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 / c / i / Solution

The nontotally symmetric longitudinal modes are the two ${\Sigma }_u^{+}$ coordinates $t$ and $a$. Both have $q_3-q_2=0$, so the middle spring is not extended. Hence its spring constant cannot alter their energies:These results are exact for every positive $\gamma$, not just perturbative. The zero mode expresses overall translation invariance.