Here the response of PhD Professor at Bologna University Italy:
"You have no clues! You are so ignorant and arrogant. The moment of inertia does not depend on the temperature! Have you ever hear about vibrational states? The moment of inertia depends on the averaged structure the molecule has in the vibrational state where it lies. Why are you talking about things you do not know? You have to study quantum chemistry, statistical thermodynamics and spectroscopy. Then, you will be entitled to discuss about this topic without bullshitting. "A diatomic molecule of a gas have its atoms near each either at very low temperature and so their moment of inertia is low": this is hte most stupid thing I hav ever heard!"
The moment of inertia of a diatomic molecule is generally considered to be temperature-independent, especially at low temperatures. This is based on the assumption that the vibrational and rotational contributions to the molecule's energy are decoupled.
So why the Cv of mono-atomic gas has a maximum at low temperature?
Read the following document about the Cv and moment of inertia at low temperature:
https://www.csbsju.edu/Documents/Natural%20Sciences/H2_MP06.pdf
The concept of moment of inertia doesn't directly apply to monoatomic gases as it does for rotating diatomic or polyatomic molecules. Instead, the specific heat capacity of monoatomic gases, like noble gases, may depend on temperature due to quantum effects and transitions between energy levels.
As long as the harmonic approximation for vibrational states is valid, temperature independence can be assumend (which is the case at low temperatures). If you get to higher states in which anharmonicity begins to play a role (i.e. at high temperatures), the mean bond length will begin to expand thus altering moments of inertia:
Article Morse potential rotation effect and effective potential for ...
Additionally, centrifugal distortion will play an increasing role at higher rotations.
When it comes to the heat capacity maximum from the slides you posted, you could just look up the source given there - the next figure explains it all: para-D2 has no maximum, but ortho-D2 does have a significant one (while it's the other way round for H2), so this is a coupling effect.
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