roles. In addition, they noted that the treatment of zero-point energy had a strong effect on the calculated lifetimes. Other related studies have considered vibration-rotation interaction in H20217 and the quasiclassical dynamics of other peroxides.IE These previous studies have been concerned with the dynamics of the isolated H202molecule. Our purpose here is to determine the mechanisms by which a finite rare gas cluster can modify the processes occurring in the gas phase, and to gain insight into the general effects of a solvent on condensed-phase molecular processes. By considering a finite rare gas cluster as our "solvent", we are able to treat the dynamics in detail, while still modeling, at least qualitatively, the essential many-body solvation effects present in condensed systems. The dynamics of clusters is an area of intense current interest, and has been the focus of many recent e ~ p e r i m e n t a l ' ~and ~*~~-~~ studies. A previous study of particular relevance to the present work is the classical trajectory study of intramolecular Vibrational energy redistribution (IVR) in CF3H and CF3H-(H20), by Tardiff et al.,33 which addressed the role of clusters in intramolecular energy transfer. We also note recent work by Li et al. on the overtone-induced decomposition of HOC1 in liquid which is also closely related to the present paper. The organization of this paper is as follows: In section 11, the methods used to calculate the classical dynamics of H202and H202-Ar13unimolecular decomposition are given. We describe the potential energy function and the calculation methodology used, including the generation of ensembles of initial conditions, trajectory integrations, and data analysis. In d o n 111, the results of the calculations are presented. A discussion of the results and a conclusion are given in section IV.
II. Method The dynamics of overtone-induced unimolecular decomposition of H202and H2O2-ArI3are modeled by classical mechanics. In this section, we describe the potential energy surface used and the methods employed to generate initial conditions, numerically integrate classical trajectories, and analyze the results. A Potential hergy Surface. The general form of the potential surface used to model hydrogen peroxide interacting with a cluster of argon atoms is v = v, + v, + v,, (1)
The Journal of Physical Chemistry, Vol. 96, No. 26, 1992 10627 TABLE I: Potential Parameters 88.6 kcal/mol 0.965 A 2.62 A-' 49.6 kcal/mol 1.452 A 2.58 A-I 1.745 rad 0.99 mdyn A/rad2 0.865 kcal/mol 3.453 kcal/mol
A2 A3 €Ar-Ar uAr-Ar %Ar UC+Ar CH-A~
aH-Ar
A
B
3.206 kcal/mol -0.488 kcal/mol 0.2379 kcal/mol 3.410 A 0.1706 kcal/mol 3.180 A 0.0638 kcal/mol 3.110 A 0.1205 A-S 1.58
Here, V, is the intramolecular potential surface of the isolated hydrogen peroxide molecule, V, is the total Ar-Ar interaction V is the interaction between hydrogen peroxide potential, and , and the argon atoms of the cluster. For studies of isolated H202 dynamics, only the first term V,,, is considered. V, and V,, are approximated by two-body interactions and are represented by pairwise potentials of the Lennard-Jones (12,6) form:35*36 Ns
N,-l
i= I j=i+ I
and
where N, = 13 and N , = 4 are the number of cluster and molecule atoms, respectively. The well depths e, and cij and the hard-sphere radii a, and uijused in the calculations are given in Table I. These values were determined using standard combination rules and the numerical values given in ref 36. A number of previous classical trajectory studies have been performed on models of hydrogen p e r o ~ i d e . ~ Several ~ ' ~ J ~potential energy surfaces have been employed and the sensitivity to variations in the potential energy surfaces in~estigated.'~.'~ We use the surface developed by Getino et al.I5 This potential is "quasiseparable", in that the only potential coupling occurs through the dependence of the bend force constants and the torsional barrier on bond extensions. This simple potential facilitates detailed analysis of mode-mode intramolecular energy transfer, as the definition of mode energies is straightforward. The form of the intramolecular potential is given by a sum of three terms, vm
=
vstrctch
+ vbmd + version
(4)
corresponding to two-body stretch, threebody bend, and four-body torsional interactions, respectively. The OH and 00 stretches are modeled by Morse potentials:
where the sum is over the first OH bond, the 00 bond, and the second OH bond of the peroxide molecule. Here, Di are the bond dissociation energies, ri are the instantaneous bond lengths, rr are the equilibrium bond lengths, and the Pi are potential scale factors. The bend potential function is a sum over two three-body interactions: 2
v b m d
k
= c s i ( r i ) si+l(ri+l)T(ei- e?)2 is 1
(6)
Here, Oi are the angles between the bonds defining the bend and 0; are the equilibrium bend angles. For bond lengths fixed at
equilibrium, the bends are taken to be harmonic, with a force constant k. These force constants are modified by switching functions Si(r),which allow the attenuation of the bend frequencies
10628 The Journal of Physical Chemistry, Vol. 96, No. 26, 1992 TABLE II: Normal Mode Freauenciea mode w (cm-I) mode OH w m str 3883 OOH asym bend OH isym str 3882 00 str OOH sym bend 1474 torsion ~~
~~~
~
w (cm-I)
1402 937 406
as bonds are stretched and, ultimately, broken. For the OH stretches, S(r) is given byIs
{
1
S,(R) = S3(R)=
r r&
while the switching function for the 00 bond is15 Sz(r) = 1 - tanh[A(r - rooe)5]
(8)
The torsional potential is expressed as a trigonometric series:15
where is the torsional angle. As in the case of Vw, the torsional potential is attenuated by switching functions as bonds in the molecule are broken. The angles 8 . and 4 are determined using standard geometrical appro ache^.^^,^^ Numerical values for the molecular potential parameters are given in Table I. The normal mode frequencies of the H202potential surface15are given in Table 11. B. Initial Conditions. In all trajectory calculations, we used classical thermal initial conditions for all modes except the initially excited OH stretch. The following procedure was used to generate the initial conditions: 1. The atoms of the system (molecule or molecule-cluster complex) were placed at a potential minimum. For the isolated molecule, this was the global potential minimum, with all internal coordinates taking on their equilibrium values. For the molecule-ciuster complex, kinetic energy quenches of trajectories were performed to find local "aof the system. The lowest potential geometry found consisted of the H202molecule at the center of the Ar13cluster, and was chosen as the initial configuration for all calculations reported here. 2. An ensemble of trajectories was generated, with all atomic positions at the minimum of (l), and with random velocities chosen from a Maxwellian distribution3' with temperature T = 10 K. The velocities were then adjusted to give zero total linear and angular momenta. Each member of the ensemble was then integrated in time for 120 ps and the locally time-averaged kinetic energy calculated. This "temperature" deviated from the desired initial temperature of 10 K due to energy flow from kinetic to potential. The velocities of the particles were periodically rtscaled to give an ensemble of initial trajectories with locally timeaveraged kinetic energies close to a value corresponding to a temperature of T = 10 K. Each ensemble consisted of 200-300 individual trajectories. 3. An initial excitation was placed in one of the OH bonds of each member of the ensemble. A bond was selected randomly, and then the relative momentum prwas adjusted to give a local mode energy equal to the corresponding quantum overtone state: PI
= [211E(u)l"2
(10)
where p is the 0-H reduced mass and E(u) is the classical energy corresponding to the local mode state with quantum number u. 4. The initial conditions were saved for subsequent integration of trajectory ensembles. It should be noted that zero-point energy was not included in the initially unexcited intramolecular degrees of freedom. In classical simulations, zero-point energy can redistribute and induce spurious dynamical processes, and this is particularly important when weak van der Waals bonds are present. A number of authors have proposed methods for treating the zero-point-energy problem;38J9we have chosen to neglect zero-point energy and treat
Finney and Martens all processes as purely classical. For a given initial overtone excitation, the absence of zero-point energy in the rest of the vibrational modes results in a molecule with total energy significantly less than the case where it is included. Thus, the unimolecular rate calculated here will be less, for a given u, than the result of calculations performed on the same potential surface which include the zero-point energy. Classical trajectory calculations performed on this surface with zero-point energy included are reported in ref 15, and are in general qualitative agreement with experiment. In the work reported here, our goal is not to reproduce previous results, or to obtain an absolute value for the unimolecular rate, but to assess the differences in behavior between the dynamics of the isolated molecule and the molecule solvated by rare gas atoms for a model system. We thus perform calculations on our purely classical models of both the cluster and the isolated molecule using the same thermal initial condition strategy in order to make meaningful qualitative comparisons. C. Trajectory Integrations. Hamilton's equations were integrated for 33 ps using the initial conditions generated as described above. The sixth-order hybrid Gear routine was used,4owhich is particularly stable for stiff equations involving a wide range of intrinsic time scales. The accuracy of these long integrations was monitored by checking energy conservation and the conservation of total linear and angular momenta. The integrations were performed in Cartesian coordinates. D. Andy& of Trajectory Data. The evolving trajectories were analyzed to determine the instantaneous partitioning of energy between the modes of the molecule and the cluster. The energies of the stretch, bend, and torsional modes was determined using the method of Wilson, as described in ref 15. The dynamical C matrix was calculated as a function of time, and the diagonal elements were used along with the single mode potential functions to calculate the mode energies: where r(r) and p(t) are the instantaneous values of the internal coordinates and conjugate momenta, respectively, and 5 is the term in the potential corresponding to modej. For the bend and torsion, the stretch-dependent switching functions are included in the definition of 5. This method neglects the kinetic cross terms coupling internal degrees of freedom. These contributions to the total energy are relatively small for individual traject~ries,'~ and oscillaterapidly about zero, so that they are of even less importance when ensemble averaging is performed. We have verified that the sum of the mode energies accurately approximates the total molecular energy for isolated H202nonrotating trajectories, and that this quantity is approximately conserved as the isolated molecule dissociates. The time histories of these energies were written out for each member of the ensemble, in addition to their ensemble-averaged values. The times conesponding to 00 bond breakage and, in the case of recombination, 00 bond reformation were also determined. Unimolecular decomposition was detected by monitoring the 00 bond distance as a function of time and using a critical extension of 11 A as a condition for dissociation. The actual time of dissociation was taken to be the time of the previous inner turning point of the 00 bond. We also employed an energy criterion, where dinsociation was taken at the time when the 00bond energy exceeded D,; recombination occurred when this mode energy dropped below D,. Both methods gave equivalent results for dissociation. The energy criterion was more convenient for detecting recombination events and was used in the results presented below. III. Results In Figure 1, we show the dependence of the unimolecular lifetimes of H202on u, the vibrational quantum number of the initially excited OH overtone. The results for both the isolated molecule and the molecultcluster complex are given, and they are labeled H202and H202-Ari3,respectively. Also shown is a curye labeled the signihnce of this quantity will be discussed
The Journal of Physical Chemistry, Vol. 96, No. 26, 1992 10629
Classical Trajectory Study of H2O2-ArI3
30000
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Figure 1. Dependence of the unimolecular lifetime of H202 on u, the vibrational quantum number of the initially excited OH stretch. Results for isolated molecule and molecule embedded in an Ari3 cluster are labeled 'H202.) and "H2O2-ArI3", respectively. A third curve, labeled q,is also shown, which gives the lifetime of moleculecluster complexes ignoring recombination events. 2.600 1 O4 -. z
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time (psec) Figure 2. Vibrational deactivation of hydrogen peroxide by the argon cluster. The curves shown are ensemble-averaged vibrational energies of H202embedded in an Ar13cluster, for u = 6-9.
later in this section. As the figure indicates, the lifetimes decrease as the initial OH energy is increased. In addition, the unimolecular lifetimes of H202in Ar13are longer than the lifetimes of the isolated molecule with the same initial excitation. For an initial OH excitation corresponding to u = 6 and 10 K thermal energy in the remaining degrees of freedom, the lifetime of the isolated species is 100 ps. In the case of the molecule-cluster complex, however, no trajectories dissociate during the 33-ps integration time, giving an effectively infinite lifetime. As initial excitation energy increases, both the isolated molecule and molecultcluster lifetimes decrease and they become comparable for u = 9 and above. As we mentioned in the previous section, the lifetime of the isolated molecule determined here is longer than previous calculations which included zerepoint energy in the classical initial condition~,l'-~~ but is qualitatively consistent with the observations of ref 10, where no zero-point energy was included. From these results, it is clear that the presence of the argon solvation shell has a significant effect on the unimolecular dissociation of H202. Although the individual trajectories of this system display a wide range of complex behavior, it is possible to identify three major effects of the Ar13cluster on the dynamics of hydrogen peroxide. We now describe each of these effects in turn. A. Vibrational Deactivationof H202. For isolated hydrogen peroxide, the total energy and angular momentum of the molecule are conserved, due to the absence of external forces and torques on the system. For the molecultcluster complex, however, interactions with the argon atoms allow both the molecular energy and angular momentum to change. Two new energy-transfer pathways are created: (1) relaxation of vibrational and rotational energy from the molecule into the cluster degrees of freedom and (2) intramolecular vibration-rotation energy transfer mediated by collisionswith cluster atoms. Both of these processes contribute
.
i
rotational
translational 0
' 'I
35
vibrational
j
4
-
2 1
dissociation lhreshold
5
10
15
20
25
30
35
time (psec) Figure 4. Partitioning of molecular energy between translation, rotation, and vibration, for the nondissociative trajectory shown in Figure 3. See the text for details.
to the decrease in unimolecular decomposition rates in the H202-Ar complexes. Figure 2 shows the average vibrational energy of HzOz as a function of time for the u = 6 to u = 9 moleculeduster ensembles. The internal vibrational energy is computed by summing the mode energies defined in eq 11. In each case, the vibrational energy of the ensemble decays to an asymptotic value; for the isolated molecule, this quantity is (on the average) conserved. The initial rate of relaxation increases with increasing initial OH excitation energy. Also indicated on the figure is the threshold for unimolecular decomposition of hydrogen peroxide." The internal energy of the D = 6 ensemble drops below the dissociation energy at approximately 23 ps, consistent with the absence of dissociation events for the molecultcluster system. The possible destinations for this energy transfer are molecular translation, molecular rotation, and the Ar degrees of freedom. To gain further insight into the details of this energy transfer, we have examined individual members of the trajectory ensembles. In Figure 3, a representative nondissociative trajectory from the u = 7 ensemble is displayed. The three labeled curves correspond to the total molecular (kinetic and potential) energy, the total van der Waals cluster energy (including argon-argon and argonmolecule interactions), and the total energy of the system, which is a constant of the motion. Over the cowseof the 33-p trajectory integration, the molecule gradually loses approximately 3500 cm-l of energy, which is transferred to the van der Waals modes of the system. Relaxation of energy from the molecular system to the cluster degrees of freedom occurs and will decrease the rate of unimolecular decomposition. The total energy of the molecule, however, remains above the dissociation threshold of 17 340 cm-I throughout the simulation for this particular trajectory. Figure 3 indicates that the total energy decrease of the H202 molecule over the course of the trajectory integration is not sufficient to deactivate the system and prevent decomposition. In Figure 4, we examine in more detail the partitioning of energy
10630 The Journal of Physical Chemistry, Vol. 96, No. 26, 1992
Finney and Martens 2.5001041
--
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1
0
5
20 25 30 3 5 0 5 10 1 5 20 2 5 30 3 5 time (psec) time (used Figure 5. Vibrational mode energies of undissociative trajectory shown in Figures 3 and 4. Also shown (on the same plot as the torsional energy) is the total vibrational energy. See the text for discussion. 0
10
15
between the translational, rotational, and vibrational degrees of freedom of H202. The translational energy was calculated by determining the kinetic energy of molecular center of mass motion. The vibrational energy is defined as the sum of single mode energies, given in eq 11. For a single trajectory, this quantity exhibits high-frequency oscillations due to the neglect of off-diagonal C-matrix elements; these have been eliminated by smoothing the data. The rotational energy was calculated by subtracting translational and vibrational energies from the total molecular energy. At the beginning of the simulation, and throughout the first 23 ps, virtually all of the energy in the molecule is localized in the vibrational degrees of freedom. This energy slowly decreases due to vibrational relaxation into the cluster. At approximately 23 p, a rapid transfer of around 4000 cm-' of energy from vibration to rotation OCCUR.Thus, although the total energy of the molecule remains sufficient to cause dissociation, the available internal energy for unimolecular decomposition has been decreased below the dissociation threshold, resulting in a deactivation of this particular trajectory. Recent analysis of the effect of rotation on statistical reaction rates4z43predicts that, for a faed total energy, the rate of decomposition is expected to decrease with increasing J." Thus, the transfer of energy from vibration to rotation made possible by the argon cluster will increase the observed unimolecular lifetime of the moleculecluster complex. In Figure 5, we show individual mode energies for this trajectory as a function of time. It can be seen that extensive energy transfer between the stretch and bend degrees of freedom occurs during
the early portion of the integration. As Figures 3 and 4 show, however, very little energy transfer between vibration and rotation or vibration and cluster degrees of freedom occurs until t = 23 ps. The sudden transfer of energy out of molecular vibrations at this time is preceded by a large increase in the energy content of the torsional mode of the molecule. This torsional energy fluctuation then decays rapidly. The profile of this decay closely matches the decrease of the total vibrational energy of the molecule, which is shown on the same plot as the torsional energy, and is mirrored by the corresponding increase of the rotational energy. The torsional energy has been transferred into overall molecular rotation. This process is "catalyzed" by a collision of the twisting H202 with an argon atom. Because of its large amplitude and low frequency, the torsional mode is particularly susceptible to energy exchange with rotation resulting from collisions of a localized part of the molecule with a cluster atom. B. MOdiiicatiOa of I n t r a d d r J?nergy RedisMbUtion. The previous example illustrates a mechanism by which the argon cluster can enhance rapid energy transfer between molecular rotation and vibration, while leaving the total molecular energy relatively constant. The Ar cluster can also modify the energy transfer between vibrational degrees of freedom by perturbing the character of the intramolecular dynamics. To examine this effect without the complication of competing unimolecular dissociation, we first consider ensembles of trajectories for an initial excitation corresponding to u = 5 , which, for our model, does not result in any decomposition events for either the isolated molecule or molecule-cluster ensembles.
8000
F'
7000
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"
'
I
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'
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5000
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4000
4
3000
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1
The Journal of Physical Chemistry, Vol. 96, No. 26, 1992 10631
Classical Trajectory Study of H2O2-Arl3
B1
i
h
5000 '8 g 4000
e
PD
3000
2000
2000
1000
1000
OH 2
0
0 0
5
10
15
20
25
30 35
40
0
10
5
Figure 6. Ensemble-averaged mode energies for the isolated molecule with initial excitation corresponding to u = 5. The labeling is as follows: OH 1 is the initially excited OH bond energy, OH 2 is the other OH bond energy, B 1 is the energy of the bend mode adjacent to the initially excited OH stretch, B 2 is the energy of the distant bend, and TOR is the torsional energy. The vertical position of these labels indicate the general ordering of the mode energy curves.
In Figure 6, we show the ensemble-averagedmode energies for the isolated molecule. The curves are labeled as follows: OH 1 is the energy of the initially excited OH stretch, OH 2 is the energy of the initially unexcited OH stretch, B 1 is the energy of the bend adjacent to the initial excitation, B 2 is the energy of the distant bend, 00 is the energy of the 00 bond, and TOR is the torsional energy. At t = 0, the OH 1 energy is approximately 18OOO cm-l, while the rest of the modes contain an excitation corresponding to a temperature of 10 K. (Note that the scale of the plot has been chosen to emphasize lower energies, and so the initial OH 1 excitation is off-scale). As the ensemble evolves, the initial local mode excitation relaxes into the other degrees of freedom. Asymptotically, energy transfer is expected to result in equipartition of energy, with each of the vibrational modes having, when averaged over the ensemble, roughly 1/6 of the initial excitation, or 3000 cm-l. As indicated by the figure, this redistribution is not complete by the end of the integration at 33 ps. The first step in the energy relaxation of the excited OH is energy transfer into the B 1 mode, as observed previously in classical trajectory studies of this system.'*15 Large offdiagonal Gmatrix elements strongly couple a 2:1 rmnance between these two modes. However, the resonance is detuned for u = 5 , and thus the energy flow is not as rapid as for higher values of v. The curves B 1 and B 2 in Figure 6 are nearly superimposable. This indicates that energy is rapidly equilibrated between B 1 and B 2 after leaving the OH 1 bond. The 00 bond lags somewhat behind the bends in excitation, but by around 20 ps, an equipartition of available energy (Le., energy not still trapped in OH 1 or in the other degrees of freedom) between B 1, B 2, and 00 is established. The remaining degrees of freedom, OH 2 and TOR, which are not strongly coupled to the other modes, do not gain energy as rapidly. Thus, for the isolated molecule, energy which has left the initially excited OH bond can redistribute over the bends and the 00 stretch. In Figure 7, we show the results obtained for the moleculecluster complex. The energy scale and labeling are the same as in Figure 6. This plot illustrates a number of differences between intramolecular energy transfer in the isolated and solvated molecules. First, the OH 1 relaxation rate is enhanced for u = 5 by interaction with the cluster. The large amplitude motion of the initially excited bond results in collisions between the OH hydrogen atom and the argon cluster, which induce energy transfer from the OH 1 stretch into the bend mode B 1. The intermolecular interactions, in effect, causes the OH periodic orbit to become unstable, and accelerates energy flow into the rest of the molecule. A second important difference between the H 2 0 2and Hz02-Ar13 systems is shown in Figure 7. Equipartition of available energy between B 1, B 2, and 00 is not complete on the time scale of the trajectory integration. Initially, energy flows from OH 1 into B 1, just as in the isolated molecule system. The mode
20 25
15
time (psec)
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40
time (psec) Figure 7. Ensemble-averaged mode energies for the molecule-cluster complex with initial excitation corresponding to u = 5. Labeling is as described for Figure 6. 200001
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B 1, however, then remains "hotter" than the other bend. Subsequent energy redistribution from B 1 into B 2 is evidently impeded by the argon shell surrounding the molecule. Interaction between the H202and cluster cage inhibits large amplitude motion of the encased molecule and interferes with the equipartition of energy between B 1, B 2, and 00. Energy transfer into the torsional mode is strongly decreased by the same mechanism. This example illustrates that the details of intramolecularenergy transfer in HzOzare modified by the presence of an argon atom "solvation shell". For u = 5 , two competing effects are at work enhancement of energy transfer out of the initially excited OH bond by the cluster and inhibition of redistribution by the cluster. The net effect of these two effects balance somewhat, and the final 00 energy is comparable in both Figures 6 and 7. For higher values of u, however, the initial relaxation of the OH is fast in the isolated system, due to the 2:l stretch-bend resonance. H-Ar collisions thus do not accelerate the initial step of relaxation, and the cluster effect of restricting IVR becomes the dominant difference between the isolated and clustered molecules. In Figure 8, the OH 1 energies for both molecule and molecule-cluster v = 6 ensembles are shown. The plot demonstrates that the relaxation of the OH bond is qualitatively similar in these two cases. Figures 9 and 10 show the corresponding B 1, B 2, and 00 ensemble-averaged energies for the molecule and molecule-cluster, respectively. Here, dissociation causes the isolated molecule 00 energy to increase with time; the molecule-cluster ensemble does not show any dissociation events. Again, a lack of energy equilibration between the bend degrees of freedom is observed for H202-Ar,3. The modification of IVR by the cluster on the overall dissociation rate is a somewhat subtle effect. On one hand, a symmetric bending motion of the molecule, involving in-phase oscillations of B 1 and B 2, is strongly coupled to the reaction coordinate.l*Is Thus, a dynamical effect which interferes with free exchange of energy between B 1 and B 2 can be expected to inhibit activation of the 00 stretch. On the other hand, Figure 10 shows that,
10632 The Journal of Physical Chemistry, Vol. 96, No. 26, 1992 10000
L
1
8000
8
wF;
Finney and Martens
4000
2000
B2
0
5
10
15
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30 3 5
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4000 W
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0 0
5
10
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20 25
30
35
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time (psec) Figure 10. Ensemble-averaged bend and 00 stretch energies for the molecule-cluster complex with initial excitation corresponding to u = 6. Labeling is as in Figure 6.
although B 2 excitation is suppressed, the energy in 00 and B 1 have equilibrated. A given amount of available energy is being shared among fewer effective degrees of freedom, and thus activation of the reaction coordinate would be expected to be enhanced by the restriction of IVR by the cluster. A third consideration, which is not illustrated by these figures, is the cluster perturbation of energy transfer from the bends into the 00 stretch. On the rapid time scale of molecular vibrations, the solvation shell of argon atoms acts as a hard-walled, static cage. Excitation of the reaction coordinate is accompanied by a fast extension of the 00 bond, which can lead to energetically unfavorable repulsive interactions between 0 and H atoms and the argon cluster. This can be expected to inhibit the final step of energy transfer into the 00 bond, irrespective of how bend-bend redistribution has been affected, and thus decrease the overall rate of unimolecular decomposition. C. Recombination. We now consider a third mechanism for increasing the observed lifetime of H202-Ar13. The previous examples focused on trajectories which did not dissociate, due to deactivation of the vibrational degrees of freedom or changes in intramolecular energy transfer pathways from initial excitation to reaction coordinate. For u I7, however, the lifetime of the molecule in the cluster predicted by our model is finite, and a significant fraction of trajectories in an ensemble experience 00 bond fission. Here, the overall observed rate of decomposition of H 2 0 2associated with an argon cluster can be decreased by subsequent recombination events. In Figure 11, four frames of a typical trajectory from the u = 7 ensemble are shown. Figure 1l a displays the initial geometry of the H202-Ar13complex. The H 2 0 2is located at the center of the cluster, with the 13 Ar atoms forming a closed “solvation shell” around molecule. In Figure 11b we show the instantaneous configuration of the system a t = 5.8 ps. The 00 bond of the molecule has broken, yielding two OH fragments. The energy released by the dissociation has caused the argon shell to open,
allowing the OH fragments to move apart. Figure 1IC displays the system after 14.5 ps of trajectory integration. Here, the OH fragments are bound to the cluster by van der Waals interactions and are “diffusing” over the surface of the partially intact argon cluster. Finally, in Figure 1Id, the configuration at t = 21.4 ps is shown. The OH fragments have migrated over the surface of the cluster and recombined to give an intact H202molecule. The energy given off by the 00 bond formation ejects the molecule from the cluster at later times (not shown). In Figure 12, we show the vibrational mode energies for this trajectory. Initially, energy is transferred from the excited OH 1 bond into the bend degrees of freedom. Energy then gradually flows into the 00 reaction coordinate. Just after t = 5 ps, the 00 energy increases above the dissociation threshold, leading to decomposition of the molecule. The bend energies drop to almost zero (a small amount of diatomic OH rotation is present), and the OH energies become virtually constant. At approximately 20 ps, the 00 energy falls back below the dissociation energy. This corresponds to recombination of the OH fragments to give the intact molecule. Energy transfer among the degrees of freedom then continues. In particular, it should be noted that recombination results in a significant energy flow into the OH 2 bond, which for nondissociative trajectories is a slow process. Figure 13 shows the total vibrational energy for this trajectory. Initially, the energy fluctuates around a value somewhat above the dissociation threshold of 17 340 cm-l (the fluctuations are due to neglect of the off-diagonal G-matrix elements). At 5 ps, the fluctuations decrease as the 00 bond breaks, leaving the fragments with total internal energy above the dissociation value. However, as recombination occurs at t = 20 ps, the vibrational energy drops sharply below the dissociation threshold. Recombination of this particular trajectory results in a deactivated molecule, and thus in this case the overall process of dissociation and recombination is equivalent to vibrational relaxation of H202, leading to a nonreactive system. Other trajectories from this ensemble show behavior that is similar to this example and also display direct recombination, where the 00 bond breaks but is re-formed before the fragments can leave their initial cage. The unimolecular lifetimes shown in Figure 1 for the moleculecluster complex, labeled “H202-h13”, are calculated for the overall dissociation reaction and do not count trajectories that dissociate and subsequently recombine as reactive. The curve
The Journal of Physical Chemistry, Vol. 96, No. 26, 1992 10633
Classical Trajectory Study of H202-Ar,,
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labeled T , shows the rate for thefirst fmion of the 00 bond. Here, a trajectory is counted as reactive when the 00 bond breaks initially, irrespective of whether it later recombines. This cuwe gives a measure of the change in unimolecular lifetime due to e f f m beyond those resulting from the recombination mechanism. As the energy content of the molecule increases, the dissociation event is expected to yield higher kinetic energy fragments, leading to a decrease in the efficiency of recombination. In Figure 14, we give the probability of recombination as a function of the initial vibrational excitation of OH. For u = 6,no net dissociations occur, and the recombination probability is unity. As u increases, this probability drops sharply. After the initial drop from u = 6 to u = 7, the decrease is approximately linear with u.
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IV. Discussion and Conclusion In this paper, we have investigated the effect of intermolecular interactions on the intramolecular dynamics and unimolecular decomposition of a polyatomic molecule embedded in a rare gas cluster. In particular, we considered a classical model of the system H,O,-Ar,, and studied the overtoneinduced unimolecular reaction resulting from initial excitation of an OH bond. For a given level of excitation, the unimolecular dissociation rate of the complex is less than that of the isolated molecule. The dynamics of this system involve many coupled degrees of freedom, and a detailed analysis and understanding presents significant challenges. Nonetheless, it is possible to gain a simplified and qualitative view of the major processes, and we have
10634 The Journal of Physical Chemistry, Vol. 96, No. 26, 1992
identified three general mechanisms at work in the system which influence the intramolecular dynamics, energy relaxation, and unimolecular lifetimes. The first process described was deactivation of molecular vibrations by molecule-cluster interactions. The rate of unimolecular reaction depends strongly on the amount of available energy in the vibrational degrees of freedom; if the total vibrational energy is less than the dissociation threshold, then (neglecting quantum effects such as tunneling) the decomposition reaction cannot occur. In the case of an isolated molecule, conservation of total energy and angular momentum restrict the energy flow out of the vibrational modes, and thus a molecule with an initial vibrational excitation sufficient to cause decomposition will eventually dissociate. The presence of a "third body"-here, in the form of an associated rare gas cluster-relaxes the constraints of energy and angular momentum conservation. The initially excited molecule can interact with the cluster atoms, and the total molecular energy can decrease by vibrational or rotational relaxation, leading to an attenuation of the dissociation rate. Cluster-molecule interactions also break the conservation of the molecular angular momentum. Large-amplitude vibrational motion (particularly of the torsional mode) can cause impulsive collisions between localized portions of the molecule and cluster atoms. Energy transfer from vibrational to rotational degrees of freedom results, with the creation of rotational excitation at the expense of vibrational energy. For a fxed total energy, statistical unimolecular rate constants are expected to decrease with increasing rotational angular Energy transfer from vibration to rotation lowers the effective energy available for unimolecular decomposition and can permanently deactivate given trajectories. The second effect we described was the modification of intramolecular vibrational energy redistribution by the cluster. Previous classical mechanical studies of energy transfer in H z 0 2 have revealed the detailed nonlinear dynamical mechanisms governing the pathways of energy flow from initial site of excitation to the reaction coordinate. In particular, strong resonance coupling between the excited OH stretch and the adjacent local bend mode has been found to be responsible for the initial step of energy transfer in the system. The rate of relaxation depends on the frequency ratio between the anharmonic stretch and bend motions, and energy transfer is enhanced when the ratio is close to 2:1, allowing a prominent off-diagonal kinetic coupling term to induce facile energy transfer.'*I5 If the resonance is detuned, the initially excited mode can exhibit a long lifetime, especially if the initial conditions are near the periodic orbit corresponding to pure local mode excitation, as in the present case, where zero-point energy is not included. This behavior has been observed previously in ref 15 for Hz02. For the complex, though, interactions between the molecule and the cluster can enhance energy transfer out of the initially excited mode. Collisions of the hydrogen atom with nearby argon atoms result in energy transfer into the adjacent bend mode. In effect, the stability of the periodic OH stretch orbit has been changed by the presence of the cluster. We observe this effect for the u = 5 ensemble. For levels of excitation where the resonance condition is closely met and the initial OH relaxation is rapid in the isolated molecule, the presence of the cluster does not cause important changes in this initial step of the energy transfer. This was illustrated in Figure 8 for the u = 6 ensemble. The cluster also has an effect on the intramolecular energy redistribution that follows the initial OH relaxation. The crowding of the molecule by the "solvent cage" of the cluster inhibits excitation of modes which require large-amplitude movement of H or 0 atoms. In particular, the rapid sharing of energy between the bend modes of isolated Hz02 does not occur in the molecultcluster complex. The overall effect of this trapping of energy in a subset of the available molecular phase space on the unimolecular decomposition rate is subtle, and it depends on the particular dynamical mechanism by which energy is transferred into the reaction coordinate in the isolated molecule. Previous studies on the classical dynamics of H z 0 2have shown that these mechanisms depend sensitively on the form of the potential energy s ~ r f a c e , ~with ' ~ ~ some ~ * ' ~models suggesting that an in-phase inward
Finney and Martens bending of the H atoms is an important step in 00 bond activation" while other systems show different energy-transfer pathways.I4J5Our results suggest that cluster restriction of IVR may actually enhance energy flow into the 00, as Figure 10 indicates that B 1 and 00 can equilibrate their energies, even if B 2 lags behind due to the effect of the cluster. However, large excitations of the 00 bond leading to dissociation are evidently inhibited by the hard repulsive potentials of the slow and heavy argon atoms, which cause rapid bond extensions to be energetically unfavorable. A third mechanism by which the cluster affects the unimolecular decay dynamics of hydrogen peroxide is cluster-induced recombination of the nascent OH fragments. In some cases, the molecule experiences direct recombination without leaving its initial solvent cage. Alternatively, the OH fragments can wander around on surface of cluster remnant until they encounter each other, as illustrated in Figures 11-13. This process involves a cluster analogue of surface diffusion, and the probability of recombination decreases with the initial excitation energy. The model adopted here for H 2 0 2dynamics is purely classical, in that both the initial condition generation and the subsequent time evolution are based on classical mechanics. The results thus provide insight into classical mechanical mechanisms of energy transfer and relaxation. Although such insight can provide qualitative understanding of molecular processes, a quantum mechanical treatment will be required to obtain quantitative results. Unfortunately, the computational difficulty of an exact quantum calculation for a system with many heavy atoms such as this is prohibitive. A compromise would be to treat the intramolecular dynamics using approximatequantum methods, while retaining a classical trajectory representation of the argon motion. Methods which combine both classical and quantum mechanical aspects in a mixed hybrid dynamics offer promise for modeling many-body chemical systems.39 We are currently investigating their application to this problem. From a fundamental viewpoint, the results of this study provide insight into the nonlinear dynamics of many-dimensional systems, and the effect of "dissipation" and "noise"-here, provided by the many-body perturbations of the cluster-on few-mode coupled nonlinear systems. Further work is required to analyze the dynamics of systems of this kind from the perspective of nonlinear science.45 Acknowledgment. We thank Robert Whitnell for helpful discussions and for communicating results prior to publication. In addition, we benefited from stimulating discussions with Prof. Jesus Santamaria and Prof. Gregory S. Ezra. We gratefully acknowledge the donors of the Petroleum Research Fund, administered by the ACS, the National Science Foundation, the Office of Naval Research, and Digital Equipment Corporation for support of this research. We also thank the University of California, Irvine, for an allocation of time on the UCI Convex C240 and the SDSC Cray Y-MP, where many of these calculations were performed.
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