  Techniques for producing cold and ultracold molecules are enabling the study
of chemical reactions and scattering at the quantum scattering limit, with only
a few partial waves contributing to the incident channel, leading to the
observation and even full control of state-to-state collisions in this regime.
A new R-matrix formalism is presented for tackling problems involving low- and
ultra-low energy collisions. This general formalism is particularly appropriate
for slow collisions occurring on potential energy surfaces with deep wells. The
many resonance states make such systems hard to treat theoretically but offer
the best prospects for novel physics: resonances are already being widely used
to control diatomic systems and should provide the route to steering ultracold
reactions. Our R-matrix-based formalism builds on the progress made in
variational calculations of molecular spectra by using these methods to provide
wavefunctions for the whole system at short internuclear distances, (a regime
known as the inner region). These wavefunctions are used to construct collision
energy-dependent R-matrices which can then be propagated to give cross sections
at each collision energy. The method is formulated for ultracold collision
systems with differing numbers of atoms.
