Chapter 1
Applications of Quantum Statistical Methods to the Treatment of Collisions

Paul J. Dagdigian1 and Millard H. Alexander2

1Department of Chemistry, The Johns Hopkins University, Baltimore, MD, 21218-2685, USA

2Department of Chemistry and Biochemistry, Institute for Physical Science and Technology, University of Maryland, College Park, MD, 20741-2021, USA

Contents

1. I. Introduction
2. II. Quantum Statistical Theory
3. III. Fine-Structure Branching in Reactive O(1D) + H2 Dynamics
4. IV. Inelastic OH + H Collisions
   1. A. OH + H Vibrational Relaxation
   2. B. OH + D Isotope Exchange
   3. C. OH + H Rotationally Inelastic Collisions
5. V. OH + O Reaction and Vibrational Relaxation
6. VI. Inelastic Collisions of the CH Radical
   1. A. CH + H2
   2. B. CH + H
7. VII. H + O2 Transport Properties
8. VIII. Conclusion
9. Acknowledgments
10. References

I. Introduction

In chemical kinetics, statistical theories were first developed to understand unimolecular reactions and predict their rates (see, for example, [1, 2]). In the field of molecular reaction dynamics, we would expect statistical models to be well suited to reactions proceeding through formation and decay of a strongly bound collision Examples would be the reaction of electronically excited atoms with [C(D), N(D), O(D), and S(D), for instance]. Here, atom M inserts into the H-H bond with the subsequent formation of a transient HMH complex, which then decays to form MH + H products.

Statistical models for reactions involving the formation and decay of a complex were first proposed in the 1950s to describe nuclear collisions [3]. These models were then applied to molecular collisions [4]. Molecular statistical theories were put on a firm theoretical footing by Miller [5], who used as justification the formal theory of resonant collisions [6-8].

Pechukas and Light [9, 10] pioneered a statistical theory to predict the rate and product internal state distribution of the reaction of an atom with a diatomic molecule. This theory was based on Light's work on the phase space theory of chemical kinetics [11, 12] but, in addition, imposed detailed balance. This work has formed the basis of modern quantum mechanical treatments of complex-forming chemical reactions [13-17]. Here, once the complex is formed, it can fall apart to yield any accessible reactant or product subject to conservation of the total energy and angular momentum.

Pechukas and Light [9, 10] made some additional simplifications: First, they assumed that the capture probability was zero or one, depending on whether the reactants had sufficient energy to surmount the centrifugal barrier for each partial wave (related to the total angular momentum J of the collision complex). Second, they assumed that the long-range potential could be described by an inverse power law, -. These assumptions, particularly the latter, were reasonable in an era where calculation of a potential energy surface (PES) was a major undertaking.

Subsequently, Clary and Henshaw [13] showed how to apply time-independent (TID) coupled-states and close-coupling methods to the determination of capture probabilities for systems with anisotropic long-range interactions. More recently, Manolopoulos and coworkers [14, 15] combined the statistical considerations of Pechukas and Light with the Clary-Henshaw TID quantum capture probabilities, using, in addition, accurate ab initio potential energy surfaces. This so-called "quantum statistical" method has been applied to a number of atom-diatom reactions that proceed through the formation and decay of a deeply bound complex. Guo has demonstrated how a time-dependent (wavepacket, WP) determination of the scattering wave function can be used in an equivalent quantum-statistical investigation of reactions proceeding through deep wells [18].

In related work, Quack and Troe [19] developed an adiabatic channel model to describe the unimolecular decay of activated complexes. This theory has been applied to a variety of processes, including the OH + O reaction [20].

González-Lezana [21] has written a comprehensive review of the use and applicability of quantum statistical models to treat atom-diatom insertion reactions. A good agreement with full quantum reactive scattering calculations has been found for properties such as the differential cross sections for the reactions of C(D) and S(D) with , while less satisfactory agreement was found for the O(D) and N(D) + reactions [15]. This comparison illustrates a limitation of the statistical theory: how to assess the accuracy of the approach without recourse to more onerous calculations. The differential cross-section of the product of a statistical reaction should have forward-backward symmetry. This is often not quite the case because the quenching of interferences between partial waves is not complete, particularly in the forward and backward directions [22].

Typically, fully quantum scattering calculations involve expansion of the scattering wavefunction in terms of all the triatomic states that are energetically accessible during the collision. The computational difficulty scales poorly with the number of these internal states. Both the large number of accessible vibrational states of a triatomic as well as the rotational degeneracy of the states corresponding to the A + BC orbital motion contribute to this bottleneck. Deep wells in any transient complexes are particularly problematic. An example is the O(D) + OH + H reaction, for which the PESs are illustrated schematically in Fig. 1. Even without taking anharmonicity into account, there are >1900 O vibrational levels with energy below the O(D)+ asymptote. And this does not include rotational levels. Thus, full quantum reactive scattering calculations for complex-mediated reactions are a heroic task [23].

Figure 1 Schematic diagram of the potential energy surfaces of the OHH system. Only the lowest () PES was taken into account in the initial quantum statistical calculations on the O(D) + reaction [14, 15]. Note that O () well lies 59,000 below the O(D) + asymptote.

Adapted from Rackham et al. 2001 [14] and Rackham et al. 2003 [15].

In the quantum statistical method, the close-coupled scattering equations are solved outside of a minimum approach distance, the "capture radius" , at which point the number of energetically accessible states (open channels) is much less than at the minimum of the complex. Also, because the point of capture occurs well out in the reactant and/or product arrangement, one does not need to consider the mathematical and coding complexities associated with the transformation from the reactant to product states [24]. It is these simplifications that make the quantum statistical method so attractive.

Formation of a transient complex does not always lead to chemical reaction. The complex may decay to the reactant arrangement, resulting in an inelastic collision. The present review outlines the application of quantum statistical theory to nominally nonreactive collisions that access PESs having one or more deep wells. Consider, the generic A + BC AB + C collision. The inelastic event A + BC(v, ) A + BC() can occur in a direct (non-complex-forming) collision, either through an encounter in which the partners approach in a repulsive geometry or at a larger impact parameter, where the centrifugal barrier prevents access to the complex. In addition, complex formation () and subsequent decay into the reactant arrangement will also contribute to inelasticity.

In general, weak, glancing collisions contribute substantially to rotational inelasticity. Thus, one might naively expect that both direct and complex-forming processes will contribute to rotational inelasticity. By contrast, vibrational inelasticity in collisions on basically repulsive PESs is very inefficient [25, 26], because the variation of repulsive PESs with the vibrational modes of the molecular moiety is weak. Thus, one might anticipate that the formation of a transient complex, in which substantial change in the bond distances might occur, could make a major contribution to vibrational relaxation.

Also, for A + BC collision systems where one of the reactants is an open-shell species, typically (as shown schematically in Fig. 1), there are a number of electronic states that correlate with the A + BC (or AB + C) asymptote. Of these states, one (or, only a few) leads to strongly bound intermediates, while the others are repulsive. The branching between the energetically accessible fine-structure levels of the products (in the case of OH, the spin-orbit and -doublet levels) will be controlled by the coupling between the various electronic states as they coalesce in the product arrangement, as the complex decays. We might predict that this branching, which can often be measured experimentally [27, 28], would be insensitive to any couplings within the complex, where the excited electronic states lie high in energy, and hence be an ideal candidate for prediction by a quantum-statistical calculation.

Reactions involving isotopologs of the same atom,

where designates an isotopolog of B, are an additional example...