Past exam of the chemistry course of the year 3 of the natural sciences course of the University of Cambridge / 2022 / Paper 3 / 49 / c / i / Solution

In the class order $(E,8C_3,3C_2,6S_4,6{\sigma }_d)$, the 15 Cartesian nuclear displacements of methane have characters $(15,0,-1,-1,3)$. Only nuclei left fixed by an operation contribute to that operation's trace. Subtract translations $T_2$ and rotations $T_1$ to obtain the methane normal-mode representationIts dimension is $1+2+3+3=9=3(5)-6$. There are four frequency families: a nondegenerate symmetric stretch, a doubly degenerate bend, and triply degenerate stretch and bend modes.