Mechanics of molecular collisions - Connor, J; Connor, J. N. L. (1969)
﻿ This thesis presents a semiclassical development of the theory required for the interpretation of thermal energy molecular beam experiments in the study of molecular interactions, and for the prediction of new effects. Chapter (1) provides a brief introduction to molecular scattering and describes the work in the following Chapters. In Chapter (2), semiclassical connection formulae based on an exact solution of the Schrandouml;dinger equation for a parabolic well and a parabolic barrier are derived and their properties developed. These results are required in later Chapters. The connection formulae for a parabolic well (barrier) are valid for energies which lie either above or-below the well minimum (barrier maximum) and provide a direct connection between one classically disallowed (allowed) region and another. These connection formulae contain important quantum correction functions which remove certain singularities from the semiclassical analysis. Chapter (3) presents a semiclassical analysis of resonance effects in molecular orbiting collisions. The explicit introduction of a complex energy is used to characterize the quasi-stationary states in the dip of the effective potential for the collision. Expressions for the resonance energies and widths of the quasi-stationary states are derived from the semiclassical wavefunction, obtained with the help of the connection formulae in Chapter (2), and a formula is given for the resonant contribution to the measurable total elastic cross section. In Chapter (4), a quantum mechanical theory is developed for an electronically adiabatic bimolecular exchange reaction, with the restriction that the three atoms are constrained to move on a straight line but with the whole system free to rotate in three dimensions. An important feature of the analysis is the use of a set of coordinates which pass smoothly from those suited to reactants to those suited to products. A vibrationally adiabatic approximation is used to reduce the scattering problem to the solution of one dimensional Schrandouml;dinger equations. Semiclassical techniques are used to evaluate the partial wave summations that occur in the theory and elastic and reactive differential cross sections are calculated for three different kinds of potential surface. An interesting feature of the calculations is the occurrence of a new kind of rainbow effect, which is named a 'cubic' rainbow since it arises when the deflection function varies cubically with impact parameter. The classical and semiclassical theory of cubic rainbows is developed. Chapter (5) investigates the effect of a dip in the activation barrier for a chemical reaction using the model developed in Chapter (4). The complex energy techniques introduced in Chapter (3) are used to relate the resonance tunnelling through the barriers to Breit-Wigner theory. A novel feature of the theory is the use of a complex energy with an imaginary part that may be positive, negative or zero. Such a complex energy is the natural consequence of applying a 'forward moving waves only' boundary condition and the sign of the imaginary term has a straightforward interpretation in terms of the population of states within the dip. Chapter (6) investigates the effect of the rotational motion of the reactants on the differential cross sections of a chemical reaction. In order to allow a tractable development of the theory, a model is adopted with the following restrictions: first the atoms are constrained to move in a plane and secondly the central atom is given an infinite mass. The use of a set of natural collision coordinates and an adiabatic separation of variables is used to reduce the scattering problem to the solution of one dimensional Schrandouml;dinger equations. Partial wave expansions for the elastic and reactive scattering amplitudes are obtained. Appendix (A) is devoted to various numerical aspects of the theory. The properties of the integral: andinfin; andnbsp; andint; exp[i(x4 + ax)]dx, andminus;andinfin; andnbsp; which characterizes the semiclassical description of the cubic rainbow effect are considered together with its numerical evaluation. Numerical methods for the evaluation of phase integrals and their derivatives are described when one of the integration limits is zero (existing methods break down for this case). Appendix (B) presents a discussion of two dimensional elastic scattering from a central potential for use in conjunction with Chapter (6). The topics considered include the partial wave expansion of the wavefunction, the semiclassical approximation for the phase shift and the correspondence with classical two dimensional scattering.