  The cross section and rate constants for the title reaction are calculated
for all the spin-orbit states of N$^+$($^3P_{J_A}$) using two statistical
approaches, one purely adiabatic and the other one mixing quantum capture for
the entrance channel and adiabatic treatment for the products channel. This is
made by using a symmetry adapted basis set combining electronic (spin and
orbital) and nuclear angular momenta in the reactants channel. To this aim,
accurate {\it ab initio} calculations are performed separately for reactants
and products. In the reactants channel, the three lowest electronic states
(without spin-orbit couplings) have been diabatized, and the spin-orbit
couplings have been introduced through a model localizing the spin-orbit
interactions in the N$^+$ atom, which yields accurate results as compared to
{\it ab initio} calculations including spin-orbit couplings. For the products,
eleven purely adiabatic spin-orbit states have been determined with {\it ab
initio} calculations. The reactive rate constants thus obtained are in very
good agreement with the available experimental data for several ortho-H$_2$
fractions, assuming a thermal initial distribution of spin-orbit states. The
rate constants for selected spin-orbit $J_A$ states are obtained, to provide a
proper validation of the spin-orbit effects to obtain the experimental rate
constants.
