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In statistical mechanics we have the equipartition theorem which can derive heat capacity simply from degrees of freedom. For example, diatomic gases have an increase heat capacity of $ \frac{7}{2} N k_B T $.

I am studying Molecular Orbital Theory and there is a big emphasis on discrete symmetries and their effects on electron configuration. I was wondering if you can find statistical effects of the discrete symmetries of molecules in such a way that you could experiment on something like the heat capacity and deduce that the molecule in question has a given symmetry.

For example, in $ AH_2 $ molecules, if they bend, then we have an additional degree of freedom that could contribute to the equipartition heat capacity. This is still a continuous symmetry that was broken, but we could imagine discrete ones like $ ACH_3 $. If A is a hydrogen atom, then we have methane and a $ E 8C_3 3C_2 6S_4 6\sigma_d $, otherwise, we just get $ E 2C_3 3\sigma_v $ symmetry (from here).

Does this increase in symmetry leave any statistical artifacts that experiment could show?

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Please correct me if I am wrong!!

After thinking for a while, I believe that you need a thermodynamic control parameter (think pressure, volume, magnetic field, etc.) that breaks the symmetry.

Consider a system with a phase space $ \Lambda $ that is symmetric under group operations $ g \in G $. If you calculate the partition function, you notice it can be broken up,

$$\begin{align} Z &= \int_\Lambda dz~e^{-\beta \mathcal{H}(q, p; A)} \\ &= \sum_{g \in G} \int_{\Lambda ~/~ G} e^{-\beta ~\mathcal{H}\big(T_g(q), ~T_g(p) ~;~ T_g(A)\big)} \\ &= \sum_{g \in G} \int_{\Lambda ~/~ G} e^{-\beta ~\mathcal{H}\big(q, p ~;~ T_g(A)\big)} \\ &= \sum_{g \in G} Z_g \end{align}$$

where $ A $ is a statistical control variable like pressure or volume.

Computing some thermodynamic quantity would be simply done by,

$$\begin{align} A &= \frac{\partial \log Z}{\partial A} \\ &= Z^{-1} \sum_{g \in G} \frac{\partial Z_g}{\partial A} \end{align}$$

If $ A $ doesn't transform under the group action, then thermodynamics will not have an artifact.

Working on an example

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