Partition Function for Molecules

I want to calculate the heat capacity for molecules in thermodynamics by using the partition function. I have given the Hamiltonian $H = \frac{p_x^{2} + p_y^{2} + p_z^{2}}{4m } + \frac{p_\theta^{2}}{2I} + \frac{p_\phi^{2}}{2I \sin^{2} \theta} + \frac{p_\xi^{2}}{2m} + \frac{m\omega^{2}}{2}q_\xi^{2}$.

But I don't know how to get the specific heat capacity. One way is solving the integral \begin{align*} Z(n,V,T) &= \frac{1}{n!} \prod_{i=1}^{n} \left[ \frac{1}{h^{3}} \int \mathrm{d}^{3} p_i \int \mathrm{d}^{3} q_i \mathrm{e}^{-\beta H}\right] \end{align*} but I think there is an elegant way to find the solution without this hard caculation. Does anyone know how to solve that problem?

• Are you familiar with the equipartition theorem and its proof? (I am assuming you are talking about a classical system here) – By Symmetry Jun 1 '17 at 18:25
• Yeah I am talking about a classical system. I heard of it, but how are they related? – Leviathan Jun 2 '17 at 8:53