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Accurately modelling cold and ultracold reactive collisions occuring overdeep potential wells, such as \ce{D+ + H2 -> H+ + HD}, requires the developmentof new theoretical and computational methodologies. One potentially usefulframework is the R-matrix method adopted widely for electron-moleculecollisions which has more recently been applied to non-reactive heavy particlecollisions such as Ar-Ar. The existing treatment of non-reactive elastic andinelastic scattering needs to be substantially extended to enable modelling ofreactive collisions: this is the subject of this paper. Herein, we develop thegeneral mathematical formulation for non-reactive elastic and inelasticscattering, photo-association, photo-dissociation, charge exchange and reactivescattering using the R-matrix method. Of particular note is that the innerregion, of central importance to calculable R-matrix methodologies, must befinite in all scattering coordinates rather than a single scattering coordinateas for non-reactive scattering. % The choice of coordinate set and basisfunction is these cases becomes more complexThis introduces substantialchallenges to the basis sets utilised in practical calculations as integralsover finite domains are often much more challenging than over infinite domainsfor this problem.

General Mathematical Formulation of Scattering Processes in Atom-Diatomic Collisions in the RmatReact Methodology