Partition functions 1: Improved partition functions and thermodynamic
  quantities for normal, equilibrium, and ortho and para molecular hydrogen
  Aims. In this work we rigorously show the shortcomings of various
simplifications that are used to calculate the total internal partition
function. These shortcomings can lead to errors of up to 40 percent or more in
the estimated partition function. These errors carry on to calculations of
thermodynamic quantities. Therefore a more complicated approach has to be
taken.
  Methods. Seven possible simplifications of various complexity are described,
together with advantages and disadvantages of direct summation of experimental
values. These were compared to what we consider the most accurate and most
complete treatment (case 8). Dunham coefficients were determined from
experimental and theoretical energy levels of a number of electronically
excited states of H$_2$ . Both equilibrium and normal hydrogen was taken into
consideration.
  Results. Various shortcomings in existing calculations are demonstrated, and
the reasons for them are explained. New partition functions for equilibrium,
normal, and ortho and para hydrogen are calculated and thermodynamic quantities
are reported for the temperature range 1 - 20000 K. Our results are compared to
previous estimates in the literature. The calculations are not limited to the
ground electronic state, but include all bound and quasi-bound levels of
excited electronic states. Dunham coefficients of these states of H$_2$ are
also reported.
  Conclusions. For most of the relevant astrophysical cases it is strongly
advised to avoid using simplifications, such as a harmonic oscillator and rigid
rotor or ad hoc summation limits of the eigenstates to estimate accurate
partition functions and to be particularly careful when using polynomial fits
to the computed values. Reported internal partition functions and thermodynamic
quantities in the present work are shown to be more accurate than previously
available data.
