Quantal scattering theory

D. C. Chatfield, D. G. Truhlar, and D. W. Schwenke, Benchmark calculations of thermal reaction rates. I. Quantal scattering theory, J. Chem. Phys. 94 2040 (1991). 

In quantal scattering theory, Chapter 4, we saw that this is equally true for the motion in the unbound part of the potential. Indeed, the shape of the vibrational wave function changes smoothly as we go to higher energies and even as the energy is hi er than the threshold for dissociation, see Figure 7.1. 

The low-temperature chemistry evolved from the macroscopic description of a variety of chemical conversions in the condensed phase to microscopic models, merging with the general trend of present-day rate theory to include quantum effects and to work out a consistent quantal description of chemical reactions. Even though for unbound reactant and product states, i.e., for a gas-phase situation, the use of scattering theory allows one to introduce a formally exact concept of the rate constant as expressed via the flux-flux or related correlation functions, the applicability of this formulation to bound potential energy surfaces still remains an open question. 

This chapter deals with quanta and semiclassical theory of heavy-particle and electron-atom collisions. Basic and useful formulae for cross sections, rates and associated quantities are presented. A consistent description of the mathematics and vocabulary of scattering is provided. Topics covered include collisions, rate coefficients, quantal transition rates and cross sections, Bom cross sections, quantal potential scattering, collisions between identical particles, quantal inelastic heavy-particle collisions, electron-atom inelastic collisions, semiclassical inelastic scattering and long-range interactions. 

The Cl + HC1 quantized transition states have also been studied by Cohen et al. (159), using semiclassical transition state theory based on second-order perturbation theory for cubic force constants and first-order perturbation theory for quartic ones. Their treatment yielded 0), = 339 cm-1 and to2 = 508 cm"1. The former is considerably lower than the values extracted from finite-resolution quantal densities of reactive states and from vibrationally adiabatic analysis, 2010 and 1920 cm 1 respectively (11), but the bend frequency to2 is in good agreement with the previous (11) values, 497 and 691 cm-1 from quantum scattering and vibrationally adiabatic analyses respectively. The discrepancy in the stretching frequency is a consequence of Cohen et al. using second-order perturbation theory in the vicinity of the saddle point rather than the variational transition state. As discussed elsewhere (88), second-order perturbation theory is inadequate to capture large deviations in position of the variational transition state from the saddle point. 

In transition state theory it is assumed that a dynamical bottleneck in the interaction region controls chemical reactivity. Transition state theory relates the rate of a chemical reaction in a microcanonical ensemble to the number of energetically accessible vibrational-rotational levels of the interacting particles at the dynamical bottleneck. In spite of the success of transition state theory, direct evidence for a quantized spectrum of the transition state has been found only recently, and this evidence was found first in accurate quantum mechanical reactive scattering calculations. Quantized transition states have now been identified in accurate three-dimensional quantal calculations for 12 reactive atom-diatom systems. The systems are H + H2, D + H2, O + H2, Cl + H2, H + 02, F + H2, Cl + HC1, I + HI, I 4- DI, He + H2, Ne + H2, and O + HC1. 

The main objective of any theory is to be able to understand and predict the results of experiments. Since the world is three dimensional one cannot limit oneself to the study of collinear systems. In section IV we show how the collinear analysis based on periodic orbits may be generalised to three dimensional systems.We provide a 3D adiabatic transition state theory which is used to analyse numerical computations as well as experimental results. A 3D analysis of quantal resonances predicts that one should hope that quantal resonances will provide a new spectroscopy of transition states. A discussion of the future role of periodic orbits and reactive scattering is given in section V. 

One of the nice aspects of classical variational TST is that the theory provides upper and lower bounds to the exact classical rate of reaction. To date, there is no such quantal theory. Quantally, transition state theory is an approximation - one of the objectives of the theorist is to optimize the approximation. The concept of a transition state remains though as an important guide towards understanding the quantal structure of the reaction probability. In this section we shall see that quantal barriers and wells and their associated properties determine both threshold behaviour and resonance phenomena in quantal reactive scattering. 

R.E, Wyatt. Reactive scattering cross sections ii Approximate quantal treatments. In Atom-Moiecule Coliision Theory. A Guide for the Experimentalist, edited by R. B. Bernstein (Plenum. New York. 1979) pp. 477-503. 

J. C. Light, Reactive scattering cross sections I General quantal theory, in reference 4, chapter 14. 

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