2971
J . Phys. Chem. 1984,88, 2911-2911
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A Quasiclassical Trajectory Study of the State-to-State Dynamics of H 4- H20 OH -t H2 George C. Schatz,* Mitchell C. Coiton, and John L. Grant Department of Chemistry, Northwestern University. Evanston, Illinois 60201 (Received: November 14, 1983; In Final Form: March 2, 1984)
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This paper presents quasiclassical trajectory cross sections and rate constants for the H + H 2 0 OH H2 reaction, including a detailed study of reagent normal-mode and local-mode excitation effects, and of product state energy partitioning. The potential surface used is based on a fit to an accurate ab initio calculation. The quasiclassical trajectory calculation used a classical perturbation theory method to define the initial H 2 0 normal-mode and local-mode eigenstates, with the normal-mode representation used to describe the lower energy eigenstates and the local-mode representation used to describe OH stretch overtones having several quanta of excitation. The resulting ground-state thermal rate constants are compared with experimental measurements and found to agree to within their respective uncertainties. Good agreement is also found in comparing our OH product vibrationfrotation distributions with the results of recent laser photolysis studies. In studying the influence of initial vibrational excitation, we find substantial rate constant enhancements when any of the three vibrational normal modes of H 2 0 are excited, with most of that enhancement due to a reduction in activation energy which comes from a lowering in the effective cross section threshold energy. The efficiency of this lowering, Le., the ratio of threshold change to vibrational excitation energy, is highest for the bend mode and lowest for the asymmetric stretch. Our studies of local-mode excitation consider specifically the effect of exciting OH stretch overtones on the reaction rate constant. Most of the emphasis is on Hz + OD where we compare the reaction rate constant when OH is excited with five quanta the reaction H + HOD to that when the nonreacting OD is excited with the same amount of energy. We find that, although the rate constant is significantly enhanced even when the “wrong” bond is excited, the enhancement is 10-103 larger when the bond being broken is initially excited.
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Introduction There are surprisingly few studies, either theoretical or experimental, or state-to-state dynamics in bimolecular reactions involving triatomic or larger molecules.’ From a theoretical viewpoint, several obstacles to this exist, including the difficulty in developing realistic potential energy surfaces, the difficulty associated with defining stationary initial and/or final states for the polyatomics involved, and the computational complexity of the many atom reaction dynamics. The last of these problems largely restricts the choice of dynamics methods to the use of quasiclassical trajectory methods, and in this context, a few exploratory studies have been made. For example, Chapman and Bunker2 have studied CH3 + H2 CH4 + H, comparing cross sections at fixed translational energy for CH, out-of-plane bend excitation with that for H2 stretch excitation. Schatz and Elgersma, have studied the effect of CS2 stretch mode excitations in a collinear model of 0 + CS2 CO + CS, and Chapman4 has studied 0, and NO vibrational excitation in 0, NO NOz 02.All of these studies have suffered from the absence of good potential energy surface information for these systems and all but the second have used initial excited states which were probably not stationary eigenstates due to anharmonic and Coriolis coupling effects in the intramolecular Hamiltonian. Despite these limitations, the results of these three trajectory studies are interesting in that they show that reagent vibrational excitation is not always channelled into reaction coordinate motions in a way which follows simple i n t ~ i t i o n . ~ - ~ In this paper we consider the reaction dynamics of H H 2 0 OH H2, using the quasiclassical trajectory method to study the details of how reagent vibrational excitation influences reactivity and to look at product-state energy partitioning, H H 2 0 is especially interesting from the point of view of reagent vibrational excitation effects because the character of the H 2 0 vibrational states changes from normal-mode-like to localmde-like as the vibrational energy changes from near zero to that associated with OH stretch overtones. Thus a detailed study of mode specific chemistry in H + H20 may reveal how changes in mode character influence reactivity. H HzO also serves as a useful protype for studying reagent vibrational effects in other
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* Alfred P. Sloan Fellow; Camille and Henry Dreyfus Teacher-Scholar. 0022-3654/84/2088-297 1$01.50/0
polyatomic systems. In particular, this is the simplest system which can be used to study local-mode specificity, which is a subject of significant current interest in studies of unimolecular reaction^.^ Although no experimental studies of mode specific chemistry have been done yet for H + HzO (and it is likely that such studies will be difficult because H 2 0vibrationally relaxes quite rapidly), two pieces of experimental data for ground-state H + H 2 0 now exist with which to compare our results: thermal rate constants,6 and product OH vibration-rotational distributions.’ The latter are the results of recent laser photolysis studies wherein fast hydrogen atoms (relative translational energy 2.52 eV) were reacted with H 2 0 and the subsequently produced O H was observed by laser-induced fluorescence. Comparison of our trajectory results with these data should provide at least an approximate calibration of our other results for which no data exists. We can also make comparisons with the results of variational transition-state theory calculationss for this reaction on the same potential energy surface. The surface available for this reaction9 has been derived from high-quality ab initio calculation^^^^^ that are known to give fairly accurate thermal and state-resolved rate constants for the reverse OH + H2 r e a ~ t i o n . ~ , ~ J ~ A preliminary report of the results of our H + H20studies was given in ref 12. In that work, we presented cross sections for H (1) For a review, see Kimel, S . ; Speiser, S . Chem. Rev. 1977, 77, 437. (2) Chapman, S.; Bunker, D. L. J . Chem. Phys. 1975, 52, 2890. (3) Schatz, G. C. J . Chem. Phys. 1979, 71, 542. Elgersma, H.; Schatz, G. C. Chem. Phys. 1981,54, 201. (4) Chapman, S. J . Chem. Phys. 1981, 74, 1001. (5) Chandler, D. W.; Farneth, W. E.; Zare, R. N. J . Chem. Phys. 1982, 77, 4447. Reddy, K. V.;Berry, M. J. Chem. Phys. Lett. 1979, 66, 223. Faraday Discuss. Chem. Soc. 1979, 67, 188. (6) (a) Dixon-Lewis, G.; Sutton, M. M.; Williams, A. Symp. (Int.) Combust. [Proc.],loth 1965, 495. (b) Cohen, N.; Westberg, K., unpublished report. See also Dixon-Lewis, G.; Williams, D. J. Comp. Chem. Kine?. 1977, 17, 1. (7) Kleinermanns, K.; Wolfrum, J. Appl. Phys. B, in press. ( 8 ) (a) Truhlar, D. G.; Isaacson, A. D. J . Chem. Phys. 1982, 77,3516. (b) Isaacson, A. D.; Truhlar, D. G. Ibid. 1982, 76, 1380. (9) Schatz, G. C.; Elgersma, H. Chem. Phys. Lett. 1980, 73, 21. (10) Walch, S. P.; Dunning, T. H. J. Chem. Phys. 1980,72, 1303. Schatz, G. C.; Walch, S . P. Ibid. 1980, 72, 776. (11) Schatz, G. C. J . Chem. Phys. 1981, 74, 1133. Schatz, G. C.; Elgersma, H. In “Potential Energy Surfaces and Reaction Dynamics Calculations”; Truhlar, D. G., Ed.; Plenum: New York, 1981; p 311.
0 1984 American Chemical Society
2972
The Journal of Physical Chemistry, Vol. 88, No. 14, 1984
Schatz et al.
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t-ioioj
/,.,,I ‘TI e”
0.5
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S Figure 1. Potential energy along reaction coordinate for H + H 2 0 OH H2 showing the energies of all the normal-mode initial states with energies less than 1.6 eV and of certain product vibrational states.
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H 2 0 in different normal-mode excited states (all with zero rotational energy), focusing our attention on the effective reaction threshold energies. In the present paper, these normal-mode excited-state cross sections have been computed with much better statistical accuracy and we have studied the influence of initial H 2 0 rotational energy on the reaction cross sections. In addition, we have analyzed product energy partitioning and have examined local-mode excitation effects, including both H H 2 0 and the H2 O D system. Both experimentally relevant H H O D cross sections and rate constants will be presented. To summarize the rest of this paper, we present in section I1 a description of how the calculations were done, while the results are presented in sections 111, IV, and V. These results include rate constants and final state distributions for the ground vibrational state (in section 111), cross sections and rate constants for normal-mode excited states (section IV), and cross sections and rate constants for local-mode excited states (section V). Section VI summarizes our conclusions.
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11. Description of Calculations Many of the details of the trajectory calculation have been given previously.12 Figure 1 summarizes the energetic considerations associated with H H 2 0 O H H2 as obtained from the potential surface which we took from ref 9. The potential surface is characterized by a large (0.923 eV) classical barrier to reaction and an overall endoergicity of 0.659 eV. Isaacson et a1.* have examined vibrationally adiabatic potential curves for this system and found that the vibrationally adiabatic ground-state barrier location is fairly close to the classical barrier location, with a barrier height (relative to the reactant zero-point energy) of about 0.94 eV. The potential energy surface is also characterized by a relatively low barrier (0.406 eV) to hydrogen exchange (Le., H’ H 2 0 H’HO H). This is analogous to the low barrier to hydrogen atom exchange on LEPS surfaces for F + H2 and, as in the F H2case, it appears that the true hydrogen exchange barrier should be much higher. Because of this we will ignore those trajectories which undergo hydrogen exchange. One other feature which is not correctly described by the potential of ref 9 is the reaction path multiplicity. On this surface, reaction with only one of the two hydrogens on H 2 0is possible, and collisions with the other hydrogen lead snly to nonreactive scattering. To correct for this we have multiplied our H + H 2 0 cross sections by 2. This problem does not occur in calculating the cross sections
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(12) Elgersma, H.; Schatz, G. C . Int. J. Quantum. Chem. Symp. 1981, 25, 611.
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for H HOD H2 O D where the D atom is put on the nonreactive site. The states listed on the left side of Figure 1 are all the normal-mode excited states with harmonic energies up to 1.6 eV. These states correspond to harmonic vibrational quanta of 0.48, 0.21, and 0.49 eV for symmetric stretch, bend, and asymmetric stretch, respectively. Rather than consider all these normal-mode states in our trajectory calculations, we concentrate instead on the states associated with single-mode excitations, i.e. (000), (OlO), (020), (loo), (OOl), (030), (200), and (002). The reaction dynamics of combination tone excitations was briefly considered in ref 12, and in all cases it was found that the results could be understood in terms of an average of those obtained by using the fundamentals making up the combination. The trajectory calculations followed the usual three-dimensional Monte Carlo approach, with the sampling of initial H,O coordinates and momenta for the normal-mode calculations taken from the procedure described in ref 13. In this procedure the canonical transformations relating the good action-angle variables to the normal coordinates and momenta are determined by second-order classical perturbation theory using the zero angular momentum molecular Hamiltonian with the quartic force field part of the H 2 0 potential. Semiclassical eigenvalues are also determined to second order. This procedure leads to “good” action variables which are close to being constants of the motion for the lower eigenstates of H20. For the ground state, for instance, the good actions are constant to within f0.05 h for H 2 0 while the corresponding harmonic actions vary by h or more. Deviation of the perturbation theory actions from being good constants of the motion is more pronounced for the excited states, particularly those perturbed by Fermi resonance, so we have chosen not to consider states higher than (002) in this analysis. Even the states chosen are not entirely free from problems when the effects of H 2 0 rotation are included, as Coriolis coupling effects (other than those for zero rotational angular momentumI3) have not been included in the perturbation calculation. However, most of our results are for low initial angular momentum where Coriolis effects are small. Since the higher O H stretch overtones of H 2 0 are better thought of as a local-mode vibrations,14we have used a local-mode partitioning of the H 2 0Hamiltonian to define initial conditions in studying reactions with highly excited states (>2.0 eV). In this partitioning, the three modes are chosen to be 0-HI stretch, 0-H2 stretch, and bend, with atom H1 chosen to be the “reactive” H atom. The full Hamiltonian is then written as a sum of a zeroth-order local-mode Hamiltonian plus a coupling, and the effect of the coupling in generating initial conditions is neglected. The zeroth-order Hamiltonian is taken as -+
where HM is an OH Morse function, with De = 0.17577 au, re = 1.80809 ao, and be = 1.26058 ao-l, and Hb is a harmonic bend potential ‘12k(B- Bo)2 with force constant k = 0.716 mdyn k and 0, = 104.6O (i.e., the quadratic bend potential on our H20surface). While it is certainly the case that both the Morse and bend potentials could be modified to generate a somewhat better local-mode Hamiltonian (see, for example, ref 14), the Hamiltonian chosen does achieve the major desired effect, i.e., it generates OH stretch excited states for which the excitation remains localized primarily in the bond initially excited for suitably chosen initial states. This is not to say that there is no transfer of energy between the excited and unexcited bonds (about one quantum in a state initially having five quanta in OH stretch excitation appears to exchange rapidly with the other modes), but the degree of localization present should be adequate to determine the effects of this localization on reactivity. It should be mentioned here that the “true” stationary quantum eigenstates of H 2 0 are either symmetric or antisymmetric with respect to interchange of the two H’s irrespective of the amount (13) Schatz, G. C. Top. Current Phys. 1983,33, 25. (14)Stannard, P.R.; Elert, M. I.; Gelbart, W. M. J . Chem. Phys. 1981, 74, 6050, and references therein.
Dynamics of H
+ H20
-
OH
+ H2
The Journal of Physical Chemistry, Vol. 88, No. 14, 1984 2973 TABLE I: Ground-State and Thermal Rate Constants (cm3/( molecule s)) log k T, K calcd" exptlb
H+H20 -+ OH(v=O,j)tH,
c
1
500 600 700 800 900 1000
-18.5 -17.1 -16.2 -15.4 -14.9 -14.4
f 0.2 f 0.2 f 0.2 f 0.2 f 0.2 f 0.2
-18.9 -17.4 -16.3 -15.5 -14.9 -14.3
f 0.3
f 0.3 f 0.3 f 0.3 f 0.3 f 0.3
Calculated rate constant from the ground vibrational state cross sections. *Measured thermal rate constant from ref 6b.
i
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Figure 2. Experimental and theoretical cross sections for H + H 2 0 OH(v = 04) Hz vs. OH vibrational quantum numberj. Normalization on cross section is such that all cross sections have been divided by the
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TABLE II: Product Energy Partitioning for H + H 2 0 QR = 1.0 f 0.2 a: average product translational energy = 1.3 1 eV
OH
+ HZa
initial reactive orbital angular momentum: ( I ) = 31.3 scattering angle ( 0 ) = 94.8O OH
total reactive cross section. The trajectory results (squares with error bars and solid line which is higher at largej) have been calculated by using a weighted average of Eo = 2.519 eV and Eo = 2.090 eV as in Table 11. The experimental results (triangles and solid line passing through the triangles) are from ref 7. The experimental solid line is a spline fit to the measured points. The theoretical solid line is a Legendre smoothed fit to the histogram cross sections. of H 2 0 vibrational excitation. Thus the localized OH stretch excited states that have been defined above classically are not stationary states of H 2 0 . We will use them anyway, however, as they do provide an important illustration of local-mode selectivity effects. Moreover, the water isotope HOD, which we shall also study, provides a real example of separately localized stretch modes (OH and OD). The comparison of localized H,O and HOD will indicate how important are isotope effects on mode selectivity. Specific parameters used in the normal-mode excited-state trajectory calculations are similar to those given in Table I of ref 12, except that the number of trajectories sampled has at most energies been doubled compared to what is in that table. In addition, we have computed cross sections for several rotationally excited states of H20. As will be discussed in section IV, the initial rotational state dependence of the reactive cross sections is quite weak for states which are important in a thermal Boltzmann average below 1000 K, so in computing rate constants, we have used only the zero rotational energy results. The formula used to determine initial vibrational state resolved thermal rate constants was adapted from standard expressions and is as follows:
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H2
N1
5 - 0-0
h N
0
a
Y
0
O
I
1-
Eo (eV) Figure 3. Quasiclassical reactive cross section (in bohr2)vs. translational energy (in eV) for H + H 2 0 with HzO initially having zero rotational kNIN2N3= ( 2 k T ) - 3 / 2 ( ~ ~0)~- 1E / o2 ~ - ~ ~ ~ ~ ~ dEo Q N , (~2 )~ ~ , (energy E o and ) vibrational quantum numbers (000) (circles), (100) (squares),
where QN1N2N, is the integral cross section associated with an initial translational energy Eo and a H 2 0 vibrational state labeled by the quantum numbers N , , N2,N3. Note that no reference to initial rotational state appears in this formula because of the assumptions concerning rotation given in the previous paragraph. Evaluation of rate constants for local-mode excited states was accomplished with the same expression but with the corresponding local-mode cross sections substituted. The evaluation of the energy integral above was accomplished to the function by fitting QNIN2N3(EO) where EoT is the effective cross section threshold for each state NlN2N3.A nonlinear least-squares procedure was used to optimize the parameters Qo, EoT,and x. Although the above formula is not very flexible, the relatively poor statistics associated with many of our trajectory results near threshold necessitated the use of a simple functional form such as that chosen. The poor statistics arose from the extremely low reaction probability (rarely more than 1%) associated with most of the cross section calculations. This low probability was obtained despite attempts to optimize
and (200) triangles. Representative 1-u error bars are displayed for representative cross sections. reactivity through importance sampling of the initial impact parameter.I5
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111. Results for Ground-State H H 2 0 As mentioned in the Introduction, all of the experimentally HzO at present relates relevant information concerning H essentially to the ground vibrational state of H 2 0 . This includes the thermal rate constants and the product OH vibration/rotation distribution. Information about these as obtained from our trajectory calculations is summarized in Tables I and I1 and Figure 2. Table I gives the rate constant at selected temperatures as obtained from the ground-state (000) cross section which is plotted in Figure 3, and compares it with the experimental thermal rate constant of ref 6. The temperature range considered is limited
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(15) Redmon, M. J.; Bartlett, R. J.; Garrett, B. C.; Purvis, G. D.;Saatzer, P. M.; Schatz, G. C.; Shavitt, I. In "Potential Energy Surfaces and Dynamics Calculations"; Truhlar, D. G., Ed.; Plenum: New York, 1981; p 771.
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The Journal of Physical Chemistry, Vol. 88, No. 14, 1984
to those temperatures where the influence of vibrationally excited states can be neglected. The only direct measurement of the H + H20rate constant is given in ref 6a. This is in good agreement with values determined by applying microscopic reversibility to the O H Hz rate constant as summarized in ref 6b. The result of ref 6b is what is summarized in Table I, and we see that the trajectory rate constant is higher than experiment at low temperatures and lower at high temperatures, with the 1-a error bars overlapping at all temperatures. Unfortunately, it is not practical computationally for us to reduce the trajectory error bars further to determine with certainty just how far off the trajectory results are. However, the observed lower activation energy in the trajectory results (0.81 vs. 0.88 eV at 500 K) is consistent with the observation that trajectory calculations for endoergic reactions often give low reaction threshold energies because of the absence of product zero-point energy constraints in classical mechanics. A least-squares estimate of the threshold energy in eq 3 based on fits to our cross sections is 0.81 eV for the ground state, which is below the adiabatic barrier of 0.94 eV. Although the adiabatic barrier is not a rigorous threshold constraint, recent quantum scattering calculationsI6 show that true thresholds are often closer to the adiabatic barriers than to the trajectory thresholds. In addition, the success of variational transition-state theory calcul a t i o n ~ depends ’~ in part on the closeness of reaction thresholds to adiabatic barriers. We also found that in many of our trajectories either the O H or Hz product would end up with less than zero-point energy. TruhlarIs has suggested that, for atom-diatom reactions, one way to correct the trajectory thresholds is to set the cross section equal to zero whenever the total energy is below the product zero-point energy. This has been successful in some a p p l i c a t i ~ n s , ’ ~but J ~ in the present context it is not since the threshold translational energy is constrained by this procedure only to be greater than 0.56 eV, well below 0.94 eV. A similar problem was encountered in ref 20 and it appears that the reasons for this behavior are analogous. Another idea is to reject trajectories which produce a product diatomic with less than zeropoint energy. Unfortunately it is not clear whether this should be applied to each diatomic separately or just to that whose bond is being formed. The first of these choices may seem more appealing, but in a few calculations which we tried using this approach, unphysically high ground-state threshold energies were obtained. Evidently, more work on this topic is needed. Table I1 and Figure 2 present information about the vibration/rotation distributions in the products OH and H2 for an average of the two translational energies (2.519 and 2.090 eV) which corresponds to that used by Wolfrum and Kleinermanns in their laser photolysis experiment^.^ The information in Table I1 includes the average vibrational and rotational quantum numand energies ( ( E , ) and ( E j ) )for each diatomic, bers ( ( u ) and (j)) the average product translational energy, the average scattering angle ( ( e ) ) , and the average of the reactive initial orbital angular momentum ( ( I ) ) . It should be noted that the average quantum numbers and energies were calculated from the histogram cross sections rather than from the individual trajectories. This is of minor consequence for the rotational information where both averages are very similar but strongly influences the vibrational information where the trajectory average (0) is -0.2 for O H and 0.0 for Hz. Table I1 shows that most of the energy available to the products ends up as translational energy (61%), with a surprisingly large amount (26%) going to product H2rotation, and relatively smaller amounts going to Hz vibration, O H vibration, or O H rotational excitation. It is also noteworthy that the molecule containing the newly formed bond (i.e., H2) receives 6 times as much energy as
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(16) Schatz, G. C. J . Chem. Phys. 1983, 79, 5386. (17) (a) Garrett, B. C.; Truhlar, D. G. J . Phys. Chem. 1979, 83, 1079. 1980,84,682 (E). 1983,87,4553 (E). (b) Truhlar, D. C.; Isaacson, A. D.; Skodje, R. T.; Garrett, B. C. Ibid. 1982, 86, 2252. 1983, 87, 4554 (E). (18) Truhlar, D. G. J. Phys. Chem. 1979, 83, 188. (19) Miller, J. A. J . Chem. Phys. 1981, 74, 5120. (20) Blais, N. C.; Truhlar, D. G.; Garrett, B. C. J. Chem. Phys. 1983, 78, 2363.
Schatz et al. the molecule containing the “old” bond. The value of ( I ) is consistent with an impact parameter equal approximately to the O H distance in H20. This, plus the preference for sideways scattering in the angular distributions and the lack of any complex forming trajectories, suggests that many reactive collisions are intermediate in character between stripping and rebound collisions. Comparison with experiments can be made in three ways. First, the measured total reactive cross section is 0.9 f 0.4 ao2,which is in good agreement with our value (Table 11) of 1.0 f 0.2 ao2. Second, in Figure 2 we see that our OH(u=O) rotational distribution is in reasonable agreement with that from ref 7. No error bars are given for the experimental data, but it seems likely that the two rotational distributions agree to within their respective (admittedly large) uncertainties. In presenting the theoretical results, we have applied a Legendre smoothing procedurez1 to generate the solid curve, and it is probably this result rather than the histogram cross section which is most meaningfully compared to experiment. Figure 2 indicates that this solid curve is a bit broader than the experimental distribution, and from Table I1 we find that the calculated ( E j ) for O H is 5% of the energy available to the products rather than the 3% estimated in ref 7. Relative to the differences between the OH rotational distributions observed in H 0, OH 0 and in H HzO OH H2,7 the presently observed differences between theory and experiment are minor. The third comparison between theory and experiment in these hot atom results refers to O H vibrational excitation. In this case, an upper bound of 0.1 was estimated’ for the ratio of u = 1 to u = 0 reactive cross sections. This implies that (0) G 0.1, which is in agreement with our estimate ( u ) = 0.02 in Table 11. Let us now discuss the mechanism whereby O H rotation receives only 5% of the energy available to the products whereas H2 rotation receives 26%. To facilitate this discussion we label the atoms as HI + HzOH3 HIHz OH3. Kleinermanns and Wolfrum7 have used an impulsive model to explain the OH, rotational distributions in which the repulsion between the H2 and 0 atoms in the bond being broken is considered to exert torque on the OH3 product directed along the breaking O-Hz bond. By taking the H20H3bond angle to be 104O (the saddle point value) at the point where the torque is exerted, they calculate the fraction of product energy going to OH3 rotation as 0.3%. Although this value is low compared to either our trajectory results or experiment, it does show how kinematic effects (namely, the short 0 to center of mass of OH3lever arm on which the torque is exerted, and the small mass of the departing H2 atom) greatly restrict how much angular momentum can appear in OH3 rotation due to repulsion between the products. It also suggests one reason why the reaction H + 0, O H 0 has a much hotter O H rotational distribution, namely, that the departing atom is 0 rather than H , so more torque is exerted. To further study this problem, we have examined individual H1 H2OH3 OH3 HIHztrajectories to determine the extent of OH3and HIHzrotational excitation while the reaction proceeds. For OH3, we find that in accord with the Kleinermanns and Wolfrum model a certain portion (sometimes all) of the rotational excitation occurs while the products depart. However, most trajectories also exhibit a stripping mechanism for rotational excitation, wherein the instantaneous angular momentum in OH3 due to H20H3bending at the time of reaction is frozen in when the incident HI atom strips off the H, atom of H2OH3. It is relatively easy to model this stripping mechanism if we approximate the 0 atom as infinitely heavy. In this case, the instantaneous angular momentum of OH3 due to H2OH3 bending is
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joH N mHrOHZb (4) where mH is the H atom mass, rOH is the OH3distance in H20H3, and 0 is the time derivative of the H201-13bending angle. Let us use a harmonic oscillator mode1 for e, i*e*
e = -(h wb/h,)l/zsin
wbt
(21) Schatz, G. C. J . Chem. Phys. 1980, 72, 3929.
(5)
Dynamics of H
+ H20
6
-
I
OH
+ H,
The Journal of Physical Chemistry, Vol. 88, No. 14, 1984 2975
,
I
N2
5-
0-0
4-
0-1 a-2
h N0
m
v
+
0-1 h N
-3
0
m
3-
Y
U
U
2 1
0 0
0.5
1.o
1.5
Eo (eV) Figure 4. Cross section vs. translational energy as in Figure 3 but for H 2 0 in the normal-mode states (000) (circles), (010) (squares), (020) (triangles), and (030) (crosses).
E, (eV) Figure 5. Cross section vs. translational energy as in Figure 3 but H 2 0 in the normal-mode states (000) (circles), (001) (squares), and (002) (triangles).
where &, is the bending moment of inertia (roughly mHroH2),wb is the bend frequency, and t is the time at which the strip occurs. Equation 5 assumes that the bend initially has zero-point energy. Substituting ( 5 ) into (4) and averaging over t gives the average energy in OH rotation as
citation are 0.59 eV for (001) and 0.36 eV for (002), corresponding to -0.22 eV or 45% per quantum of asymmetric stretch excitation. The greater efficiency of symmetric stretch excitation in lowering the reaction cross section was previously noted in ref 12 where it was argued that energy in the higher frequency asymmetric stretch mode would tend to remain locked up adiabatically as vibrational energy along the reaction coordinate more readily than that in the symmetric stretch mode. In addition, the asymmetric stretch mode frequency does not change significantly from the reagents to the saddle point along the reaction coordinate while the symmetric stretch mode frequency decreases substantially, thereby freeing up substantial adiabatic energy to motion along the reaction coordinate. Examination of Figure 4 indicates that the threshold energy drops as the bend mode is excited from 0.81 eV for (000) to 0.59 eV for (010) to 0.38 eV for (020), corresponding to about 0.2 eV or 96% per quantum of excitation. This extremely high efficiency of threshold reduction does not persist for higher overtones, however. Figure 4 indicates that the (030) threshold is about the same as for (020), or at least to within the rather substantial error bars associated with each cross section. Although considerable effort was made to improve the precision of the calculations, the very low reaction probabilities made it very difficult to determine just how close the (020) and (030) thresholds are. If the efficiency of threshold reduction is really 96% for bend excitation then this would be the most efficient mode to excite in enhancing the rate constant. One possible reason for this would follow from our discussion above concerning high-frequency motions and vibrational adiabaticity. Since the bend is the lowest-frequency motion, it would be expected to be the least adiabatic and thus the most active in lowering thresholds if changes in adiabatic frequencies along the reaction coordinate are not too important. Unfortunately there is another explanation for this which is less desirable. This is that exciting the bend initiates the flow of a substantial amount of energy out of the zero-point symmetric stretch motions into the bend and then into the reaction coordinate motions during the reaction due to the Fermi resonant coupling between the symmetric stretch and bend. If important, this would be an artifact of the use of trajectories, since zero-point energy should not be convertable into energy in other degrees of freedom as it leads to violation of the uncertainty principle. We have looked at product energy partitioning in these collisions and find that essentially all of the initial bend excitation ends up as product translation. This differs from what is obtained for the symmetric or asymmetric stretch excited states where some of the energy ends up as product vibration, but either of our two explanations for the influence of bend excitation on reaction threshold is consistent with this behavior. To study the influence of the initial H,O rotational excitation on the reactive cross sections, we have calculated cross sections for H + HZO(NlN2N3) in various rotational states specified by
This formula predicts (EjoH) = 0.05 eV, which is about half of what is calculated in the trajectories (Table 11). Note that the value of j,, in eq 4 is consistent with what we observed in trajectory calculations of j,, for isolated H20H3. Our H1 + H20H3 trajectories also indicate that the product H,Hz rotational excitation comes from a combination of product repulsion and reagent excitation effects. In this case, the reagent excitation comes from the orbital angular momentum of the incoming HI atom about the H2 atom. Crudely speaking, this angular momentum is given by
j" = r"P0 sin (7) where Po is the initial HiHzrelative momentum, "r is the HIH, distance at the point of "reaction", and a is the angle between the initial Hi atom momentum vector and the HIHzinternuclear vector. Taking "r = 1.4 a. and averaging j,, over a hemisphere (0 < a < r / 2 ) with a cos2 a weight to favor reaction near a = 0 gives ( j H H =)10.8. This is similar to but somewhat larger than that found in Table 11. That it is larger is reasonable since the cos2 a weight factor probably overestimates contributions to reaction from a's near r / 2 . In considering the H 0, reaction, we would expect that this reagent orbital angular momentum model will also be appropriate and that it will explain at least part of the OH rotational excitation.
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IV. Normal-Mode Excited-State Cross Sections and Rate Constants In Figures 3-5 are presented cross sections as a function of translational energy for the normal-mode excited states of H H,O. Included in each figure are the fits of these cross sections to eq 4, along with error bars on representative points. The ground-state (000) cross section is reproduced in each figure for reference, with Figure 3 showing the cross sections for the symmetric stretch excited-states (100) and (200), Figure 4 the bending states (OlO), (020) and (030), and Figure 5 the asymmetric stretch excited-states (001) and (002). Each of Figures 3-5 shows a steady increase in reaction cross section and steady decrease in reaction threshold energy with increasing vibrational excitation in each mode. For symmetric stretch excitation, the threshold energy associated with the least-squares fit drops from 0.81 eV for (000) to 0.49 eV for (100) to 0.18 eV for (200). This shift of -0.31 eV per quantum of excitation is 65% of the energy added to the reagent H 2 0 per quantum. The analogous thresholds for asymmetric stretch ex-
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2916 The Journal of Physical Chemistry, Vol. 88, No. 14, 1984
Schatz et al.
-:Of0
Pbil I
0.80
1.oo
I
,
1.20
I
I
1.40
I
1000
T (K) 700
500
4eO
1
1.60
Y 0,
E, (eV) Figure 6. Cross section vs. translational energy as in Figure 3 but for HzO in the (000) state and with rotational quantum numbers jH20= kH20 = 0 (circles),jH20= kHlo = 5 (triangles), and j,, = kH20= 13 (squares). Error bars for the jH20= kH,O = 0 results are given. Those for the other
t
states are comparable. the quantum numbers jH20 and kHIOwhere j H I Ois the HzO rotational angular momentum and kH20its projection along the czu symmetry axis of HzO. A description of how the rotational sampling is actually done is given in ref 13. Figure 6 presents some typical results, in this case the cross sections for HzO(OOO) in thejH20 = kH20= 0, 5 , and 13 states. These states correspond to rotational energies of 0, 0.09, and 0.61 eV, respectively. jH20 = 0-5 represents the approximate range of states important in thermal rate constant calculations while jHIO = 13 corresponds to rotational excitation which is slightly larger than one quantum of symmetric or asymmetric stretch excitation. Figure 6 indicates values agree to within their that the cross sections for all three jH20 respective error bars, indicating that the effect of rotation can be neglected, as we did in section 111, in calculating thermal rate constants. Similar results were obtained in calculations done with other initial vibrational states. The j,, = 13 result is especially interesting, for the relative insensitivity of cross section to rotational excitation exhibited by jHzO = 13 stands in sharp contrast to the strong influence of similar amounts of vibrational excitation in Figures 3-5. In Figure 7 we present thermal rate constants obtained from eq 2 using the cross sections in Figures 3-5. Approximate uncertainties in log k at each temperature are all about 0.3. The ground-state result from Table I is also included for reference. The figure indicates that the rate constant is enhanced by factors of 60, 200, and 2500 at 500 K upon excitation of the fundamentals (OlO), (OOl), and (loo), respectively. These enhancements arise mostly from decreases in activation energy, with E, being 0.81, 0.62, 0.60, and 0.49 eV for (000), (OlO), (OOl), and (loo), respectively. The corresponding A factors are 4.3 X lo-”, 3.6 X lo-”, 7.0 X lo-”, and 6.7 X lo-“ cm3/(molecule s). The enhancements associated with the overtones (020), (002), and (200) are all about the squares of those for the corresponding fundamentals and are again mostly due to activation energy reduction. The lowering in activation energy for excitation of the asymmetric stretch state (001) has previously been estimated by Truhlar and Isaacsonsa as 0.09 eV using an extension of variational transition-state theory. This is only about half of the 0.21-eV value (i.e,, 0.81-0.60 eV) which we calculate on the same potential energy surface using classical trajectories. Presumably the difference between these two numbers is due to vibrationally nonadiabatic effects, but it is not clear at this point how accurately the influence of such effects is described by trajectories. V. Local-Mode Excited-State Cross Sections and Rate Constants In this section we consider cross sections and rate constants associated with local-mode excited overtones of HzO and HOD as defined in section 11. Specifically we confine our discussion
\u
(ODO)\
1.o
2.0
1000KIT Figure 7. Arrhenius plot of rate constant vs. inverse temperatures for H HzO with H,O initially in the normal-mode states (000), (OlO), (OOl), (loo), (020), (002), and (200).
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12
I
A
A
A
’O 8I
A
0 . 1
n n
Y
(ooo)9 m
r-L
Eo (eV) Figure 8. Cross section vs. translational energy for H HzO as in Figure 3 but for HzO initially with zero rotational energy and in the local-mode
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states (000)1, (500)1, and (050)1. to HzO in the (500),and (050),states and HOD in the (5Oo)l and (070)1 states. The notation used here is that the first quantum number is the “left” O H stretch, the second is the “right” O H stretch, and the third is the bend, with the subscript “1” used to make it clear that a local-mode state is implied. In HOD, the OH bond is chosen to be “left”, and the OD to be “right”. We also choose the “left” H atom always to be the reactive one on our potential surface. The vibrational excitation energy associated with (5Oo)l and ( 0 5 0 ) in ~ H 2 0 is 2.66 eV, while that for (5Oo)l in HOD is 2.58 eV, and for (070)1in HOD is 2.63 eV. The HOD states were chosen to be close in energy so that the effect of similar amounts of excitation in the reactive and nonreactive bonds could be studied. Likewise in HzO, the effect of localizing energy in both the reacting and nonreacting bonds was modeled. The cross sections for H HzO and H + HOD in the above-described local-mode excited states, all for jH20= 0, are given in Figures 8 and 9, respectively. Included for reference in each plot are the corresponding vibrational ground-state cross sections. The local-mode Hamiltonian was used in defining these ground-state cross sections, but comparison of Figures 3 and 8 indicate that the local- and normal-mode ground-state cross sections are very similar.
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Dynamics of H
+ H20
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OH
+ H2
The Journal of Physical Chemistry, Vol. 88, No. 14, 1984 2911
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+ HOD Hz + OD with HOD initially with zero rotational energy and in the local-mode states (OOO),, (500)1, and (070),. Figure 9. Cross section vs. translational energy for H
Examining Figure 8, we see that excitation of the (500)l state in H H 2 0 leads to a cross section which has a slightly lower threshold energy and a much larger magnitude (factor of 10) than (050),. Both cross sections have much lower threshold energies than the ground state; in fact the threshold shifts relative to the ground state are larger than any of those observed in Figures 3-5. The results for H H O D in Figure 9 are similar to those for H H 2 0in Figure 8, with the (500)~cross section much larger in magnitude (factor of - 6 ) and with a lower threshold than (070)1. The only really important difference between HOD and H 2 0is that the threshold for (500)1is about 0.3 eV below (070)1 in HOD while that for (SOO)l in H20 is only 0.03 eV below (050)l. This indicates that H O D shows changes in reactivity which are more mode specific than H 2 0 , a result which appears to be due to the frequency difference between the OH and O D bonds in HOD. This frequency difference causes intramolecular transfer of energy between the excited O H or O D local modes and the other degrees of freedom to be less efficient in HOD than in H20. We have confirmed this in trajectory comparisons of isolated HOD and H20. Figure 10 presents rate constants calculated from the local-mode cross sections in Figures 8 and 9. Here we include the normalmode ground-state rate constant from Table I for reference. Evidently, the rate constant enhancements relative to the ground state for all of the excited states are quite substantial (over lo7 for HOD(500)). More relevant to possible experiments, however, is the relative enhancement associated with exciting a reactive or nonreactive bond, and in the case of H H O D we find that k for (500)1is 1-3 orders of magnitude higher than for (070)1. The corresponding difference between (SOO), and (OSO), for H + H 2 0 is about 1 order of magnitude. For H H 2 0 , essentially all of the rate constant difference is due to a larger preexponential factor (reflecting the cross section magnitudes in Figure 8) while for H HOD, both preexponential factor and activation energy are appreciably different.
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VI. Conclusions In this paper we have explored three interrelated aspects of the state-to-state chemistry of H + H 2 0 O H Ha: the groundstate product-state distributions, the influence of normal-mode excited states on reactivity, and the influence of local-mode overtone states on reactivity. In studying the ground state kinetics we found that our trajectory rate constants agreed with experiment to within their respective uncertainties and that the product OH rotational distributions are in agreement with recent laser photolysis results. Both of these conclusions suggest that the use of classical dynamics to study H H 2 0 on our assumed potential surface is reasonable. They do not, however, imply that our
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1.o
2.0
1000KIT
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Figure 10, Rate constant vs. inverse temperature for H H20 in the (000) normal-mode state (dotted line), H H20in the (500), and (050)l local-mode states (solid lines), and H HOD in the (SOO), local-mode states (dashed lines).
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excited-state trajectory results are necessarily free from error. In fact, systems are known16 where excited-state trajectory cross sections are less accurate than ground-state cross sections. Our normal-mode excited-state results show that the reaction cross section and rate constant are enhanced by exciting any of the vibrational modes of H20. Most of that enhancement is due to reduction in threshold energy, but the efficiency of threshold reduction varies significantly from mode to mode, with the bend being the most efficient mode and the asymmetric stretch the least. A qualitative understanding of these results can be developed on the basis of how vibrational adiabatic effects transform reagent excitation either into product excitation or into motion along the reaction coordinate. Unfortunately, there is also the possibility of artifacts in some of these results due to the inability of classical mechanics to constrain at least zero-point energy in each mode during the reaction. This problem is relatively easy to correct in atom-diatom collisions but is fraught with ambiguities in application to polyatomic systems. Further work on this is needed in order to establish the reliability of classical mechanics for studying state-to-state reactions involving polyatomics. Our local-mode excited-state results were aimed at answering an important question: is there a significant difference between the reaction rate constant when reaction involves breaking the bond which is initially excited vs. that involving an unexcited bond in the same molecule? Our results for H + HOD H2 + OD suggest that there indeed is, with a 1-3 order of magnitude difference between reaction with the excited O H vs. that with an unexcited O H in the presence of excited OD. This mode specificity was more pronounced for H HOD than for H H20 as might be expected in view of the frequency difference between the modes in HOD which is absent in H,O. It is interesting to note, however, that, even for H HOD, reaction with the unexcited O H bond when O D is excited with seven quanta is enhanced by several orders of magnitude over the ground-state rate constant. Thus even for well-localized bonds, mode specific effects are not “all or nothing”.
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Acknowledgment. This research was supported by NSF Grant CHE-8 1 15 109. Registry No. H, 12385-13-6; H2, 1333-74-0, H20, 7732-18-5.
