Past exam of the chemistry A course of the University of Cambridge / 2023 / Paper 1 / A4 / b

The D4h molecular point group contains an inversion centre, so infrared and Raman mutual exclusion applies. A normal mode of $u$ parity can have vibrational infrared activity, whereas the quadratic polarizability tensor components responsible for Raman vibrational activity have $g$ parity. No individual fundamental can be active in both spectra.

The Td molecular point group lacks an inversion centre. In particular, $T_2$ normal modes transform both as electric-dipole components and as off-diagonal polarizability tensor components, so they occur in both spectra. Coincident IR and Raman fundamentals therefore favour the tetrahedral structure; symmetry requires no such coincidences for the square. Weak or unresolved features can conceal coincidences experimentally, while accidental equality of different frequencies is not a violation of mutual exclusion.

There are four internal C–N stretching coordinates. A stretch length is unchanged in sign when a molecular symmetry operation carries a bond to an equivalent bond, so these coordinates have the same permutation group representation as the four ligand sigma functions.

For the square,Only $E_u\sim (x,y)$ has vibrational infrared activity. The $A_{1g}$ and $B_{1g}$ stretches have Raman vibrational activity. Thus the ideal square gives one IR feature and two Raman features, with no common fundamental. The two components of $E_u$ have one frequency, not two different lines.

For the tetrahedron,The $T_2$ stretch is IR active, while both $A_1$ and $T_2$ are Raman active. Thus the tetrahedron also gives one IR feature and two Raman features, but one feature, $T_2$, is shared between the spectra. Its three components likewise describe three independent displacements at one frequency.

The separate line counts alone do not distinguish the geometries. Comparing their frequencies does: the tetrahedral IR C–N feature coincides with one Raman C–N feature, whereas the square has no symmetry-enforced coincidence. This assumes resolved fundamentals and excludes accidental equality of unrelated frequencies.

The nine-atom nonlinear ion has $3N-6=21$ vibrational degrees of freedom. To construct its displacement group representation, only atoms left in place by an operation contribute to the character. Each such atom contributes the trace of the operation on $(x,y,z)$:For example, reflection in the square's molecular plane fixes all nine atoms and has trace $1$, giving character $9$; inversion fixes nickel alone and has trace $-3$, giving $-3$. Molecular character reduction usesFor the D4h molecular point group, with primed twofold axes along the Ni–C directions, the class order and displacement characters areReducing and removing translations $A_{2u}+E_u$ and rotations $A_{2g}+E_g$ givesThe vibrational infrared activity belongs to $2A_{2u}+4E_u$, giving six features. The Raman vibrational activity belongs to $2A_{1g}+2B_{1g}+2B_{2g}+E_g$, giving seven features. The remaining $A_{2g}+2B_{2u}$ modes are silent. There are zero shared fundamentals, by infrared and Raman mutual exclusion. Counting dimensions gives $10$ IR-active, $8$ Raman-active and $3$ silent vibrational coordinates, totaling $21$.

For the Td molecular point group, the corresponding calculation starts fromand gives ${\Gamma }_{3N}=2A_1+2E+2T_1+5T_2$. Subtracting translations $T_2$ and rotations $T_1$ leavesThe $4T_2$ fundamentals give four IR features. The $2A_1+2E+4T_2$ fundamentals give eight Raman features, of which four coincide with the IR features. The $T_1$ fundamental is silent. The dimensions are $2+4+3+12=21$.

Thus the predicted IR/Raman/coincidence counts are $6/7/0$ for the square and $4/8/4$ for the tetrahedron. These are numbers of distinct ideal fundamental frequencies: each $E$ or $T$ multiplet contributes one feature, and different copies of the same irreducible representation generally have different frequencies. Accidental degeneracy, overtones and combination bands are outside this count.