ABSTRACT We have critically reviewed the literature pertaining to reactions that are significant for the chemistry of hydrogen-, deuterium-, and helium-bearing species in the homogeneous early Universe. For each reaction rate coefficient, we provide a fit in the modified-Arrhenius form, specifying the corresponding uncertainty and temperature range. This new network, limited to 21 reactions, should be the most reliable to date. Combined with accurate state-to-state rate coefficients for inelastic and reactive collisions involving H2 and HD, it allows us for the first time to follow the evolution of the abundances of atomic and molecular species, level populations of H2 and HD, and the ortho:para ratio (OPR) of H2, in a self-consistent fashion during the adiabatic expansion of the universe. The abundances of H2 and HD change only marginally compared to previous models, indicating that the uncertainties on the main reaction rate coefficients have essentially been removed. We also find that the adiabatic expansion has a dramatic effect on the OPR of H2, which freezes-out at redshifts z ≲ 50. In contrast, at higher redshifts, the populations of the rotational levels of H2 and HD are predicted to be fully thermalized at the temperature of the cosmic background radiation field, a result that conflicts with some recent, independent calculations. This new network allows the chemistry of primordial gas to be followed during the early phase of collapse towards Population III star progenitors.