  The radiative cooling of shocked gas with primordial chemical composition is
an important process relevant to the formation of the first stars and
structures, as well as taking place also in high velocity cloud collisions and
supernovae explosions. Among the different processes that need to be
considered, the formation kinetics and cooling of molecular hydrogen are of
prime interest, since they provide the only way to lower the gas temperature to
values well below $\sim$10$^4$~K. In previous works, the internal energy level
structure of H$_2$ and its cation has been treated in the approximation of
rovibrational ground state at low densities, or trying to describe the dynamics
using some arbitrary $v>0$ H$_2$ level that is considered representative of the
excited vibrational manifold. In this study, we compute the vibrationally
resolved kinetics for the time-dependent chemical and thermal evolution of the
post-shock gas in a medium of primordial composition. The calculated
non-equilibrium distributions are used to evaluate effects on the cooling
function of the gas and on the cooling time. Finally, we discuss the dependence
of the results to different initial values of the shock velocity and redshift.
