J. Phys. Chem. 1992, 96, 57-63
57
Quantized Transition-State Structure in the Cumulative Reaction Probabilities for the Ci HCi, I H I , and I D I Reactions
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David C. Chatfield, Ronald S. Friedman, Gillian C. Lynch, and Donald G . Truhlar* Department of Chemistry, Supercomputer Institute, and Army High Performance Computing Research Center, University of Minnesota, Minneapolis, Minnesota 55455-0431 (Received: June 19, 1991)
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The cumulative reaction probabilities of Schatz for C1 HCl, I + HI, and I + DI are analyzed by averaging them over an energy interval wider than the trapped state and rotational threshold structures and narrower than the quantized transition-state spacings. This analysis shows that the total reactivity is dominated by short-time dynamics in these cases and clearly reveals quantized transition-state structure underlying the other features. In all three cases we present a spectroscopic fit to the transition-state spectrum, and we compare the spectroscopic constants to those extracted from a variational transition-state theory analysis based on the adiabatic theory of reactions.
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1. Introduction
we have found clear evidence that the miIn recent crocanonical ensemble rate constants for the H H21-3and 0 H23.4bimolecular reactions are globally controlled by quantized transition states. The transition-state structure is most clear in the fixed total angular momentum cumulative reaction probability, NJ(E),where J is total angular momentum and E is total energy; "(E) may be regarded as a unitless form of the fixed-J microcanonical ensemble rate constant obtained by multiplying by the reactant density of states. We analyzed "(E) for J = 0, 1, and 4 for H H2 and for J = 0 for 0 H2, and we were able to assign the quantum numbers to the energy levels of the transition state^.^ For both H H2 and 0 H2, we found very good agreement with variational transition-state energies calculated as the maxima of vibrationally adiabatic potential c u r ~ e s , ~indicating -~ that the quantized transition states may be quantum mechanical analogues of locally adiabatic dynamical bottlenecks in phase space. We were also able to determine the transmission coefficients for these transition ~ t a t e s , l -and ~ we found that most of them were about 0.8-1 -0, in good agreement with the fundamental assumption of transition-state theory that all transmission coefficients are unity.6*10 In comments" communicated to the recent Faraday Discussion on Reactive Transition States we also pointed out-on the basis of a very crude analysis of quantum mechanical J = 0 cumulative reaction probability curves published by S~hatz'~-'~--that quantized transition-state structure may also be observable in the
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(1) Chatfield, D. C.; Friedman, R. S.;Truhlar, D. G.; Garrett, B. C.; Schwenke, D. W. J. Am. Chem. Soc. 1991, 113,486. (2) Chatfield, D. C.; Friedman, R. S.;Truhlar, D. G.;Schwenke, D. W. Faraday Discuss. Chem. Soc., in press. (3) Chatfield, D. C.; Friedman, R. S.; Lynch, G. C.; Truhlar, D. G.; Schwenke, D. W. Poster paper at Faraday Discussion 91, Nottingham, England, March 25-27, 1991. Full article in preparation. (4) Chatfield, D. C.; Friedman, R. S.;Schwenke, D. W.; Truhlar, D. G. J. Phys. Chem., in press. ( 5 ) Our earlier low-energy 0 + H2 data were also analyzed by Bowman. (a) Haug, K.; Schwenke, D. W.; Truhlar, D. G.; Zhang, Y.; Zhang, J. 2.H.; Kouri, D. J. J. Chem. Phys. 1987, 87, 1892, and unpublished data. (b) Bowman, J. M. Chem. Phys. Lett. 1987, 141, 545. (6) Eliason, M. A.; Hirschfelder, J. 0. J. Chem. Phys. 1959, 30, 1426. (7) Hofacker, L. 2.Naturforsch. A 1963, 18, 607. (8) Marcus, R. A. J. Chem. Phys. 1967, 46,959. (9) Truhlar, D. G.J. Chem. Phys. 1970, 53, 2041. (10) Truhlar, D. G.;Isaacson, A. D.; Garrett, B. C. In Theory of Chemical Reaction Dynamics; Baer, M., Ed.; CRC Press: Boca Raton, FL, 1985; Vol. 4, p 127. (11) Chatfield, D. C.; Friedman, R. S.;Lynch, G. C.; Truhlar, D. G. Faraday Discuss. Chem. SOC.,in press. (12) Schatz, G . C. J . Chem. Phys. 1989, 90, 3582. (13) Schatz, G.C. J. Chem. Phys. 1989, 90, 4847. (14) Schatz, G.C. J. Chem. SOC.,Faraday Trans. 1990, 86, 1729.
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hvdroaen atom transfer reactions C1+ HCl, I HI. and I DI. sbmewhat speculatively we tentatively assigned this structure and found semiquantitative agreement with variational transition-state energies again calculated by the adiabatic theory of reactions. The results for these heavy-light-heavy systems are qualitatively different from those for H H2 and 0 + H2 though, in that the quantized transition-state structure is almost completely obscured by structures which have been attributed to trapped-state resonances and rotational thresholds. In contrast, although the stateto-state reaction probabilities for H + HZ1JJ5and 0 + H?% show a variety of trapped-state effects'5J6 and a complicated dependence on energy and initial and final state, almost all this structure either disappears or coalesces with quantized transition-state structure at the level of microcanonical rate constant^.'^ Both canonical and microcanonical rate constants depend only on the sum of all state-to-state reaction probabilities over both initial and final state i n d i c e ~ ; ~ - ~this J ~ -quantity '~ is called'* the cumulative reaction probability. The cumulative reaction probability corresponds to the total flux from reactants to products, and it is the total flux that is gated with unit or near-unit transmission coefficients by the quantized transition states, while the state-testate flux shows a more complicated energy dependency. Very recently, Darakjian et calculated the quantum mechanical J = 0, even permutation symmetry cumulative reaction probability for the He + H2+ HeH+ + H reaction. The cumulative reaction probability for this reaction shows a very complicated resonance structure, with peaks and shoulders spaced by about 0.005-0.01 eV over the whole energy range studied. They noticed, though, that this structure was superimposed on a coarser grain steplike structure similar to what was seen in H + H2 and 0 HI. Then, in a very stimulating computation, the authors averaged the raw cumulative reaction probability over an energy interval F of 0.01 eV, and the resulting finiteresolutioncumulative reaction probability, N'(E;F), revealed the quantized transitionstate structure surprisingly clearly, with an average spacing between transition states of about 0.07 eV. In the present paper we apply such an averaging procedure to C1+ HCl, I HI, and I DI. This, combined with an analysis of cWJ(E;F)/dE, allows us to improve the crude analysis of these systems that we presented previously.
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(15) Zhao, M.; Mladenovic, M.; Truhlar, D. G.; Schwenke, D. W.; Sun, Y.; Kouri, D. J.; Blais, N. C. J. Am. Chem. SOC.1989, Ill, 852. (16) Cuccaro, S. A.; Hipes, P.G.;Kuppermann, A. Chem. Phys. Lett. 1987, 157, 440. (17) Eyring, H.; Walter, H.; Kimball, G. Quantum Chemistry; Wiley: New York, 1g44. (18) Miller, W. H. J. Chem. Phys. 1975, 62, 1899. (19) Garrett, B. C.: Truhlar. D. G.J. Phvs. Chem. 1979.83. 1052. 1079. (20) Darakjian, Z.; Hayes, E. F.; ParkerIG. A.; Butcher, E. A,; Kress, J. D. J. Chem. Phys. 1991, 95, 2516.
0022-365419212096-57$03.00/0 0 1992 American Chemical Society
58 The Journal of Physical Chemistry, Vol. 96, No. 1, 1992
Chatfield et al.
2. Computations
considerations of the coordinate system and the anharmonicity.
The published cumulative reaction probabilities of Schatzi2-14 were digitized with an optical scanner. (This introduces negligible error.) They were then convoluted with a Gaussian resolution function of adjustable width. For I HI we ignore the two trapped-state resonanm below 0.16 eV, and for I DI we ignore the two trapped-state resonances below 0.14 eV. A Gaussian convolution simulates experimental instrument resolution. To carry out the convolution, we fit the accurate NJ(E) with cubic splines and convoluted it as follows
3. Theory Looking at the cumulative reaction probability, or any function of energy, with a finite resolution F corresponds to looking at a time scale of less than about Af ~ L / F ,where ~' h is Planck's constant divided by 27. Thus,finite-resolution cumulative reaction probabilities calculated with broad-resolution functions show structure due to short-time dynamics, but not long-time dynamical features, such as those due to long-lived trapped states or to entrance channel or exit channel couplings, which are washed out. Since, in many cases, we expect that the total rate constant, summed over initial and final states, is controlled by short-time dynamics in the vicinity of the transition ~ t a t e p finiteresolution ~-~~ convolutions should be particularly appropriate in searching for quantized transition-state structure. (Important exceptions may occur in reactions with low barriers, wells, or two or more dynamical bottlenecks, where reflections from potential energy surface features far removed from the best dynamical bottleneck may contribute significantly to recrossing. When this occurs, the total rate constant is not controlled by short-time dynamics.) The separation of time scales in terms of energy resolution is familiar in spectroscopy, where it has been especially emphasized by Heller. For e~ample:~'"a spectrum taken at ultrahigh resolution (and containing long-time information) contains within it all lower-resolution information (Le., shorter-time information), We can choose to examine it at lower resolution and extract the dynamics corresponding to shorter and shorter times." In order to make these ideas more concrete in the present context, we note that if the full width at half-maximum of a resonance feature is r and At is the time interval over which the dynamics responsible for the resonance occurs, thed8
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where the resolution function is f(E,E';F) = A exp{-[2(E - E?/F12 In 2)
(2)
and where the normalization factor A is determined by A =
(1"' dE'exp{-[2(E - E?/F'J2In 2])-' Ei
(3)
F is the full width at half-maximum of the resolution function, and Ei and Efare the initial and final energies for the energy range of NJ(E). The finite-resolution density of reactive states pJ(E;F) is defined by d pJ(E;F) = -NJ(E;F) dE
(4)
and it was calculated by analytically differentiating a cubic spline fit to NJ(E;F) computed at an evenly spaced set of energies. All calculations in the paper are for J = 0. The values used for F i n the analysis below were chosen by trial and error as those which best average out the structure due to rotational thresholds and trapped states while preserving the structure due to quantized transition states. We also calculated vibrationally adiabatic potential c ~ r v e s ~ - ~ in natural collision coordinates2' for all these systems. The methods are the same as we have used p r e v i ~ u s l y . ~ ~In- ~par~ ticular, stretching motions are treated by the WKB method23and bending motions are treated by a quadratic-quartic Taylor's series in the bond angle22 with the WKB centrifugal oscillator app r o ~ i m a t i o n . ~We ~ , ~carefully ~ checked that the Taylor's series fits provide a good representation of the bend potentials over the whole relevant range for bond angles, even for the highly excited bend states. The vibrationally adiabatic potential curves are labeled by the quanta of vibrational excitation, v , in the stretch and v2 in the bend. The maxima in the vibrationally adiabatic potential curves, V,(vi,u2,s),as a function of reaction coordinate2629 s are identified as the vibrationally adiabatic thresholds ~ ( v l r v 2 ) . These calculations were carried out with the ABCRATE3' computer code. To avoid misinterpretation of our intent, we note that our goal in carrying out the vibrationally adiabatic calculations is to check their qualitative consistency with the analysis of the quantum results. We do not believe that predictions of the quantized transition-state spectrum based on vibrationally adiabatic curves can be made quantitative for these systems without more detailed (21) Marcus, R.A. Faraday Discuss. Chem. SOC.1967, 44, 7. (22) Garrett, B. C.; Truhlar, D. G.; Grev, R.S.; Magnuson, A. W. J . Phys. Chem. 1980,84, 1730. (23) Garrett, B. C.; Truhlar, D. G. J . Chem. Phys. 1984, 81, 309. (24) Garrett, B. C.; Truhlar, D. G. J . Phys. Chem., in press. (25) Natanson, G. J . Chem. Phys. 1990, 93, 6589. (26) Shavitt, I. University of Wisconsin Theoretical Chemistry Laboratory Technical Report WIS-AEC-23, Madison, WI, 1959. (27) Marcus, R. A . J . Chem. Phys. 1966, 45, 4493. (28) Marcus, R. A. J . Chem. Phys. 1968, 49, 2610. (29) Truhlar, D. G.; Kuppermann, A. J . Am. Chem. SOC.1971,93, 1840. (30) Garrett, B. C.; Lynch, G. C.; Truhlar, D. G. ABCRATE computer program, in preparation for submission to an international program library.
f
FAt = h (5) Thus, we associate a time constant Af with each finite-resolution cumulative reaction probability by the relation At = h/F,and we note that any features dependent on dynamical processes requiring appreciably longer than Af should lead to features with widths appreciably less than F. These should be washed out in the finite-resolution cumulative reaction probability NJ(E;F) and hence in p'(E;F) as well. Since our ideas about transition-state thresholds are still evolving and since they conflict in part with traditional language, a word of caution on the semantics may be in order. A "transition-state threshold" is an increase in the cumulative reaction probability associated with a quantized dynamical bottleneck to reactive flux; it should not be confused with an energetic threshold where a new state of the reactants or products opens. Rather, a transition-state threshold is a resonance, and typically it is broad when compared to the resonances which have so far been widely discussed39in the literature. To distinguish these traditional resonances from transition-state thresholds and to emphasize their invariable identification with a system trapped in some sort of effective well, we call them trapped states. The model for transition-state thresholds is very different; it is of a system dallying at some sort of an effective barrier. We note though that some resonances (3 1) Sakurai, J. J. Modern Quantum Mechanics; Addison-Wesley: Redwood City, CA, 1985; pp 78-80. (32) Miller, W. H. J . Chem. Phys. 1974, 61, 1823. (33) Tromp, J. W.; Miller, W. H. J . Phys. Chem. 1986, 90, 3482. (34) Tromp, J. W.; Miller, W. H. Faraday Discuss. Chem. SOC.1984,84, 441. (35) Day, P. N.; Truhlar, D. G. J . Chem. Phys. 1991, 94, 2045. (36) Day, P. N.; Truhlar, D. G. J . Chem. Phys. 1991, 95, 5097. (37) Heller, E. J. In Potential Energy Surfaces and Dynamics Calculations; Truhlar, D. G., Ed.; Plenum: New York, 1981; p 103. (38) Das, A.; Melissinos, A. C. Quantum Mechanics: A Modern Introduction; Gordon and Breach: New York, 1986; p 533. (39) See,e.g.: (a) Toennies, J. P. Comments At. Mol. Phys. 1979,8, 137. (b) Truhlar, D. G.; Wyatt, R. E. Annu. Rev. Phys. Chem. 1976, 27, 1 . (c) Kupperman, A . In Potential Energy Surfaces and Dynamics Calculations; Truhlar, D. G., Ed.; Plenum: New York, 1981; p 375. (d) Resonances in Electron-Molecule Scattering, van der Waals Complexes, and Reactive Chemical Dynamics; Truhlar, D. G., Ed.; American Chemical Society: Washington, DC, 1984.
The Journal of Physical Chemistry, Vol. 96, No. 1, I992 59
Quantized Transition-State Structure 7 ,
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Energy (eV) Energy (eV) , the value Figure 1. C1 + HCI: (a) " ( E ) ; (b) p'(E); (c) N ' ( e 0 . 0 2 7 ) ; (d) p ' ( e 0 . 0 2 7 ) . E is the total energy, and J is 0. Assignments are [ u , u ~ ]and of N J ( E F )is indicated at each minimum in p'(E;F). partake of both characters, especially in cases where zero-order models predict a trapped state at an energy close to a barrier 4. Analysis CMCI. The accurate cumulative reaction probability for the C1 HCl reaction is shown in Figure la. Some quantized transition-state structure can already be identified before the averaging procedure is app1ied.l' The CRP exhibits a marked steplike rise between 0.40 and 0.45 eV, so we assign this feature as [OOO]. (As before,'-4 we use quantum numbers [ u , u ~for ~] quantized transition states, where u1 is the bound stretch quantum number, u2 is the bend quantum number, and K is the vibrational angular momentum.) The sharp rise near 0.66-0.67 eV has been shown to be due to a dynamical threshold for formation of vibrationally excited products,41and so we associate this feature with [lo0]. The sharp peak a t 0.641 eV has been identified previouslyI2as a trapped-state resonance, and the broader features between 0.5 and 0.6 eV have been associated with rotational levels of the asymptotic diatom in reactants and products. The density of reactive states calculated from the accurate NJ(E)is shown in Figure lb. The density of reactive states without a resolution function has high-frequency oscillations due to long-time dynamics but perhaps also in part to the limits of convergence of the quantum calculations,I2 the spacing between points in these calculations, the interpolation scheme used to generate the original plot, and the scanning process. The most prominent feature is a high-amplitude oscillation associated with a trapped state. Some threshold features are also discernible, but the many rapid oscillations make this figure difficult to interpret. The finiteresolution cumulative reaction probability and density obtained with F = 0.027 eV are shown in Figures lc,d. This resolution function corresponds to At = 24 fs. The broader features in the original NJ(E)are still present, but the narrow trapped-state peak has disappeared in this short-time picture. The finite-resolution density of reactive states shows seven well-resolved
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(40) Friedman, R.S.; Truhlar, D. G.Chem. Phys. Lerr. 1991, 183, 539. (41) Schatz, G.C.;Sokolovski, D.;Connor, J. N. L.J . Chem. Phys. 1991, 94, 4311.
peaks and one discernible shoulder. We p r d to interpret this spectrum in light of the increment in NJ(E;F)associated with each peak. The value of NJ(Eflis shown at each minimum between peaks. The first peak is clearly correlated with the initial rise in NJ(E), which we associate with [@I. Since NJ(E;F)rises by about unity to 1.06 at the minimum of pJ(E;F)following its first peak (0.416 eV), this threshold is a nearly ideal dynamical bottleneck to the reactive flux. The two peaks following the [OOO] feature are associated with the broad rotational threshold structures identified in NJ(E)between 0.5 and 0.6 eV. Between the minima of pJ(E;F) at 0.469 and 0.594 eV, NJ(E;F)rises by 1.15 to reach a value of 2.21. Therefore, we associate both of these peaks with the next higher energy level of the transition state, [02O]. The [02O] feature is apparently split in this way because the reactive flux is channeled through rotational thresholds, and some of the channeling into different rotational manifolds occurs at short enough times to affect even the 24-fs averaged cumulative reaction probability. The overall rise of NJ(EF)through these two peaks in pJ(E;F) along with the consistency of the higher energy part of the spectrum with transition-state theory, indicates that primarily one new level of the transition state is accessed; however, since the value of NJ(E;F)rises to a value slightly greater than 2 by 0.594 eV, some flux is beginning to be channeled through [04O] by the upper end of this energy range. Between 0.594 and 0.775 eV, NJ(E;F) rises by 3.14 to 5.35. We believe that four transition state levels lie in this energy range. The narrow peak in pJ(E;F)at 0.665 eV is readily identified as [lo0] from comparison with "(E). This shows a pleasing consistency with our previous results14 for H H2and 0 H2where the [ lo0] peak was higher and narrower than preceding [ O V , ~ ] peaks. Although the various peaks overlap strongly so that they all represent composites of contributions from more than one level, we note that NJ(E;F) rises by about unity (1.02) between the minima of p J ( E ; F ) at 0.644 and 0.689 eV on either side of this feature. With the [ lo0] state assigned, we can identify the peak at 0.629 eV as [04O]. We note that although a trapped state has been identified near this energy, we correlate this feature with a transition-state threshold because it contributes to an overall increase in NJ(E;F)rather than an oscillation and a return to the
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60 The Journal of Physical Chemistry, Vol. 96, No. 1, 1992
Chatfield et al.
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Energy (eV) Energy (eV) Figure 2. 1 + HI: (a) "(E); (b) p'(E); (c) N'(E;0.020 eV); (d) p'(E;0.020 eV); (e) N'(E;0.027 eV); (f) p'(R0.027 eV). E is the total energy, and J is 0. Assignments are [ u , u ~ ~and ] , the value of N'(E;F) is indicated at each minimum in p'(E;F). background value as expected for traditional resonances. There are two features in pJ(E;F) between 0.689 and 0.775 eV: a peak at 0.719 eV and a shoulder at slightly higher energy. The energy spacings between the ground stretch (u, = 0) thresholds and the increase in NJ(E;F)over this energy range, 1.47, are consistent with associating the [06O] and [12O] levels with these features. On the basis of fits with a spectroscopic equation, described below, we assign the peak at 0.719 eV as [06O] and the shoulder as [ 12O]. It is more difficult to assign the final feature in the finite-resolution spectrum, at 0.800 eV, but from the energy spacings we expect that it is due to [14O], [OS0], or both. We conclude this analysis of ClHCl transition-state dynamics by noting that by 0.775 eV the value of NJ(E;F),5.35, leads us to predict six levels of the transition state if the levels are effective dynamical bottlenecks. In the same energy range there are six peaks and one shoulder in the finite-resolution density. We find a 1 to 1 correspondence between the features in p'(E$') and levels of the quantized transition state, except for the two peaks between 0.5 and 0.6 eV which are due to rotational thresholds both associated with the [02O] transition state level. This method of analysis clearly reveals that the dynamics of reaction for C1 + HCl is predominantly controlled by quantized transition states. IHI. The cumulative reaction probability for the I + HI reaction, shown in Figure 2a, exhibits overall steplike features suggestive of quantized transition-state structures. Superimposed on these are oscillations which have been shown to be due to
rotational thresh01ds.l~ The density of reactive states in Figure 2b shows many oscillations, which makes it difficult to interpret. We simplify these spectra by applying a resolution function. It is difficult to know a priori what time interval A? is most relevant to the transition-state dynamics that control the reactive flux,and so we will examine convolutions for two different values of F. A convolution with F = 0.020 eV is shown in Figures 2c,d. This convolution corresponds to A? = 33 fs. Analysis of vibrationally adiabatic curves, a procedure which yields very accurate [ lo0] energies for other reactions,ls3 predicts that the [ lo0] threshold for this reaction occurs at 0.422 eV, higher than the range of the calculated "(E). Therefore, we only need to consider the ground stretch manifold when making assignments. The first peak in the density is clearly associated with the initial unit rise in the cumulative reaction probability and is assigned [OOO]. The value of NJ(E;F) rises by about unity to 1.03 at the first minimum of p J ( E ; J ) at 0.187 eV. The origin of the next two peaks in the convoluted density, between 0.187 and 0.226 eV, is easily traced back to oscillations in the unaveraged "(E) that Schatz13 associated with rotational thresholds. By 0.249 eV, NJ(E;F)reaches a value of 1.97. Therefore, just as for ClHCl, we associate both of these peaks with a [02O] transition-state feature that is split due to rotational thresholds. Over the energy range of the next three peaks, 0.2264.329 eV, N'(E;F) rises by 0.94 to 2.91. We attribute this increase of nearly unity to [04O]. In this case, rotational thresholds apparently associated with the [04O] level
The Journal of Physical Chemistry, Vol. 96, No. I, 1992 61
Quantized Transition-State Structure
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0.25 0.30 0.35 0.40 0.45 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 Energy (eV) Energy (eV) Figure 3. I + DI: (a) "(E); (b) p'(E); (c) N'(lC0.015 eV); (d) p'(E;O.OlS eV); (e) N'(l20.025 eV); ( f ) p'(e0.025 eV). E is the total energy, and J is 0. Assignments are [ v , u ~ ~and ] , the value of N'(E;F) is indicated at each minimum in p'(E;F). 0.15
0.20
of the transition state give rise to three peaks in the density. (The concept of doorway appears to emerge naturally in interpreting these split features.) The value of NJ(E;F)rises by about unity (1.07) to 3.98 over the energy range of the next peak, and so we assign it [06O]. The final feature in the spectrum is incomplete, but from energy spacings we assign it as [08O]. A convolution with a Gaussian having F =: 0.027 eV is shown in Figures 2e,f, and the rotational threshold structures are washed out. This convolution corresponds to At = 24 fs. The two peaks associated with [02O] in the earlier density plot merge; the most prominent of the peaks associated with [04O] remains while the other two become shoulders to the peak for [06O]. Thus, at this level of averaging, i.e., on this time scale, one peak is associated with each quantized transition-state level. For this reaction, as for CIHCl, a finite-resolution density of reactive states clearly reveals that a quantized transition state exerts a controlling influence on the chemical reactivity. IDI. The [OS]threshold is clearly observable as the initial rise to 1 in the unaveraged NJ(E)shown in Figure 3a. The figure is suggestive of further steplike features at higher energy, but these are partially obscured by rotational and trapped-state structures identified previously14 in the energy ranges 0.16-0.23 and
0.23-0.30 eV. Additionally, we identify the sharp rise near 0.32 eV as due in part to the [ IO0] threshold because the production of vibrationally excited products rises rapidly in the energy range 0 . 3 0 . 3 4 eVI4 and because the vibrationally adiabatic [ IO0] level of transition state is at 0.298 eV. The unaveraged density function shown in Figure 3b is again not very instructive because of the many rapid oscillations. When we decrease the resolution to 0.01 5 eV, corresponding to At = 44 fs, though, the density brings out many features, as shown in Figure 3d. The first peak is [OOO]; at the first minimum NJ(E;F) reaches a value of 0.94, corresponding to a transmission coefficient of very close to unity. Over the course of the next two peaks, NJ(E;F)rises by 0.95 to 1.89, and so we associate both peaks with [02O]. The splitting is caused by the rotational thresholds identified previ0us1y.l~We attribute the next set of peaks up to the minimum at 0.283 eV, where NJ(E;F)reaches 3.03, to the [04O] threshold. In this case the multiplicity of peaks is possibly due to trapped states and possibly also to rotational thresholds. By the minimum a t 0.35 eV, N'(E;F) rises further by about 2 to 4.96. We tentatively associate the [ 1001 threshold identified in Figure 3a with the prominent peak at 0.330 eV and the [06O] threshold with the peak at 0.295 eV. The next major feature, between 0.35 and 0.41 eV, is broad and has (42) Feshbach, H,; K ~ A. K,;~ L ~ ~ R. H. ~ ~ ~phys. ~ (, ~ ,~ y , ) ~ , longtime structure (one dmdder and a slight splitting of the peak) probably associated with trapped states identified 1967, 41, 230. (43) George, T. F.; Ross, J. J . Chem. Phys. 1972, 56, 5786. earlier by Schatz.I4 By the minimum following this feature,
Chatfield et al.
62 The Journal of Physical Chemistry, Vol. 96, No. 1, 1992 TABLE I: Spectroscopic Constants of Quantized Transition States from Quantal and Vibrationally Adiabatic Calculations
ClHCl present quantal" 0.231 previous quantalb 0.241 0.271 adiabatic IHI present quantal C previous quantal d 6.65 (-3) adiabatic ID1 present quantal 8.80 (-3) previous quantal 7.16 (-3) 1.72 (-2) adiabatic
2010 2000 1920
497 429 671
C
-11.2 17.4
-269
2200
235 216 253
26.8
-154
1560 1520 1450
230 263 192
14.2
d
+
-78.5
'This paper. bReply to comments on Faraday Discussion paper." cVeffQTs+ OShcw, = 0.139 eV = 1124 cm-I. dVellQrs + 0.5hcwl = 0.141 eV = 1133 cm-I. NJ(E;F)rises by 0.79 to a value of 5.75. There are two possible assignments for this feature: [12O] and [08O]. On the basis of fits to a spectroscopic equation presented below, we nominally assign this feature as [12O]. However, both the spectroscopic fits and the energy spacings of other bend levels in the uI = 0 manifold lead us to.believe that [08O] also contributes to this feature. We note that if in fact there are two transition-state levels associated with this feature, then at least one of the transmission coefficients must be deviating significantly below unity. Since in other cases, both previ~uslyl-~ and above, we find that highly bend excited states tend to have smaller transmission coefficients, this is reasonable here and probably serves more as a confirmation of the assignments than as a troublesome point. Nevertheless, further work is needed to understand this feature better. The last feature in the spectrum is difficult to interpret because it is incomplete. On the basis of energy spacings and the spectroscopic fits presented below, we tentatively associate it with [0 lo0] and [14O]. With a wider resolution function, F = 0.025 eV, corresponding to Af = 26 fs (Figure 2e,f), the rotational threshold structure and trapped-state structure are smoothed out, and only the structure due to the quantized transition state levels remains clear. We note that the first peak is primarily due to [OOO] but is also due in part to the first rotational threshold associated with [02O] (these features are distinct in Figure 3a,d, but they are close enough in energy that they coalesce into a single peak in Figure 3e). With the exception of the peak between 0.35 and 0.41 eV, which is assigned as [ 12O] and [OS0], each peak and shoulder can be nominally associated with a single transition-state threshold. As for the other two chemical systems analyzed here, the use of a finite resolution to emphasize the short-time dynamics clearly brings out the quantized transition-state levels for the I + DI reaction. 5. Spectroscopic Constants
The vibrationally adiabatic thresholds, V3uI,u2),were fit with the spectroscopic series44
E(l)(u,u*)=
V,ffQTS
+ hc[w,(u1 + y2) + w2(u2 + 1) + +
~ 1 2 ( ~ 1 Y2)(u2
+ 1) + x22(u2 +
(6)
where c is the speed of light, V", is the effective potential energy of the quantized transition state, wi is an effective harmonic frequency for mode i, and xij is an anharmonicity constant. The five constants on the right-hand side were extracted by fitting the thresholds for [uIu2K]= [OOO], [02O], [04O], [loo], and [12O]. We did not include the xII(uI '/2)2 term in (6) because it would require fitting to the [2@] level, which occurs at a higher energy than the range covered by the quantum mechanical calculat i o n ~ ' analyzed ~ - ~ ~ here. We emphasize that, as usual in analyses involving a limited number of energy levels, all spectroscopic constants are effective values corresponding to a given truncation of the spectroscopic series rather than to the true Taylor's series coefficients.44a The resulting constants are given in Table I.
+
We also fit eq 6 to peak pitions in the finiteresolution densities of reactive states of section 4. These are called E(ul,uZ).In these fits, however, we do not believe we have accurate enough values to include xI2,so we set it to zero. In two cases we set xzzequal to zero as well, again because the accuracy of the energy levels does not justify its inclusion; and for I + HI we do not know E( 1,0) so we cannot determine VefiQmand uIseparately but only the sum VcffQTS 0.5hcwl. First, consider C1+ HCl. Here we obtained four spectroscopic constants from E(O,O), E(0,4), E(0,6), and E(1,O). Table I compares these to the values from the adiabatic analysis and to the estimates we made in ref 11. Recall that ref 11 was based directly on Schatz's "(E) values, whereas the present values are based on p'(E;F), in which the structures are clearer. Thus, the present analysis is preferred. Nevertheless, there is good agreement between the two analyses (0.5% for w1 and 15% for wz). Agreement with the adiabatic values is also encouraging, although the deviations are larger (5% for wland 30% for wz). Notice the large value of xI2in the adiabatic predictions. This is a conseoccur at the quence of the fact that the maxima of Va(0,v2,s) symmetric saddle point, whereas the maxima of Va(1,u2,s) with u2 = 0 or 2 are much earlier, where the average difference in the H-Cl bond lengths is 0.70 A. The bending frequencies for such very different structures are expected to be quite different, resulting in a large xI2. Table I1 compares the peak positions from Figure 1b and the values obtained from the spectroscopic series with the present quantal values of the constants. The agreement is very good. The spectroscopic series predicts a [02O] level at 0.528 eV, one-third of the way between the two peaks in Figure Id. The energy of [08O] is predicted to lie within 0.002 eV of the feature labeled [08O] in Figure Id, addmg confidence to this assignment. Furthermore, the shoulder at 0.75 eV correlates well with the predicted energy of the [12O] state. Similar results for I + H I are also given in Tables I and 11. In this case, the quantal peaks are located on the curve with F = 0.027 eV and the spectroscopic constants are based on E(0,O) and E(0,6). The estimates we made in ref 11 and the values from the adiabatic analysis both agree with the present value for w2 based on pJ(E;F)to within 8%. The peak positions for the [02O], [04O], and [08O] levels agree with the spectroscopic series within 0.022, 0,019, and 0.020 eV, respectively, which are small discrepancies compared to the average spacing of about 0.058 eV. Presumably the agreement would be even better if the time scale for the quantized transition-state structure were better separated from that for the rotational threshold structures. The final sections of Tables I and I1 give results for I + DI. Here, six of the E(ul,u2)values are peak positions in pJ(E;0.025 eV), but two of the energy levels are taken as peak maxima in pJ(E;0.015 eV). One of the latter is the [OOO] level, which-as discussed in section 4-seems to be better isolated from [02O] with the narrower resolution function, The feature we assign as [06O] is a shoulder in pJ(E;0.025 eV) so its value too is taken from pJ(E;0.015 eV). The energies E(O,O), E(1,0), and E(1,2) were used to calculate spectroscopic constants. In the I DI case, there is again good agreement between the adiabatic and present quantal values for wl(7%) and wz (17%). For both I + HI and I + DI, the agreement for w2 is better than for C1 + HCl. Note also that x12is smaller for I + HI and I + DI than for C1+ HCl. This is again better understood when we consider the structures of the variational transition states. In particular, for I + H I and I + DI, the maxima of V,(ul,u2,.s)occur at asymmetric geometries both for u1 = 0 and for u1 = 1, so there is less difference in the bending force constants. More specifically, we note that the difference in D-I bond lengths at the [OOO] and [02O] levels of the variational transition state is 0.51A and this difference for the [lo0] and [12O] levels is 0.86 A, only 0.35 A more, as compared to 0.70 A for ClHCl. The agreement between
+
(44) Herzberg, G. Molecular Spectra and Molecular Structure. [I. Infrared and Roman Spectra of Polyatomic Molecules; D. Van Nostrand: Princeton, NJ, 1945; p 210, (a) 206-207.
The Journal of Physical Chemistry, Vol. 96, No. 1 , 1992 63
Quantized Transition-State Structure
TABLE Ik Quantized Transition State Energies (eV) from Finite-Resolution Quantal Densities of Reactive States and SpectroscopicSeries Fib to the Quantal Results ClHCl IUIV,KI
[ooO1 [02OI [0401 [06OI [08O1 [O 1001 [ 1oO1 [ 12OI [1401
E(uI,u,) 0.4 16 0.510, 0.565' 0.629 0.719 0.800 0.665 0.75e
ID1
IHI
E(')(uirud 0.416 0.528 0.629 0.719 0.798 0.866 0.665 0.777 0.878
E(UIPI) 0.168 0.205 0.266 0.343 0.381
E(')(uI.u~) 0.168 0.227 0.285 0.343 0.401 0.460
E(u7 ,V,) 0.134" 0.191 0.23 1 0.295" 0.39c 0.435d 0.328 0.385 0.435d
E(')(UI,U?) 0.134 0.191 0.248 0.306 0.363 0.420 0.328 0.385 0.442
a From quantal density with F = 0.015 eV. Other values for I + DI from quantal density with F = 0.025 eV. bThere are two peaks in the quantal density which are associated with [02O]. 'This threshold is believed to contribute to the peak at 0.385 eV. dTentatively assigned features. 'This feature is a shoulder in Figure Id but occurs at 0.752 eV in pJ(E;0.020 eV).
the peak positions for [02O], [04O], [06O], [0lo0],and [14O] and the values from the spectroscopic series is very good, with an average difference of 0.010 eV compared to the averaging spacing of 0.033 eV. We also calculated spectroscopic constants using E(O,O),E(0,8), and E(1,0), with E(0,8) taken as 0.385 eV, the energy of the maximum of the next to last peak. The energy levels predicted by this fit were generally in good agreement with the energies of the peaks in pJ(E;0.025),but the agreement from the previous fit was slightly better. On this basis we nominally assign [ 12O] to the peak maximum at 0.385 eV, with the understanding that [084] also contributes to this feature. We also note that much previous analysis has indicated that heavy-light-heavy systems are better understood in terms of hyperspherical coordinates,4549coordinates based on the distance between heavy particles,50or Cartesian axes parallel to the dominant tunneling paths.10*51*52 In the present paper we have assigned the quantized transition states using quantum numbers based on natural collision coordinates. It would be interesting to see if assignments based on coordinates more suitable for separating the motions of light and heavy nuclei would provide further insight. 6. Concluding Remarks The method of averaging the cumulative reaction probability over a resolution function allows us to see quantized transition-state structure more clearly. Convoluting with a Gaussian of small width averages out the rapid oscillations in the density, clearly bringing out trapped-state structures, rotational threshold structures, and quantized transition-state features. Convoluting (45) Babamov, V. K.; Marcus, R. A. J . Chem. Phys. 1981, 74, 1790. (46) Aquilanti, V.; Grossi, G.;Lagang, A. Chem. Phys. Letr. 1982,93, 174, 179. (47) Romelt, J.; Pollak, E. ACS Symp. Ser. 1984, 263, 353. (48) Manz, J. Comments At. Mol. Phys. 1985, 17, 91. (49) Romelt, J. NATO ASI Ser. C 1986, 170, 77. (50) Kubach, C.; Vien, G.N.; Richard-Viard, M. J. Chem. Phys. 1991, 94, 1929. (51) Garrett, B. C.; Truhlar, D. G.;Wagner, A. F.; Dunning, T. H., Jr. J. Chem. Phys. 1983, 78,4400. (52) Garrett, B. C.; Abusalbi, N.; Kouri, D. J.; Truhlar, D. G. J . Chem. Phys. 1985,83, 2252.
with a Gaussian of larger width averages out narrow trapped states and rotational thresholds, thereby bringing out the pattern of energy levels of the quantized transition state which underlies them. This method of analysis allows us to clearly observe the influence of quantized transition-state levels on the cumulative reaction probability even when this influence is partially obscured by features such as trapped states and rotational thresholds whose effects are local in energy. Clearly this will work best when there is an appropriate separation of time scales in the dynamics; one of the main conclusions of the present letter is that indeed such a separation may exist even in difficult systems of current interest. The present analysis adds to the list of reactions where quantized transition states control the structure of the microcanonical ensemble rate constant as a function of en erg^.^^,^^ The averaging procedure combined with an analysis based on the density of reactive states may be an important tool in understanding other chemical systems as well. Acknowledgment. We are grateful to George Schatz for helpful discussions. This work is supported in part by the National Science Foundation (quantum theory of transition-state dynamics), U.S. Department of Energy, Office of Basic Energy Sciences (adiabatic transition-state theory), Army High Performance Computing Research Center (Graduate Fellowship to D.C.C.), and Minnesota Supercomputer Institute (Research Scholarship to R.S.F.). Registry NO. C1, 22537-15-1; HCI, 7647-01-0; I, 14362-44-8; HI, 10034-85-2; DI, 14104-45-1. (53) Neumark has also raised the possibility that some of the structure in the photodetachment spectrum he measured for FH, may be due to one or more quantized transition-state levels for the F + H2 H F + H reaction. Neumark, D. Paper 13.4, Spring Meeting of the American Physical Society, Washington, DC, April 24, 1991. (54) Note added in prmJ In an exciting new development, Moore has now reported that he and his co-workers have observed quantized transition-state structure in the photodissociation of ketene in the triplet channel. This differs from earlier work of this group on photodissociationof ketene in the singlet channel, where the potential apparently rises monotonically out to infinite separation of the fragments. In the singlet case they observed structure correspondingto the opening of new product states, but in the triplet case he reported structure apparently associated with quantized energy levels of a transition state at finite interfragment distance. Choi, Y. S.;Kim, S.-K.; Lovejoy, N.; Moore, C. B. Paper PHYS 165,4th Chemical Congress of North America/ZOSnd ACS National Meeting, New York, NY, Aug 29, 1991.
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