Quantitative comparisons of various approximations within multichannel quantum defect theory (MQDT) for ultracold alkali collisions

We have performed a suite of calculations describing homonuclear two-body ultracold collisions of alkali atoms. Within the framework of MQDT, we identify three distinct approximations. These approximations are primarily characterized by the way in which the short-ranged K-matrix is calculated: (1) The energy-independent frame transformation, (2) The energy-dependent frame transformation, and (3) a rigorous multichannel boundary condition. We compare these three approximations to fully converged coupled channels calculations for collisions of Li-6, Li-7, Na-23, K-39, K-40, Rb-85, Rb-87 and Cs-132, identifying and comparing the positions and widths of s-wave magnetic Feshbach resonances up to 1200G. Frame transformation methods provide a profound computational advantage because they completely avoid numerical solutions to the coupled channels equations, even at short range. By performing a systematic study of homonuclear collisions in the aforementioned species, we are able to identify the conditions under which the frame transformation methods begin to fail, and a rigorous short-range multichannel boundary condition becomes necessary.