  We have calculated accurate quantum reactive and elastic cross-sections for
the prototypical barrierless reaction D$^{+}$ + H$_2$($v$=0, $j$=0) using the
hyperspherical scattering method. The considered kinetic energy ranges from the
ultracold to the Langevin regimes. The availability of accurate results for
this system allows to test the quantum theory by Jachymski et al. [Phys. Rev.
Lett. 110, 213202 (2013)] in a nonuniversal case. The short range reaction
probability is rationalized using statistical model assumptions and related to
a statistical factor. This provides a means to estimate one of the parameters
that characterizes ultracold processes from first principles. Possible
limitations of the statistical model are considered.
