Chemical Physics 22 (1977) 341-366 0 NorthHolland PubIishing Company ON STATISTICAL MODELS AND PRIOR DISTRIBUTIONS IN THE THEORY OF CHEMICAL REACTIONS A. BEN-SHAUL Department of Physical Chemistry, The Hebrew University,Jerusalem, Israel Received 3 January 1977 _ Statistical models which yield rate expressions involving the asymptotic channel states are classifiedaccording to thiee main characteristics:(i) The ensemblein which frameworkthe basic statisticalassumption is formulated.This assumption states that the initial and final states of the collision are, partly or completely, independent. (ii) The “measure in phase” associated with the product state distriiution. (iii) The extent to which dynamicat considerations are incorporated into the &deI. In this connection the discussion wih be confmed to the effects of angular momentum restrictions. The diversity in statistical models is demonstrated by comparing the product vibrational and rotational distributions obtained from several models for atom-diatom exchange reactions. Numerical results are presented for two cxoergic reactions which involve dif- ferent mass combinations. Particular attention is paid to a comparison between the objectives, assumptions and the final expressions for product state distributions corresponding to the so called phase space theory on the one hand and the prior (zero surprisal) limit of the information theoretic approach to moIecular dynamics on the other. I. Introduction inversion. In such cases one may use the statistical models as standards against which actual results can Many theories of molecular [l-4] and nuclear [.5, be compared [ 181 or as bases for the development 61 collisions employ (various) statistical assumptions of more elaborate models. In the information theoretic to describe collision processes and to predict their approach to molecular collision dynamics [42--721 outcomes. In molecular physics the main success of such standards can serve as the prior distributions these theories is in the area of large molecules where (rate constants, cross sections) used in surprisal and many degrees of freedom are involved so that .detailed entropy analyses of experimental data. When supple- dynamical calculations are impractical while statistical mented by appropriate dynamical constraints the approximations seem reasonable (see, however, e.g. prior distributions play a major role in the predictive refs; [7,8]). Since detailed calculations are difficult scheme (synthesis) based on the maximum entropy to pitfo~rm, partly due to .#te lack of accumte potential principle. energy surfaces, even for small systems, such as atom- The central assumption of the statistical models is diatomic molecule, various statistical models were that past and future with respect to the collision formulated in order to account for at least some as- event are partly or completely independent; subject to pects of the coilision [g-41]. For example, the sym- symmetry and conservation requirements. The sym- metric product angular distributions observed in certain metry requirements refer in particular to microscopic bimolecular reactions can be interpreted as the result reversibility while the conservation rules concern the of intermediate complex formation followed by total (center of mass) energy E, total angular momen- “statistical” breakdown probabilities [3,4,9-I 1,19, tum K and the normalization of probabilities. Approxi- 271. The statistical models are less successful in pre- mate dynamical information, e.g. on the range of dicting, for instance, the product vibrational and allowed anguIar momenta, is usually incorporated rotational energy distributions in exoergic chemical into the mod& in order to increase their dynamical reactions; especially in those ending with population character. The concepts of intermediate, “long lived” 342 A. Be;r.Sl:aul/Ti:eorJ’ of chemical reactions (see, e.g. refs. [I-4, 7-1 I ] ) or “strongly coupled”. discussion. Yet, even within the limited scope of [ 14-191, collision complexes play important r’ resin “asymptotic channel statistical approaches” there is various statistical theories. They serve to account for room for diverse, actually an infinity of, different “quasi-equilibrium” (‘energy randomization”) [l-d, models. 7,18,33,34] or “strong coupling” [4,11,14-191 The general purpose of this paper’is to characterize postulates on the basis of which the neglect of corre- the various aspects of diversity in a systematic fashion. lation between reactants and products can be justified. Particularattention will be paid to a comparisonbe- In the traditional formulations of transition state theory tween two familiar rate expressions.Namely, the cross a quasi-equilibrium is assumed to exist between the sections and related quantities obtained in the phase reactants and the systems crossing the activated com- space theory of Light, Pechukas and Nikitin [ 14-281 pies hypersurface (see e.g. refs. [4,7,37] ). Recent and the prior distributions appearing in the informa- work [36,37j (utilizing variational principles to locate tion theoretic approach developed by Levine, Bernstein the transition hypersurfacc, see e.g. refs. [12,38-411) and others [42-723. Unlike the phase space theory indicates that transition state theory is appropriate the prior distributions do not contain any dynamical to a “direct”, as opposed to “complex”. reaction information and therefore do not attempt to predict mechanism, whereas the well known [ 14-271 phase experimental results. In fact, they are meant to re- space theory of’light, Pechukas [14-191 and present the dynamically “least biased” (or “most Nikitin 129,211 (LPN) is only applicable to reactions random”) distributions [42-501. As such they can proceeding-via the formation of long lived complexes. either be used as standards for entropy-surprisal anal- This interpretation of the phase space theory is some- yses or else as starting points in the predictive scheme what different from the one suggested in some of its (synthesis) which constitutes another direction of the original formulations [14-191 where a strongly information theoretic method. Although the LPN and coupled complex defined by the region of strong in- the prior distributions involve different statistical teraction in configuration space rather than by its ensembles (E, K versus E) and different measures in lifetime was assumed to be formed. phase space (“uniform” versus “flux”), one can also To conclude this background survey it should be observe certain similarities. We shall elaborate upon mentioned that the LPN theory, to which considerable these differences and similarities within the general attention will be given in the following, can be derived classification framework of statistical models presented as a limiting case of other statistical-dynamical ap- in the following sections. Among the special models proaches. For example. it can be viewed as transition considered, we have included an extended version state theory for “loose complexes” [37,41]. Also, OFthe prior distribution which explicitly includes the and this is of greater interest in the present context, conservation of angular momentum (i.e., appropriate with the aid of a random phase type approximation to the E, K ensemble). As specific examples illustrating the phase space model can be derived through formal the consequences of different statistical assumptions, assumptions on the structure of the scattering matrix we have chosen the Cl + HI + HCI + I and the H f Cl2 [4.11,19,1-21: similar to those made in the statistical + HCl t Cl reactions. theory of rotational excitation [ 131. Two more remarks should precede the discussion. The main concern of this paper is the statistical, First, some points related to the objectives of this rather than the dynamical, aspects of statistical- paper have already been dealt with in the literature dynamical approaches to small collision systems. We [ 11,27,43,45,49-551 (to a greater or lesser extent). shall only consider models that lead to rate expressions In our opinion, however, no clear distinctions have (e.g., cross sections or product state distributions) been provided. Hence the motivation for this paper. involving the asymptotic reactant and product states, Second, for reasons to be clarified throughout the or equivalently the phase space regions corresponding paper, the presentation of the various models and the to the asymptotic channels_ Thus, many (interesting) analyses of their physical significance will not be questions regarding, for example, the behaviour and accompanied by judgements or evaluations. structure of the colliding system at intermediate. The paper is organized as follows: The basic defini- separationscare beyond the perspectives of the present tions and the general structure of statistical models A. Ben-Shul/Theory of chemical reactions 343 are presented in the next section. Different choices of more specific ensemble, I’t C I‘,, representative of statistical ensembles, measures in phase space and collisions with well defined total energy E and total dynamical constraints are considered in sections 3,4 angular momentum K. (Averaged, not summed, over and 5, respectively and tested for two specific exam- the 2K f I equivalent values of MK.)-The “canonical” ples in section 6. Section 7 provides some remarks on ensemble I’3 = T, (r2 C lY3),which is a Boltzmann the derivation based on the assumption of complex superposition of microcanonical ensembles will formation and the beginning of the derivation based