Theory of Reactive Scattering. I. Homogeneous Integral Solution Formalism for the Rearrangement τ-Operator Integral Equation

Abstract

The integral equation for the rearrangement $\tau$ operator is used as the basis for a discussion of reactive scattering. The concept of the amplitude density introduced by Johnson and Secrest is extended to reactive collisions by means of the $\tau$ operator. The homogeneous integral solution method of Sams and Kouri is used to develop a noniterative solution to the integral equation satisfied by the reactive‐scattering amplitude density. This method is characterized by very stable behavior and is capable of quite accurate solutions with a relatively large step size. We consider the formulation of the method first for a general reactive collision. Both open and closed channel contributions to the reactive scattering amplitude density are considered. We employ the Feshbach projection operator formalism to treat the effects of complex formation due to virtual excitation of internal degrees of freedom. After discussing the general reactive collision, we next deal with the specific problem of atom–diatom reactions. Notation and techniques developed by Miller are used to treat problems associated with coordinates. Unlike Miller's discussion of reactive scattering, the present approach does not require inclusion of square integrable functions since the integral equation for the $\tau$ operator (and the equation for the amplitude density) already takes explicit account of the effects described by such terms. Further, the present approach allows one to easily study the effects of vibrational and/or rotational excitation on the reaction rate since the initial state appears explicitly as an inhomogeneity in the integral equation for the reactive scattering amplitude density. Finally, we discuss the manner in which the $T$ matrix may be obtained from the reactive scattering amplitude density. It results that the reactive $T$ matrix may be expressed in terms of the nonreactive $T$ matrix.