Quantum Mechanical Studies of Molecular Scattering

Author: Dwyer, John Patrick

Year: 1978

Degree: Dissertation (Ph.D.)

Advisor: Kuppermann, Aron

Committee Member: Unknown, Unknown

Option: Chemistry

DOI: 10.7907/f6qz-1472

Abstract

This thesis is not about changing the world. Its scope is much more modest; we wish only to describe a very small part of it. In fact, we will usually restrict ourselves to only three or four atoms at any one time.

This thesis is a collection of seven theoretical studies on elementary chemical processes. In each of the studies we have used a quantum mechanical framework to describe a particular chemical system. In nearly every case we have formulated the problem exactly. That is, we have made no mathematical approximations to the governing equations. Because these problems are too complex to be solved analytically, we have turned to computers to solve them numerically. Once the appropriate equations have been derived, each study becomes an exercise in numerical techniques and computer programming. The crucial step, interpretation of the numbers, comes last.

In the first chapter we examine a model that describes energy transfer in the collinear collision of two diatomic molecules. In this system the diatomics are homonuclear Morse oscillators and interact through an exponentially repulsive potential. We compare our results with previous quantum calculations and find that the older results were not converged with respect to the number of basis functions. A comparison with the adiabatic approximation of Clarke and Thiele shows that the approximate results are reasonably accurate at all but the highest collision energies.

The second chapter is a study of the importance of low energy reaction probabilities in calculating low temperature rate constants. It limited to the collinear H + H2 reactive system using a realistic potential energy surface. We show that the integral which relates probabilities to rate constants must be calculated carefully in order to avoid numerical errors. We show that this was not done by other workers and has probably led to an incorrect conclusion concerning the isotope dependence of the transmission coefficient.

The third chapter is a survey of resonances in the reactive scattering of collinear H + H2 and its symmetric isotopic variants. From the survey we note trends and differences in the resonance strengths and positions caused by the various mass combinations. This leads us to propose a simple model for collinear scattering which should predict resonances. In fact, it correctly predicts their relative strengths and their positions to better than 0.08 eV.

The fourth chapter is an extensive study of the effect of barrier heights on vibrational deactivation in collinear scattering. The five surfaces used were for the H + FH system with barriers ranging from 1.5 kcal/mole to 40 kcal/mole. The motivation for the study comes from recently published work which indicates that the correct barrier in the HFH arrangement channel is much closer to 40 kcal/mole than the often used 1.5 kcal/mole. The results indicate that the large barriers significantly reduce the rate of collisional deactivation via reaction, but that nonreactive deactivation is roughly constant for a wide range of barrier heights.

In Chapter 5 we propose a method to describe collinear breakup collisions of the form A + BC → A + B + C. The method is general and can be applied to systems in which the energy is too low to allow dissociation. It has been applied to the H + H2 reaction and found to accurately reproduce reaction probabilities at low energies. The central idea is to use a discrete pseudo-vibrational basis set which can describe both bound and continuum states naturally. This preliminary work points the way to further calculations.

In the last two chapters we move to three-dimensional collisions. Chapter 6 is a comparison of two independent calculations of reactive scattering in an actual chemical system, one by Elkowitz and Wyatt (EW) and the other by Schatz and Kuppermann (SK). The comparison shows that the results are qualitatively and quantitatively different. It further shows that differences lie in the separate formulations of the problem and that the SK method is numerically converged while the EW is not. We note that the EW method used certain approximations that were not investigated and conclude that their results are in error.

In the last chapter we consider the difficulty of doing exact quantum calculations for even simple reactive systems and propose a mathematical approximation that preserves the accuracy of the exact results. The approximation arises from the neglect of certain off-diagonal angular momentum terms in the Hamiltonian and thus reduces the number of coupled equations to be solved simultaneously. We find that the computation time can be reduced by as much as a factor of 20.

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