Quantum defect theory with full close-coupling method at short-range: Applications to ro-vibrational transitions in ultracold H$_2$-H$_2$ collisions and beyond

An accurate characterization of both short-range dynamics and long-range physics continues to be a major challenge in describing ultracold collisions. While quantum close-coupling (CC) is the preferred approach, it is computationally intractable for many systems of current experimental interest. A well-tested and computationally tractable approach for ultracold collisions is multi-quantum defect theory (MQDT). It is adept in handling both long-range forces and external field dependencies. Here we describe a hybrid approach that uses explicit CC method at short-range and the MQDT formalism at long range. The CC calculations needs to be carried out only at one energy, essentially at zero collision energy, to yield a short-range K-matrix from which accurate cross sections can be evaluated for energies up to a kelvin using the MQDT formalism. We illustrate the approach for the benchmark case of rovibrational transitions in H$_2$-H$_2$ collisions. Potential applicability of the approach to reactive collisions is discussed.