Past exam of the chemistry A course of the University of Cambridge / 2025 / Paper 1 / A4 / b / ii / Solution

For operations ordered as $(E,C_2(z),{\sigma }_{xz},{\sigma }_{yz})$, the characters of the twelve Cartesian displacements areOnly atoms left fixed contribute: all four for $E$ and ${\sigma }_{xz}$, but only C and O for $C_2$ and ${\sigma }_{yz}$. Multiplying their counts by the Cartesian traces $3,-1,1,1$ gives these numbers. Subtractto obtain ${\chi }_{\mathrm{vib}}=(6,0,4,2)$. The character table reduction givesThe dimensions sum to $6=3N-6$, as required. The $B_2$ mode is the out-of-plane wag with this axis convention.