the role of tunneling and quantum reflection from the centrifugal barrier and show the corrections to classical results. We then focus on the van der Waals potential ($n=6$), which describes interactions of atoms or molecules without electric or magnetic dipole moments, and describe predictions for the collision rates and the role of shape resonances.

This paper is organized as follows. In Sec. II we briefly review the MQDT formalism. In Sec. III we describe inelastic collisions for isotropic power-law potential at long range using the two-channel quantum defect model. In Sec. IV we consider the case when the exit channel is far below threshold and derive general formulas for the complex scattering length and collision rates using MQDT functions. Sec. V describes the threshold limits for the rates, while Sec. VI focuses on the high energy limits. Finally, in Sec. VII we apply the theory to systems with van der Waals interactions at long range, relating our results to recent experiments. We discuss the results and conclude in Sec. VIII.

Quantum defect formalism
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Our goal is to develop a simple model of a reactive collision which will capture the essential physics. To this end we introduce a multichannel scattering 