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

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.

An infrared fundamental must transform like a component of the electric dipole moment; a Raman spectroscopy fundamental must transform like a component of the symmetric polarizability tensor. For this centrosymmetric D4h molecular point group, the resulting classification is:

Here parallel and perpendicular refer to the principal fourfold axis. The table obeys infrared and Raman mutual exclusion.

Three separated infrared fundamentals: one parallel $A_{2u}$ band and two perpendicular $E_u$ bands. Each $E_u$ frequency belongs to a doubly degenerate mode, but degeneracy does not create two distinct band origins.

The square-planar rotor is an oblate symmetric top. A parallel band has $\Delta K=0$ and $\Delta J=0,\pm 1$, producing P and R progressions with approximate spacing $2\tilde{B}$ and a central Q contribution from $K\ne 0$; the $K=0$ parallel Q branch is absent. At modest resolution many nearly coincident $K$ components merge.

A perpendicular band has $\Delta K=\pm 1$ and $\Delta J=0,\pm 1$. Its Q subbranches produce a characteristic series of sharper features on the broader unresolved P/R envelope. Neglecting changes of rotational constants on vibration, successive perpendicular Q features are separated by approximately $2|\tilde{B}-\tilde{C}|$. These are rotational contours rather than extra normal-mode fundamentals.