Imbedded matrix Green’s functions in atomic and molecular scattering theory

This paper presents a general, simple, and unified treatment of Green’s function approaches to the solution of the sets of coupled second order differential equations encountered in the theory of inelastic and reactive scattering. The standard scalar theory of Sturm–Liouville problems is extended to systems of equations without recourse to spectral resolutions of the Green’s operator. A nonstandard approach to invariant imbedding is then developed to yield the most general kind of recursion relation for Wigner R matrices possible; the R-matrix propagation of Light and Walker and the variable interval-variable step algorithm of Parker, Schmalz, and Light are shown to be particularly simple examples of this more general scheme. Extensions and simplifications are offered. Also recurrence relations, stabilization of solutions, adiabatic and diabatic representations, and the use of nonorthogonal bases are all treated in a transparent manner.