We develop a general quantum theory for reactive collisions involving power-law potentials (−1/r n ) valid from the ultracold up to the high-temperature limit. Our quantum defect framework extends the conventional capture models to include the non-universal case when the short-range reaction probability P re < 1. We present explicit analytical formulas as well as numerical studies for the van der Waals (n = 6) and polarization (n = 4) potentials. Our model agrees well with recent merged beam experiments on Penning ionization, spanning collision energies from 10mK to 30K [Henson et al, Science 338, 234(2012)].PACS numbers: 34.50. Cx, 03.65.Nk, 34.10.+x, 34.50.Lf Much recent work involves the inelastic and reactive collisions of cold atoms or molecules with one another [1][2][3][4] or with ions in hybrid traps [5][6][7]. These could involve the ultracold regime with translational temperature on the order of µK or less or the cold regime between a few mK and a few K. Systematic theoretical principles for understanding the quantum dynamics of such collisions are needed. Much work has already been done in this area, as reviewed by Ref. [8]. One class of theories based on Quantum Defect Theory (QDT) allows us to systematize and develop tools for understanding such collisions [9][10][11][12][13][14][15]. One special limiting case is that of highly reactive collisions, where simple classical trajectory capture models known as the Langevin (n = 4) [16] or Gorin (n = 6) [17] models apply when the long range potential takes on the form −C n /r n (n > 3). These familiar models assume that every classical trajectory contributes to the collision cross section that is captured by the long range potential so the particles spiral in to short distance where they react or relax with probability P re . We use the term "universal" to describe capture models with P re = 1, since they do not depend on any details of the strong short-range chemical interactions.In the cold and ultracold regimes, it is essential to build in quantum corrections to these classical models due to quantum threshold laws [18][19][20]. This has been done using QDT for both the Langevin [21,22] and Gorin [23,24] universal models where P re = 1. Here, we will follow the formalism by Idziaszek et al. [22,23] and generalize the previous results to the nonuniversal regime with P re < 1, and to the arbitrary collision energy. In the limiting cases of low and high temperatures we give analytic formulas that are valid for power-law potentials (−1/r n ). We apply our theory to interpret the ionization rate constants measured by recent merged beam experiments in the cold regime [25]. Using a single complex QDT parameter found by fitting low energy data only, we are able to reproduce the experimental data over four orders of magnitude in energy including about twenty partial waves in the calculation.Generalized complex scattering length. We consider reactive collisions of particles interacting via power-law potential V (r) = −C n /r n (n > 3) at large distances r R 0 ...