averaging and $p$-wave contribution.](KelvsT){width="\linewidth"}

In addition to analytical low- and high energy limits, within our model it is possible to obtain the reactive and elastic rate constants at any collision energy. This can be done either by finding the MQDT functions analytically (for example, in terms of Z functions for van der Waals potential developed by Gao [@Gao1998]) and using formulas  and , or by numerical treatment. In the latter case we perform scattering calculations, propagating the wave function using Numerov algorithm and extracting the phase shift. The key point here is to set proper boundary conditions at short range, given by . A possible way to do it is to use solutions of $1/r^n$ potential at zero energy (at short range the kinetic energy is negligible compared to the well depth), combined to reproduce the zero energy limit of the scattering length .

At finite energies the reactive rate constant can be greatly enhanced by shape resonances, which are due to the presence of quasibound states behind the barrier. In particular, analytic theory [@Gao1998; @Gao2000] predicts a $p$-wave resonance for $s=2$ and a $d$-wave resonance for $s=1$. This is confirmed by the low energy behavior of MQDT functions for 