Noniterative Solutions of Integral Equations for Scattering. V. Auxiliary T(kj) Matrix Formalism

Abstract

The homogeneous integral solution method is used to develop a scheme for direct computation of the T matrix. The method involves calculation of auxiliary matrices T (kj) in terms of which the true T matrix is obtained. The auxiliary T (kj) matrices can be obtained by solving either integral or first‐order linear non‐homogeneous differential equations. The relationship of calculations based on these equations to those based on the wavefunctionintegral equations is discussed. Numerical studies of the general behavior of the homogeneous integral solution procedure are presented. These studies deal with rotationally inelastic collisions of an atom and a rigid rotator interacting via a Morse‐type potential. An investigation of some higher‐order numerical schemes for solving the integral and differential equations for the auxiliary matrices and the integral equations for the wavefunction is reported. The results indicate that the use of higher‐order procedures such as the Runge–Kutta method or Simpson's quadrature scheme permit one to use step sizes 5–10 times larger than those required with the trapezoidal quadrature scheme (or equivalently, the uncorrected Euler method for the differential equation) to achieve a given accuracy. The role of the potential function is studied by a comparison of the | T | 2 matrix elements for Lennard‐Jones (12–6) and Morse potentials having the same well depth, curvature, and equilibrium position. In addition, hybrid potentials constructed by splicing together Morse and Lennard‐Jones potentials are studied. The results indicate that, at the energy considered, the long‐range portion of the potential has a dominant effect on the elastic and inelastic scattering.