Molecular partition function ($q$)

The single-molecule state sum $q={\sum }_s{e}^{-\beta E_s}$ for specified energy references and external constraints. In a dilute classical ideal gas, indistinguishability gives $Q_N=q^N/N!$; translation supplies the factor proportional to volume.

If energies are measured upward from a species reference ${\epsilon }_0$, its absolute single-particle sum is $q_{\mathrm{abs}}=e^{-\beta {\epsilon }_0}q$. The ideal-gas chemical potential is then ${\epsilon }_0-k_{B}T\ln (q/N)$. A zero-point factor can be included either in $q$ or in the reference energy, but not both.

For a free molecule in a uniform ideal gas, $q_i$ is proportional to volume. The ratio $f_i=q_i/V$ is independent of the container volume at fixed temperature and has units of inverse volume.