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.
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