Franck–Condon factors in studies of dynamics of chemical reactions. I. General theory, application to collinear atom–diatom reactions

Abstract

We derive and show the utility of an approximate theory of chemical dynamics based on a generalized Franck–Condon factor. We begin by showing how the general expression for the transition matrix for an electronically adiabatic reaction may be rewritten in terms of a transition between two surfaces through the use of a quasiadiabatic representation. This exact transition matrix may be reduced to a Franck–Condon overlap integral in a variety of ways, and one possible sequence of approximations for accomplishing this reduction is outlined. We neglect terms due to virtual transitions to excited electronic states, make a Born–Oppenheimer approximation, neglect terms involving gradients of the nuclear wavefunction (low kinetic energy approximation), and finally make a Franck–Condon approximation. The overlap is then evaluated for the special case of collinear exoergic atom–diatom reactions for the purpose of studying product state vibrational distributions in these reactions. The evaluation is done approximately by using physical arguments to estimate the general appearance of the reagent and product quasiadiabatic surfaces, and assuming separable solutions to the Schrödinger equation on each surface. The overlap integral is then further approximated by expanding the integrand about the nuclear configuration of maximum overlap. This enables us to obtain a simple analytical result for the product state distribution, using either harmonic or Morse oscillator vibrational wavefunctions. We then use the resulting expressions to study the dynamics of the collinear F+H₂(D₂) and H(D)+Cl₂ reactions. In both applications we find that the Franck–Condon overlap is capable of a qualitatively correct description of the product state distributions, including dependence on reagent translational energy, mass ratios, and various features of the potential energy surface. Furthermore, a physical description of the origin of a dynamic threshold effect in the F+H₂(D₂) reaction is provided, as is a simple interpretation of the role of potential energy release behavior in the determination of product state distributions.