J. Phys. Chem. 1995,99, 2512-2519
2512
Vibrational Predissociation and Intramolecular Vibrational Energy Redistribution: Three-Dimensional Quantum Dynamics of ArI2 Stephen K. Gray* Theoretical Chemistry Group, Chemistry Division, Argonne National Laboratory, Argonne, Illinois 60439
Octavio Roncero” Inst. Matemhticas y Fisica Fundamental, C.S.I.C., Serrano 123, 28006 Madrid, Spain Received: August 18, 1994; In Final Form: October 13, 1994@
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+
A detailed theoretical study of an empirical potential model for ArI2(v’) Ar 12(v R,. (There is neither a unique absorption procedure nor a unique parametrization of a given procedure. One chooses a procedure and parametrization such that calculated observable properties are good approximations to those that would be obtained in a hypothetical, infinite grid calculation without a b ~ o r p t i o n .Generally, ~~ a range of parameter values is acceptable; i.e., the results are not sensitive to
+
+
If one has a time sample sufficient to separate neighboring resonances, e.g., if one has S(t) up to a time greater than 2z/ Allmi,,, where AEmin is the minimum separation between resonances, then accurate estimates of the resonance positions and widths can usually be obtained. The resonance widths may even be consistent with lifetimes (l r , ) significantly longer than 2z/AEmi,,. As discussed in the Appendix of ref 36, one must carefully check the results inferred from the Rony analysis for convergence. With this proviso, it has proven to be successful in many cases where the model of eq 7 is appropriate. Of course, a long time sample of S(t) is quite desirable no matter what approach is taken to infer resonance properties. It is thus worthwhile to point out a simple and extremely useful trick:40 if one has propagated a wave packet to time t, one can actually construct a correlation function to time 2t using the identity, valid for time-independent Hamiltonian operators and a real initial wave packet, S(2t) = (Y*(t)lY(t)). Thus, a 150 ps propagation can actually provide S(t) up to 300 ps. For several cases we also carry out time-independent scattering calculations. The close-coupling calculations are performed as described in ref 13, but with the same vibrational and rotational basis set size as used in the time-dependent calculations above. The absorption cross section as a function of energy is mapped out in certain energy regions, and resonance parameters are obtained by fitting spectral peaks to Lorentzians. 111. Results
A. vdW Ground State Propagations. In one set of calculations, we propagated wave packets corresponding to an initial metastable state with vf vibrational quanta in I2 and zero-point energy in the vdW modes, Le., Y(R,r,y,r=O) = e ! ) ( R , y ) ,,,(r). The correlation functions from each propagation were analyzed (section IIB) to determine resonance energies, widths, and the relative contribution of each resonance. (See eq 7 and discussion.) Typical results are displayed in Table 2. Only resonances with b, ? 0.01 are listed. A given wave packet may contain many other contributions with smaller weights at a variety of energies. In each vf case in Table 2 one resonance is dominant, Le., contributes more than 50% to the wave packet. This resonance therefore has a dominant quasibound contribution from the associated bright state. However, one frequently sees contributions, with smaller b, values, from resonances within -1 cm-l of the main resonance. The smaller b, resonances in Table 2 would not be expected to occur on the basis of just the v‘ manifold vdW states (section IIA) and can be associated as resonances that contain some bright state contribution, but larger amounts of dark contributions associated with I2 having v’ - 1 or vf - 2 quanta and the vdW modes being excited. (It is more accurate to say the secondary resonances are superposition states with v’, v’ - 1, and v’ - 2 character.) The vf = 19 case is an exception of the four vdW ground state cases shown in Table 2 in that it has an extremely small main resonance decay width. This can be partly explained
x
2516 J. Phys. Chem., Vol. 99, No. 9, 1995
Gray and Roncero
TABLE 2: Resonance Decomposition of Selected Propagation@
v = 17 component of w.p.
initial metastable state
(Es - E,')/cm-'
b,
rdcm-'
v' = 18, zero-point vdW modes
-222.52 -222.19 -222.01* -221.96* -221.05 -220.58 -223.73 -222.33 -222.04 -221.76* -222.25 -221.87 -221.50* . -199.52 -198.21 -194.84 -193.63* -192.71
0.085 0.094 0.796 0.965 0.012 0.010 0.013 0.019 0.025 0.920 0.095 0.219 0.608 0.013 0.240 0.022 0.619 0.011
0.088
v' = 19, zero-point vdW modes v' = 20, zero-point vdW modes
v' = 21, zero-point vdW mode
v' = 21, 50/50 superposition of first two excited vdW states
0.098 0.0077 0.0006 0.165 0.055 0.038 0.028 0.156 0.003 1 0.095 0.113 0.023 0.049 0.029 0.030 0.013 0.058
a 12 vibrational energies E,, are in Table 1. An asterisk is placed by the dominant resonance of each propagation. The vdW zero-point region resonances for v' = 18 and 21 differ slightly from those of ref 16, owing to the use of slightly different initial conditions and propagation parameters.
by noting that the metastable state zero-point vdW levels of the v' = 18, 20, and 21 manifolds all lie within 0.25 cm-' of a corresponding excited vdW state of the v' - 1 manifold. In the v' = 19 case, however, the closest v' - 1 manifold level is about 1.6 cm-I away from the v' manifold zero-point level. Thus, there is less mixing of brighvdark states in the v' = 19 case, which considerably inhibits the vibrational predissociation. It is interesting to contrast the time-dependent results with those inferred from time-independent, close-coupling calculations. We looked for resonances in the vicinity of the zeropoint vdW excitation, for v' = 18-21. The same resonance positions as those in Table 2, to within f 0 . 0 2 cm-I, were obtained. Larger discrepancies were obtained in the associated decay constants, although we consider the level of agreement to be acceptable and, indeed, in some cases remarkable in view of the fact that S(t) is computed up to 300 ps or less. The main resonances for v' = 18-21 were found to have decay constants of 0.0076,0.0005,0.0038, and 0.021 cm-', which contrasts with the time-dependent estimates of 0.0077, 0.0006, 0.0031, and 0.023 cm-' from Table 2. The smallest of these decay constants corresponds to a lifetime of ~ 1 ns.0 One can inspect the wave packet vibrational components in time, which may show the influence of underlying dark states. A useful quantity to inspect is the reduced probability density,
6.0
6.8
7.6
8.4
9.2
10.0
Rho Figure 3. Density associated with the v = v' - 1 = 17 projection of the v' = 18, zero-point vdW metastable state propagation, at t
* 60
PS.
metastable state, t:!?,, with the normalized v = v' - 1 vibrational projection of the wave packet (proportionate to eq 9 with v = v' - 1) is %0.8. This indicates how zero-order dark states act as intermediates in the decay process. See Figure 3 of ref 16 for typical appearances of the other vibrational densities. It suffices to note here that eV,-2(f),while bounded in R, involves bending excitations above the collinear inversion barrier and so appears highly excited and unbounded in y . The complicated nodal patterns makes explicit visual identification of 'v - 2 dark states in ev,-2(t)more difficult. No particular v' - 2 dark state has an absolute square overlap with the v' - 2 wave packet projection of more than %O.l. Nonetheless, only a few states have overlaps as large as %O.l, and they occur in an energy regime ( k = 70-80) close to the zero-point level of the v' manifold. The discussion above suggests it is worthwhile to analyze the nature of the resonances from the simplest approximate picture of three interacting states. (This is a three-state variation on the two-state model discussed in refs 13 and 14.) Let Ev, denote the bright state associated with zero-point energy in the vdW modes and 'v quanta in 12. Let tv,-land E"t-2 represent highly excited vdW mode states in the v' - 1 and v' - 2 manifolds that are almost degenerate with tv,.We assume tv, is directly coupled to a single &,,-I state, which in turn is also directly coupled to a single tY,-2 state. The t " l - 2 state is the only state that will be assumed to be coupled to the Ar Iz(vIv'--3) continuum. (Obviously, from the discussion in the above paragraph, the assumption of a single v' - 2 dark state is a gross simplification.) The resonance wave functions will then be of the form
+
(9) ev,(t=O) is simply the initial metastable state density; e.g., if the vdW ground state is being propagated, it has the appearance of a 2D Gaussian in (R,y) space. It turns out that ev,(t>O) always retains its shape, although its integrated area changes. At t = 0, there is no density in the v f v' channels, but as time evolves density grows in them. Figure 3 displays ev,-l(t)= @17(t) corresponding to a propagation of the v' = 18, zero-point vdW state at a time of about 60 ps. e,*-l(t)also retains this appearance throughout time. It is remarkably similar to the density of the 16th metastable state of the v = 17 manifold, pictured in Figure 2. Indeed, the overlap of the 16th v' - 1
(More precisely, Y, is a projection onto the relevant closed vibrational channels and not the full wave function.) Equation 10, with s = 1,2, or 3, could represent the result of diagonalizing a 3 x 3 complex, non-Hermitean Hamiltonian matrix.4' If rv,-2 is the hypothetical decay width associated with Ev,-2, then the resonance decay widths are approximately Ts IC,12~,~-2.(The nonzero matrix elements of the model Hamiltonian consistent with the above description are H,,,J = ev', Hv,-t,v,-l = ev'-l, Hv*-2,vt-2 = ~ ~ ' --2 irvt-2/2, H v ~ . y= ~ Hv~--l,Y~ -~ = a, and Hv#-~.vf-2 - Hv'-~,v!-~ = /3, where E ~ ~ , E ~ ~ - I ,a, E ~/3,~ -and ~ , r"1-2 are real numbers.) Our wave packet calculations correspond to propa-
Three-Dimensional Quantum Dynamics of ArI2
J. Phys. Chem., Vol. 99, No. 9, 1995 2517
TABLE 3: Resonance Wave Function Properties and Three-State Model Analysis for Sets of Resonances near the Zero Point in the vdW Modes 'v
( E~ E,,)/cm-'
p",.
p",,-l
plV,-*
TJcm-'
rmodel/cm-l
18
-222.52 -222.19 -222.01* -222.25 -221.87 -221.50*
0.028 0.155 0.797 0.158 0.326 0.618
0.504 0.073 0.135 0.433 0.191 0.221
0.468 0.772 0.068 0.410 0.483 0.161
0.088 0.098 0.0077 0.095 0.113 0.023
0.053 0.087 (0.0077) 0.059 0.069 (0.023)
21
gating a state, and if eq 10 is approximately true, one would expect that such a propagation would show contributions from the three resonances. Table 2 shows that the v' = 18-21 zeropoint vdW mode propagations appear to be roughly consistent with this picture, although in the case of v' = 20 one also sees a fourth resonance with b, 2 0.01. An idea of the actual resonance wave-functions may be obtained by carrying out time Fourier transforms of the wave packet at the various energies E,:
Y,
a
f dt exp(iE,t) Y(R,r,y,t)
(11)
The vibrational probabilities are
where n is a normalization constant. Restricting attention to the relative probabilities associated with the closed (v', v' - 1, v' - 2) channels, we have the approximate associations A: B: and Accurate Y,estimates require somewhat longer propagations than those used to estimate the resonance energies and decay widths. For 'v = 18 and 21 we carried out ~ 2 0 ps 0 propagations that explicitly evaluated eqs 11 and 12. Of course, all three resonances for each case can be determined from the same propagation. The results, displayed in Table 3, are still not completely converged and are probably only reliable to about &lo%. The three resonances for each case are of a mixed vibrational character. The purest resonance is always the dominant resonance, which is more than 60% v' character. The two other resonances are not as pure as might naively be expected on the basis of the simple sequential mechanism, but it should be noted that this is not inconsistent with the type of three-state, sequential model outlined here. Depending on the model Hamiltonian parameters, a great variety of A,, B,, and C, values are possible. Table 3 also lists what the three-state model would predict for the two mostly dark resonance decay constants assuming that r"t-2is determined from the main resonance decay constant (r"P-2 rmai,,/e!i; rmdel = pC,-2rv,-2). The model predictions are in reasonable, although not perfect agreement with the exact resonance decay constants r,. An accurate description of the quantum dynamics can still be obtained with models involving just a few coupled states.42 In some cases it is necessary to allow for v'/v' - 2 coupling and to allow direct decay from the v' - 1 channel. It is also sometimes necessary to include one or two more dark states. Consistent with a few-state IVR mechanism, we find that the I2 rotational state product distributions can vary considerably with initial metastable state 12 vibrational quantum number v'. (The vibrational distributions are less sensitive, with it typically being the case that 95% of products are in the first open v = v' - 3 channel.) The simple three-state model outlined above, like its two-state c o ~ n t e r p a r t , ' would ~ , ~ ~ be consistent with all three resonances in the vicinity of the vdW zero-point excitation
e, - e,-,,
-e!-2.
0.0
10.0
20.0
40.0
30.0
Rotatlonal Quantum Number
Figure 4. 12 rotational state distributions in the v = v' - 3 channel associated with three resonances in the energy region of ground state vdW energy and v' = 21 quanta in 12. The filled circles connected by solid lines correspond to the -221.5 cm-' resonance, open circles connected by dots correspond to the -221.9 cm-I resonance, and open squares connected by a dashed line correspond to the -222.3 cm-I resonance. The distributions depicted here were obtained using the close-coupling method;I3J4very similar distributions were obtained by analysis of our wave packet propagations.
-
\
' \
/
\
' \I
d
.O
I
I
I
16.0
18.0
20.0
I
I
22.0
24.0
V'
Figure 5. Aain resonance decay constant associated with the ground state vdW level vs initial I2 vibrational excitation v' (open circles connected by dashed lines). Also shown, as two filled circles, are the experimentally based estimates of Burke and Klem~erer.~'
of a given v' excitation having a similar rotational distribution. (This is because in the simple three-state model all rotational products result from decay of the same v' - 2 dark state.) Figure 4 displays the three associated rotational product distributions for resonances near the v' = 21, zero-point vdW excitation. It is clear that while there are certain similarities, e.g., frequently maxima occur in the same j regions, there are also differences, and this again points to the role of a more complicated m e c h a n i ~ m . ~ ~ Figure 5 shows the main resonance decay constant vs initial vibrational excitation v' for a much wider range of excitations than those shown in Table 2. As would be expected of the few-ctate T V R nrnrew12-14 nntlinerl nhnve
VPN
lnroe vnriatinnc
2518 J. Phys. Chem., Vol. 99, No. 9, 1995
in the decay constant are seen, and the results do not monotonically increase with v’. Figure 5 also displays the two vibrational predissociation decay constants inferred by Burke and Klemperer3’ on the basis of a combination of their measurements with those of Zewail and c o - w ~ r k e r sand ~ ~ a kinetic model for competition between vibrational predissociation and electronic predissociation. Only two such experimentally deduced vibrational predissociation decay constants, at v’ = 18 and 21, are available for comparison with theory. (Table I1 of ref 31 actually lists vibrational predissociation rate constants for many v’ values. However, aside from v’ = 18 and 21, these values are interpolations or extrapolations.) The experimentally deduced v’ = 18 value is 0.023 cm-’, in contrast with our main resonance decay at v’ = 18 of 0.008 cm-’, and the experimental v‘ = 21 value is 0.041 cm-’, in comparison with our value of 0.023 cm-I. Thus, the theoretical results appear to underestimate the vibrational predissociation decay by factors of =3 (v’ = 18) and 2 (v‘ = 21). The level of theoreticaVexperimenta1 discrepancy noted so far is not extremely serious, particularly in view of the fact that the experiments involve averaging over a number of total angular momentum states and our study is just the J = 0 limiting case. A potentially more serious (and interesting) kind of theoretical/experimental discrepancy is the observation of Burke and Klemperer that their vibrational predissociation efficiencies do not change much if one looks at those associated with higher vdW excitation^.^' (The efficiencies do change, and relative maxima and minima as a function of v’ are slightly different, but qualitatively there are not any really gross differences.) Burke and Klemperer took this to imply that the actual vibrational predissociation rate constants must vary monotonically with v’ in the usual energy gap law fashion. In the few-state IVR picture, the excited vdW states should overlap with rather different dark states than the zeropoint vdW states overlap with, and so the vibrational predissociation widths may be very different. Since the vibrational predissociation efficiency includes the vibrational predissociation decay constants, and assuming electronic predissociation does not change fortuitously, the vibrational predissociation efficiency should tum out to be very different for an excited vdW state in comparison with a zero-point vdW state. It apparently is not. Section IV will discuss this further. B. Excited vdW Stretch/Bend States. We also considered propagations of initial states corresponding to a superposition of the first two vdW excited states of each v’ manifold. That is, we propagated wave packets with initial conditions y(R,r,y,t=O) = 2-”2(i$) These first two excitations form a Fermi resonance pair: the lower component can be assigned to be approximately gy) = 0.8Y1.o- O.6Yo,2, and the upper component can be assigned to be approximately t$)= 0.6Y1,o O.8Yo,2, where Y,J, denotes a vdW stretch, bend state. Such propagations should yield correlation functions dominated by at least two resonances. As with the tb!)= YO,O propagations discussed above, we also find dark resonances contributing to the corresponding wave packets and a few-state IVR mechanism also applies to the Fermi resonance state decays as well. Table 2 lists an example (v‘ = 21) of the resonance decomposition of such a propagation. Although the initial state is a 50150 superposition of the zero-order lower and upper components, the actual wave packet contains about 24% of one resonance that can be associated with mostly the lower component and 62% of a resonance associated mostly with the upper component. Other resonances in the vicinity of the lower and upper components are evident. (While propagating the superposition state allows us to conveniently obtain both lower and upper resonance components at once, it is a little harder,
+ @)x
+
Gray and Roncero
upper lower
Fermi Resonance Pair
I
14.0
16.0
I
I
20.0
18.0
I
22.0
1
24.0
VI
Figure 6. Resonance decay constants associated with the Fermi resonance pair of vdW states just above the ground vdW state.
due to the lower intensities involved, to locate all the minor resonances in the vicinity of the lower and upper components.) Figure 6 displays the lower and upper resonance decay constants inferred from such analyses for a variety of v’ runs. Each such propagation yielded results similar to those for v’ = 21, and so there was no difficulty in determining the two resonances associated with the Fermi resonance pair. Figure 6 serves to demonstrate that the vdW excitations exhibit quite different decay pattems in relation to the zero-point vdW states (Figure 5), and the two components of the Fermi resonance pair themselves can behave quite differently. This is consistent with the expectations for a few-state IVR process but again points out how our present theoretical model disagrees with the qualitative conclusions of Burke and Klem~erer.~’
IV. Conclusion We have presented a detailed, three-dimensional quantum dynamics study of an empirical potential model for Ar12, with total angular momentum restricted to be zero. The vibrational predissociation process within the model is dominated by a sequential process that may be interpreted with an IVR picture similar in spirit to that developed for Arc12.12-14The main vibrational predissociation decay constants were found to vary quite dramatically and in a nonmonotonic fashion as one varies the initial vibrational excitation of 12. This model system and the accurate quantum dynamics that we have mapped out should provide useful benchmarks for the development of approximate theories and/or testing out new quantum mechanical methods, since the dynamics is far from trivial. It should also prove quite interesting to investigate the corresponding classical dynamics of this relatively complex model vdW system. We anticipate, for example, that some nonmonotonic behavior (although not as dramatic as in the quantum dynamics) in the unimolecular rate constants, calculated via straightforward quasiclassical methods,23could result. As for comparison with experiment, our results agree, albeit crudely, with the two experimentally inferred estimates of the vibrational predissociation decay constants. However, there appears to be a qualitative discrepancy in that Klemperer and Burke3I have noted that their results are not consistent with a simple N R resonance mechanism. Indeed, on the basis of their inferences, they postulated that the vibrational predissociation
Three-Dimensional Quantum Dynamics of ArI2 rate constant in ArIn(v’) should increase monotonically with v’. We mention three possible sources for resolving this difficulty. First, the IVR effect, because its details depend strongly on the positions of certain, relatively high-lying vdW stretch and bend excitations, will depend on the potential parametrization. We have in fact carried out several calculations with alternative choices for the ArI Morse parameters, including choices that increase the v’ = 18 and 21 theoretical decay constants so that they are reasonably close to the experimental estimates. However, while details such as maxima and minima change, the qualitative observation of a nonmonotonic and dramatically varying decay constant as a function of v’ does not change. Nonetheless, a more thorough investigation of the sensitivity of the dynamics to the potential is required. Another possible source of the discrepancy between theory and experiment is the fact that our theoretical calculations have neglected coupling between VP and electronic predissociation (EP). While EP experimental results for ArI2 have been qualitatively interpreted30b with a model that decouples EP and VP, it still may be the case that EP and VP are more directly coupled than suspected and further study would certainly be merited. However, in our view the most likely possibility is that total angular momentum plays a significant r01e.I~ The experimental results are averages over a distribution of total angular momentum states, and the present theoretical results involve just the J = 0 state. The IVR effect may be present in the dynamics of different J states, but because of differences in decay patterns as one changes J , the average decay constant is smoothly varying. We are currently studying these and other possibilities in detail:* It would also be very useful if additional experiments, e.g., real-time experim e n t ~ , *could ~ be performed at a variety of v’ excitations and, ideally, at different rotational temperatures. Acknowledgment. S.K.G. was supported by the Office of Basic Energy Sciences, Division of Chemical Science, U.S. Department of Energy, under Contract W-31-109-ENG-38. O.R. was supported by DGICYT, Grant PB92-0053, and Comunidad Automata de Madrid, Grant 064192. This collaboration was initiated at a CECAM meeting held in Orsay in 1992, and we are both very grateful to CECAM and the organizer of the meeting, N. Halberstadt. References and Notes (1) Rice, S. A. In Dynamics ojPolyatomic Van der Waals Complexes; Halberstadt, N., Janda, K. C., Eds.; Plenum Press: New York, 1990; p 189. (2) Stephenson, T. A.; Rice, S. A. J. Chem. Phys. 1984, 81, 1083. (3) Rosman, R. L.; Rice, S. A. J. Chem. Phys. 1987, 86, 3292. (4) Jacobson, B. A.; Humphrey, S.; Rice, S. A. J . Chem. Phys. 1988, 89, 5624.
J. Phys. Chem., Vol. 99, No. 9, 1995 2519 (5) 0, H.-K.; Parmenter, C. S.; Su, M. C. Ber. Bunsen-Ges. Phys. Chem. 1988, 92, 253.
(6) Heppener, M.; Kunst, A. B. M.; Bebelaar, D.; Rettschnick, R. P. H. J. Chem. Phys. 1985, 83, 5341. (7) Semmes, D. H.; Baskin, J. S.; Zewail, A. H. J . Chem. Phys. 1990, 92, 3359. (8) Johnson, K. E.; Wharton, L.; Levy, D. H. J . Chem. Phys. 1978, 69, 2719. (9) Gutmann, M.; Willberg, D. M.; Zewail, A. H. J. Chem. Phys. 1992, 97, 8037. (10) Beswick, J. A.; Jortner, J. Adv. Chem. Phys. 1981, 47, 363. (11) Gray, S. K. Faraday Discuss. Chem. Soc., in press. (12) Evard, D. D.; Bieler, C. R.; Cline, J. I.; Sivakimar, N.; Janda, K. C. J. Chem. Phys. 1988, 89, 2829. (13) Halberstadt, N.; Sema, S.; Roncero, 0.;Janda, K. C. J . Chem. Phys. 1992, 97, 341. (14) Roncero, 0.; Villarreal, P.; Delgado-Barrio, G.; Halberstadt, N.; Janda, K. C. J. Chem. Phys. 1993, 99, 1035. (15) Bixon, M.; Jortner, J. J . Chem. Phys. 1968,48,715; 1969,50,3284. (16) Gray, S. K. Chem. Phys. Lett. 1992, 197, 86. (17) Gray, S. K.; Rice, S. A,; Noid, D. W. J . Chem. Phys. 1986, 84, 2649. (18) Davis, M. J.; Gray, S. K. J . Chem. Phys. 1986, 84, 5389. (19) Gray, S. K.; Rice, S. A.; Davis, M. J. J . Phys. Chem. 1986, 90, 3470. (20) Gray, S. K.; Rice, S. A. Faraday Discuss. Chem. Soc. 1986, 82, 307. (21) Zhao, M.; Rice, S. A. J. Chem. Phys. 1993, 98, 2837. (22) Tersigni, S. H.; Gaspard, P.; Rice, S. A. J . Chem. Phys. 1990, 92, 1775. (23) Wozny, C.; Gray, S. K. Ber. Bunsen-Ges. Phys. 1991, 94, 2648. (24) Gillilan, R. E.; Ezra, G. S. J . Chem. Phys. 1991, 94, 2648. (25) Gray, S. K. J . Chem. Phys. 1987, 87, 2051. (26) Blazy, J. A.; DeKoven, B. M.; Russell, T. D.; Levy, D. H. J. Chem. Phys. 1980, 72, 2439. (27) Johnson, K. E.; Sharfin, W.; Levy, D. H. J . Chem. Phys. 1981, 74, 163. (28) Jortner, J.; Levine, R. D. Adv. Chem. Phys. 1981, 47, 1. (29) Breen, J. J.; Wilberg, D. M.; Gutmann, M.; Zewail, A. H. J. Chem. Phys. 1990, 93, 9180. (30) (a) Goldstein, N.; Brack, T. L.; Atkinson, G. J. J. Chem. Phys. 1986, 85, 2684. (b) Roncero, 0.;Halberstadt, N.; Beswick, J. A. Chem. Phys. Lett. 1994, 226, 82. (31) Burke, M. L.; Klemperer, W. J. Chem. Phys. 1993, 98, 6642. (32) Barrow, R. F.; Yee, K. K. J. Chem. Soc., Faraday Trans. 2 1973, 69, 684. (33) Gray, S. K.; Wozny, C. E. J. Chem. Phys. 1989, 91, 7671. (34) Gray, S. K.; Wozny, C. E. J. Chem. Phys. 1991, 94, 2817. (35) Tran, L. B.; Huffaker, J. N. J . Math. Phys. 1983, 24, 397. (36) Gray, S. K. J. Chem. Phys. 1992, 96, 6543. (37) Park, T. J.; Light, J. C. J. Chem. Phys. 1986, 85, 5870. (38) Kosloff, R. J . Phys. Chem. 1988, 92, 2097. (39) Marple, Jr., S. Digital Spectral Analysis with Applications; PrenticeHall: Englewood Cliffs, NJ, 1987. (40) Engel, V. Chem. Phys. Lett. 1992, 189, 76. (41) Heller, D. F.; Elert, M. L.; Gelbart, W. M. J. Chem. Phys. 1978, 69, 4061. (42) Roncero, 0.;Gray, S. K. Manuscript in preparation.
JP942221V
