  At low temperatures, the main coolant in primordial gas is molecular
hydrogen, H2. Recent work has shown that primordial gas that is not collapsing
gravitationally but is cooling from an initially ionized state forms hydrogen
deuteride, HD, in sufficient amounts to cool the gas to the temperature of the
cosmic microwave background. This extra cooling can reduce the characteristic
mass for gravitational fragmentation and may cause a shift in the
characteristic masses of population III stars. Motivated by the importance of
the atomic and molecular data for the cosmological question, we assess several
chemical and radiative processes that have hitherto been neglected: the
sensitivity of the low temperature H2 cooling rate to the ratio of ortho-H2 to
para-H2, the uncertainty in the low temperature cooling rate of H2 excited by
collisions with H, the effects of cooling from H2 excited by collisions with H+
and e-, and the large uncertainties in the rates of several of the reactions
responsible for determining the H2 fraction in the gas.
  We show that the most important of the neglected processes is the excitation
of H2 by collisions with protons and electrons. This cools the gas more rapidly
at early times, and so it forms less H2 and HD at late times. This fact, as
well as several of the chemical uncertainties presented here, significantly
affects the thermal evolution of the gas. We anticipate that this may lead to
clear differences in future detailed 3D studies of first structure formation.
Finally, we show that although the thermal evolution of the gas is in principle
sensitive to the ortho-para ratio, in practice the standard assumption of a 3:1
ratio produces results that are almost indistinguishable from those produced by
a more detailed treatment. (abridged)
