  The reaction between D$^+$ and H$_2$ plays an important role in
astrochemistry at low temperatures and also serves as a prototype for simple
ion-molecule reaction. Its ground $\tilde{X}~^1A^{\prime}$ state has a very
small thermodynamic barrier (up to 1.8$\times 10^{-2}$ eV) and the reaction
proceeds through the formation of an intermediate complex lying within the
potential well of depth of at least 0.2 eV thus representing a challenge for
dynamical studies. In the present work, we analyze the title reaction within
the temperature range of 20 $-$ 100 K by means of ring polymer molecular
dynamics (RPMD) and quasi-classical trajectory (QCT) methods over the
full-dimensional global potential energy surface developed by Aguado et al. [A.
Aguado, O. Roncero, C. Tablero, C. Sanz, and M. Paniagua, J. Chem. Phys., 2000,
112, 1240]. The computed thermal RPMD and QCT rate coefficients are found to be
almost independent of temperature and fall within the range of 1.34 $-$
2.01$\times$10$^{-9}$ cm$^3$ s$^{-1}$. They are also in a very good agreement
with the previous time-independent quantum mechanical and statistical quantum
method calculations. Furthermore, we observe that the choice of asymptotic
separation distance between the reactants can markedly alter the rate
coefficient in the low temperature regime (20 $-$ 50 K). Therefore it is of
utmost importance to correctly assign the value of this parameter for dynamical
studies, particularly at very low temperatures of astrochemical importance. We
finally conclude that experimental rate measurements for the title reaction are
highly desirable in future.
