Past exam of the chemistry A course of the University of Cambridge / 2024 / Paper 1 / A5 / b / i

Choose the phase of each inward-pointing ligand sigma donor orbital positive toward its own Mo centre. The eight donors then transform by site permutation. In the usual D4h molecular point group class order their characters areOnly $E$ and a chloride-containing vertical mirror plane leave ligand functions individually fixed. For example ${\sigma }_{xz}$ fixes the four donors at $(\pm a,0,\pm c)$ and exchanges the other four; the four fixed functions retain their signs.

Molecular character reduction givesThe dimensions sum to $1+1+2+1+1+2=8$. Equivalently, take each square's four combinations: the all-in-phase pattern, the alternating x-versus-y pattern, and a two-dimensional pair. Combine corresponding patterns between upper and lower squares to obtain the six irreps above.