Reaction dynamics product quantum numbers

The OH radical is produced in a particular vibrational and rotational quantum state specified by the quantum numbers n and j. The corresponding energies are denoted by tnj. The probabilities with which the individual quantum states are populated are determined by the forces between the translational mode (the dissociation coordinate) and the internal degrees of freedom of the product molecule along the reaction path. Final vibrational and rotational state distributions essentially reflect the dynamics in the fragment channel. They are one major source of information about the dissociation process. 

At the fundamental level the course of such a reaction between an atom A and a diatomic molecule BC is governed by quantum mechanics. Thus, within this theoretical framework the reaction dynamics at a given collision energy can be analyzed for reactants in a given quantum state (denoted by the quantum number n) and one can extract the transition probability for the formation of products in various quantum states (denoted by the quantum number m). At this level one considers the state-to-state dynamics of the reaction. 

Figure 3 Surprisal plots (18) for the HF vibrational state distribution from the exoergic H atom abstraction reaction F + (CH,)4C - (CH,),CCH2 + HF(v). (Bottom panel) The observed (by D. J. Bogan and D. W. Setser, J. Chem. Phys. 64 586 (1976)) distribution, P(v), open dots connected by a line, and the (so called, prior) distribution, P (v) full symbols, vs. the HF vibrational energy. The prior distribution is the one expected when all products final states are equally probable (18). The observed distribution is qualitatively different from the prior one and their deviance, the surprisal, —In(P(v)/P"(v)) is plotted vs. E/Ev, where Ev is the HF vibrational energy and E is the total energy, in the upper panel. One can interpret the linear dependence of the surprisal on the HF vibrational energy as reflecting the presence of a quantity which is conserved by the dynamics. (See, for example, ref. (108)). In this sense, surprisal analysis is analogous to the search for quantum numbers that are not destroyed by the intramolecular couplings.

Another valuable use of accurate quantum dynamics calculations is testing the validity of classical simulations for predicting product-state distributions, and reduced-dimensionality studies of this issue are available for both Cl + H2 [67] and H + CI2 [104], In the present case extensive quasiclassical trajectory (( CT) calculations have been carried out for the full-dimensional Cl + D2 reaction by Aoiz and Bahares [105]. An example of how the QCT results compare to the accurate quantum ones is given in Fig. 4, which shows differential cross sections for Cl + D2(v=0J=1) —> DCl(v ) + D, where v and v are initial and final vibrational quantum number, respectively, j is initial rotational quantum number, and the results are summed over final rotational quantum number j. The comparison in Fig. 4 is for an initial relative translational energy of 10.1 kcal. The agreement is quite good. Notice, however, that the QCT method overestimates the amount of vibrationally excited product. 

The surprisal is the same for all the states in the sum. So the sum is the number of such states. The prior distribution is just this number, divided by a normalization (the total number of states) to render the number into a probabihty. If there is no dynamic constraint the numerical value of ky is zero and the distribution is the prior one. We reiterate that the prior distribution is not the same as a uniform distribution. Rather, it depends on how many products quantum states fall into the group of states of interest. For example, in Figure 6.13 the prior vibrational distribution falls rapidly with increasing vibrational excitation and so looks thermal-hke. Problem H shows that in the hmit where the products have many atoms so that the fraction of energy in any particular vibrational mode is hkely to be small, the prior distribution is exactly thermal. But in an A + BC reaction there is only one vibrational mode in the products and so the correct form of the prior distribution is required. 

The present grid-based quantum dynamics simulations retrieve quantum effects such as recrossings and interferences but are too limited in terms of number of degrees of freedom to provide the reaction pathways to form rearrangement products. This smdy could thus be complemented by less accurate trajectory-based semi-classical dynamics simulations or Gaussian-based quantum dynamics simulations in full dimensions like those presented in the next application cases. 

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