  The adiabatic index of H$_2\,$ ($\gamma_{\mathrm{H_2}}$) is non-constant at
temperatures between $100-10^4\,\mathrm{K}$ due to the large energy spacing
between its rotational and vibrational modes. For the formation of the first
stars at redshifts 20 and above, this variation can be significant because
primordial molecular clouds are in this temperature range due to the absence of
efficient cooling by dust and metals. We study the possible importance of
variations in $\gamma_{\mathrm{H_2}}$ for the primordial initial mass function
by carrying out 80 3D gravito-hydrodynamic simulations of collapsing clouds
with different random turbulent velocity fields, half using fixed $\gamma_{\rm
H_2} = 7/5$ in the limit of classical diatomic gas (used in earlier works) and
half using an accurate quantum mechanical treatment of $\gamma_{\mathrm{H_2}}$.
We use the adaptive mesh refinement code FLASH with the primordial chemistry
network from KROME for this study. The simulation suite produces almost 400
stars, with masses from $0.02 - 50$ M$_\odot$ (mean mass $\sim
10.5\,\mathrm{M_{\odot}}$ and mean multiplicity fraction $\sim 0.4$). While the
results of individual simulations do differ when we change our treatment of
$\gamma_{\mathrm{H_2}}$, we find no statistically significant differences in
the overall mass or multiplicity distributions of the stars formed in the two
sets of runs. We conclude that, at least prior to the onset of radiation
feedback, approximating H$_2$ as a classical diatomic gas with $\gamma_{\rm
H_2} = 7/5$ does not induce significant errors in simulations of the
fragmentation of primordial gas. Nonetheless, we recommend using the accurate
formulation of the H$_2\,$ adiabatic index in primordial star formation studies
since it is not computationally more expensive and provides a better treatment
of the thermodynamics.
