Rotational partition function

Rotational energies are quantized. For a diatomic molecule like CO or HCl, the allowed rotational energies are $E_{rot}={\frac {{\mathbf {J}}^{2}}{2I}}={\frac {j(j+1)\hbar ^{2}}{2I}}=j(j+1)\epsilon$

The number of degenerate states g_j for each level j is 2j+1. The rotational partition function is therefore $Z_{rot}=\sum _{j=0}^{\infty }g_{j}e^{-E_{j}/kT}=\sum _{j=0}^{\infty }(2j+1)e^{-j(j+1)\epsilon /kT}$