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Aims. In this work we rigorously show the shortcomings of varioussimplifications that are used to calculate the total internal partitionfunction. These shortcomings can lead to errors of up to 40 percent or more inthe estimated partition function. These errors carry on to calculations ofthermodynamic quantities. Therefore a more complicated approach has to betaken.  Methods. Seven possible simplifications of various complexity are described,together with advantages and disadvantages of direct summation of experimentalvalues. These were compared to what we consider the most accurate and mostcomplete treatment (case 8). Dunham coefficients were determined fromexperimental and theoretical energy levels of a number of electronicallyexcited states of H$_2$ . Both equilibrium and normal hydrogen was taken intoconsideration.  Results. Various shortcomings in existing calculations are demonstrated, andthe reasons for them are explained. New partition functions for equilibrium,normal, and ortho and para hydrogen are calculated and thermodynamic quantitiesare reported for the temperature range 1 - 20000 K. Our results are compared toprevious estimates in the literature. The calculations are not limited to theground electronic state, but include all bound and quasi-bound levels ofexcited electronic states. Dunham coefficients of these states of H$_2$ arealso reported.  Conclusions. For most of the relevant astrophysical cases it is stronglyadvised to avoid using simplifications, such as a harmonic oscillator and rigidrotor or ad hoc summation limits of the eigenstates to estimate accuratepartition functions and to be particularly careful when using polynomial fitsto the computed values. Reported internal partition functions and thermodynamicquantities in the present work are shown to be more accurate than previouslyavailable data.

Partition functions 1: Improved partition functions and thermodynamic quantities for normal, equilibrium, and ortho and para molecular hydrogen