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 / 21 / a / i / Solution

The nine normal coordinates transform as $A_{1g}+B_{1g}+B_{2g}+A_{2u}+B_{2u}+2E_u$. Place the molecule in the $xy$ plane, with its four outer atoms on the $x$ and $y$ axes, as specified for the vertical reflection planes. In the class orderthe fifteen Cartesian displacements have characters of a representationFor each operation this character is the number of fixed atoms times the trace of its polar-vector matrix. Subtract translations $A_{2u}+E_u$ and rotations $A_{2g}+E_g$, givingMolecular character reduction in the D4h molecular point group yields the stated decomposition. Its dimensions sum to $1+1+1+1+1+2(2)=9=3(5)-6$, providing an independent degree-of-freedom check.