  Accurately modelling cold and ultracold reactive collisions occuring over
deep potential wells, such as \ce{D+ + H2 -> H+ + HD}, requires the development
of new theoretical and computational methodologies. One potentially useful
framework is the R-matrix method adopted widely for electron-molecule
collisions which has more recently been applied to non-reactive heavy particle
collisions such as Ar-Ar. The existing treatment of non-reactive elastic and
inelastic scattering needs to be substantially extended to enable modelling of
reactive collisions: this is the subject of this paper. Herein, we develop the
general mathematical formulation for non-reactive elastic and inelastic
scattering, photo-association, photo-dissociation, charge exchange and reactive
scattering using the R-matrix method. Of particular note is that the inner
region, of central importance to calculable R-matrix methodologies, must be
finite in all scattering coordinates rather than a single scattering coordinate
as for non-reactive scattering. % The choice of coordinate set and basis
function is these cases becomes more complexThis introduces substantial
challenges to the basis sets utilised in practical calculations as integrals
over finite domains are often much more challenging than over infinite domains
for this problem.
