A new model for calculating the average energies in product rotational, vibrational, and translational degrees of freedom in photodissociation reactions involving substantial exit barriers is presented. The model is discussed as a logical extension of the impulsive model that includes statistical partitioning of energy in excess of the barrier. Calculations are performed using the new model on the photodissociation of acetone and acetic acid. The results are compared to experiments and to other models for unimolecular dissociation.This work has been published as an appendix to a paper on Acetone photodissociation (by the authors above, Dave Blank, Cheryl Longfellow, and Y. T. Lee). Look for it in the Journal of Chemical Physics, volume 102, number 11, pp. 4447-4460.
Another class of models seeks to explain the experiments in terms of a statistical distribution of energy either in the products or at the transition state. For experiments which involve large barriers to recombination, however, these models fail to reproduce the interesting physics of the dissociation event. It seems then, that neither set of models has satisfactory explanatory (or even predictive) power for many possible photodissociation events.
We seek a model that will reflect both the impulsive nature of reactions with large barriers to recombination as well as the statistical distribution of energy in excess of the barrier height. We also want to restrict our model so that it requires only readily available information about the transition state and the reactants. Commonly known quantities are the normal mode frequencies of the reactant and product molecules, geometries of the transition state, and the forward and reverse barrier heights. We believe that the model presented in this paper represents a simple extension of the impulsive and statistical models that are currently in common use. We hope that the physical picture that we present will allow a physical understanding of the unimolecular reaction that gives qualitative agreement with experiments.
The barrier impulsive model (BIM) is conceptually very simple. The available energy is separated into two "energy reservoirs", one which is denoted statistical and the other impulsive,
Eimp(tot) is chosen to be the height of the exit barrier (Figure 1). In this way the model reduces to the impulsive model at the dissociation threshold and to a simplified statistical model in the absence of a barrier. Furthermore, since the translational energy of the products from the impulsive reservoir is fixed by the barrier height, the total translational energy increases with available energy staitistically. This behavior is consistent with experimental observation.(3) The partitioning of energy into product rotation, vibration, and translation in each reservoir independently conserves linear momentum, angular momentum, and total energy. The impulsive reservoir can be partitioned among the fragment degrees of freedom according to either the soft or the rigid fragment models. The energy in the statistical reservoir is partitioned by a new method that is outlined in the section II. The average energies from each reservoir are combined to obtain the final R,V, and T energies of the products. There are a number of inherent assumptions in this model. We assume that once the molecule is beyond the transition state the localized release of the potential energy from the exit barrier can be adequately described by the impulsive model. In order for the statistical treatment of Estat(tot) to be justified, the redistribution of vibrational energy must occur prior to dissociation but cease at the moment of the impulsive energy release.
We now consider the photodissociation of a polyatomic molecule A-B. The initial bond cleavage, produces fragments A and B which can themselves be polyatomic. We assume that neither fragment is electronically excited upon dissociation and that the A-B parent contains no internal energy prior to absorption of the photon.
Other statistical theories predict product state distributions using information about the product states themselves. The prior distribution,(5) and Phase Space Theory (PST)(6) rely solely on information about the product states to partition energy. A significant problem with these methods is that they tend to overestimate product rotational excitation above the vibrational threshold(7) while underestimating product vibrations.(8) This has been understood in terms of the vibrational modes of the parent molecule that can develop asymptotically into both product rotations and vibrations. The Separate Statistical Ensembles theory (SSE)(9) attempts to correct these deficiencies by using some of the information about energy partitioning in the parent molecule to obtain product state partitioning, but below the vibrational threshold, SSE and PST do not differ at all.
The basic assumption of our method of partitioning the statistical reservoir is that the energy is distributed statistically in the parent molecule. The breaking of the bond is viewed as an instantaneous event, which prevents any further rearrangement of energy. Following this reasoning, it makes sense to partition the statistical energy into T, R, and V using information only from the parent, and not information from the products as in PST and the prior distribution.
The method we use to divide the statistical reservoir into T, R, and V is very similar to the way it is done in SSE. Product vibrations can develop out of all parent vibrational modes, while product rotations and translations develop only out of those modes of the parent molecule that disappear during the course of the reaction. A schematic of this idea is shown in Figure 3.
We utilize three ensembles. The vibrational ensemble includes all vibrational modes of the parent and overlaps with the other two ensembles. The rotational ensemble includes those disappearing modes that lead to rotational excitation of the products. These modes can include methyl torsions, skeletal bends, etc. The third ensemble is a translational ensemble that includes modes that disappear into product translations. Identification of which modes belong in which ensemble can be a subtle matter for large parent molecules, but three basic rules can be easily applied:
where rP , rR , and rT are the densities of states for the P (parent), R, and T ensembles of parent vibrational modes. The P ensemble is made up of all vibrational modes of the parent molecule. In the harmonic approximation, the densities of states can be computed easily using the Beyer-Swinehart algorithm. The expressions for 
and 
are similar to the one for the rotational energy that is given above.
Essentially, we are assuming here that the energy above the barrier is going to be distributed statistically between the three ensembles, where the ensemble for vibrations can sample from the other two ensembles. The statistics (and hence the energy distribution) are governed by the vibrational frequencies of the parent as well as a judicious choice of modes for membership in each ensemble.
Dividing 
between the two fragments is easily accomplished by conserving linear momentum:
is similarly partitioned by requiring conservation of angular momentum:
In these equations, the moments of inertia, IA and IB, are calculated by approximating both fragments are spherical tops, with moments of inertia that are the averages of the real moments of inertia for that fragment.
The only remaining difficulty is to divide the vibrational energy from the statistical reservoir between the two fragments. It seems reasonable to view the impulse as an instantaneous event, so that the energy is frozen in the parent modes. The energy in the modes that develop into fragment vibrations should then be assigned to the appropriate fragment. Identifying these modes and obtaining their frequencies seems to be impossible for all but the simplest of molecules. By approximating the frequencies of these modes by the frequencies of the fragment modes themselves, one obtains the following expression for the vibrational energy partitioned into fragment A from the statistical reservoir:
where rA and rB are the vibrational densities of states of fragments A and B respectively.
Partitioning of the statistical reservoir is made with a number of important assumptions. We use vibrational densities of states in the harmonic approximation, and we assume that both of the fragments can be well approximated by totally symmetric tops. The harmonic approximation to the vibrational density of states does moderately well, due in part to fortuitous cancellation of the effects of coupling and anharmonicity. Anharmonicity tends to raise the density of states at energies near the dissociation limit, and coupling between modes tends to separate nearly degenerate vibrational levels. The cancellation of the two effects makes the harmonic approximation to the density of states a reasonable approximation.
The approximation of both fragments as spherically symmetric tops is a necessary approximation if we are trying to satisfy conservation of angular momentum while being subject to the constraint of having minimal information about the geometry of the transition state. An additional approximation is that the parent is assumed to have no rotational motion prior to the dissociation, which is a good approximation for the rotationally cold conditions of most molecular beam experiments.
The available energy is therefore partitioned only between fragment rotation and translation, where g is the relative velocity of A and B.
Since the dissociation must conserve angular momentum (initially assumed to be zero) the translational energy is constrained as follows:
where bA and bB are the exit impact parameters. Hence, the translational energy of the products predicted by the rigid fragment model is intimately dependent on the choice of the dissociative geometry. Once the translational energy has been determined, the rotational energy of the fragments can be expressed in terms of ETimp(tot),
The two reservoirs are then combined to give average translational energies for the two fragments as follows:
Similar equations are used for rotational and vibrational energies.
For the impulsive reservoir of the BIM model, a barrier height of 13.4 kcal/mole was used based on the measurements of Zuckermann et al.(12) The rigid fragment extension, described in section IIB, was chosen with a non-planar dissociative geometry consistent with the geometry of the 1,3(n,p*) excited state. The impulse was assumed to be through the C3v symmetry axis of the CH3 group and therefore, resulted in no CH3 rotational excitation. Both the soft fragment impulsive model (SFIM) and RFIM calculations utilized the same exit impact parameters and moments of inertia as were used in the BIM calculation. The statistical reservoir was partitioned using ground state acetone vibrational frequencies(13) as an approximation to those of the 1,3(n,p*) excited state. The assignments of the vibrational modes were also taken from ref 10. Of the 24 modes of acetone, 6 disappear upon dissociation, evolving into product translation and rotation. We assigned 2 of these modes the symmetric and antisymmetric C-C stretch to the translation ensemble. Two methyl rocking modes, a methyl torsion and the C-C-C skeletal bend were assigned to the rotational ensemble. The Beyer-Swinehart algorithm was used to calculate the vibrational density of states for all ensembles. The moments of inertia for the CH3CO fragment were determined from the ab initio equilibrium geometry calculated by Baird and Kathpal.(14) Vibrational frequencies were taken from RRKM calculations of Watkins and Word.(15)
Figure 4 shows the comparison of the two limiting impulsive models, SFIM and RFIM, and BIM with the experimentally determined average translation energies at three wavelengths. The predictions of RRKM theory is also shown as a representative barrierless statistical calculation. Both pure impulsive models rise linearly with available energy as expected. In acetone photodissociation at 266 nm (Eavail=23.5 kcal/mole) the average translational energy was measured to be 13.9 kcal/mole and the SFIM prediction is 12.8 kcal/mole.(16) Although the SFIM calculation underestimates <ET> at 266 nm it overestimates it at 248 nm where the SFIM predicts 16.9 kcal/mole but the experimental <ET> is only 14.1 kcal/mole.(17) At both these wavelengths the RFIM partitions far too much available energy into translation. It is interesting to note that at an available energy near 14 kcal/mole, assuming the trend continues, then the product translational energy would agree more with the RFIM prediction. The barrier impulsive model successfully predicts the magnitude of <ET> at each wavelength but also shows the correct dependence of <ET> on the available energy. There are several points worth commenting on. The first is the choice of the rigid fragment model for treating the impulsive reservoir. This suggests that the impulse is not effectively coupled into the vibrational modes of the products which is reasonable given the geometry at the transition state. Secondly, the fact that <ET> changes gradually with available energy suggests that either there is complete randomization of energy or the unrandomized energy is not localized in the reaction coordinate.
There is very good agreement between BIM and experiment. Not only does the model accurately predict the partitioning at 218 nm, high product translational energy and low OH internal excitation, but it is also consistent with the additional energy at 200 nm being deposited primarily in the ro-vibrations of the CH3CO fragment. Although the SFIM and RFIM also predict low OH rotation and vibration they grossly overestimate the fraction of available energy appearing as product translation.
Acknowledgments. The Authors would like to thank R. Harris, S. Smith, C.B. Moore, and H. Reisler for many helpful discussions.