At $\gamma=0$, each independent diatomic spring has one translation and one stretch. Consequently the angular frequencies are
$$
\boxed{\omega=0,\ 0,\ \sqrt{2k/m},\ \sqrt{2k/m}.}
$$
The two zero modes are independent centre-of-mass translations of the two molecules. There are no transverse coordinates in this model; adding transverse degrees of freedom would be a different problem.


Solution

Past exam of the chemistry course of the year 3 of the natural sciences course of the University of Cambridge / 2021 / Paper 2A / Section B / 23 / b / i / Solution

 Solution (source code)