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The angular factors of the nickel $3d$ atomic orbitals transform like quadratic Cartesian functions. In the D4h molecular point group, the assignments areAll five are even under inversion symmetry, hence the $g$ suffixes. The pair $(d_{xz},d_{yz})$ is mixed by a quarter-turn and spans a two-dimensional irreducible representation.

In the Td molecular point group, the corresponding result isThere is no inversion label because $T_d$ has no inversion centre. In both cases the dimensions add to five.

For the ideal tetracyanonickelate(II) geometries, the square planar molecular geometry has $D_{4h}$, the D4h molecular point group, while the tetrahedral molecular geometry has $T_d$, the Td molecular point group. Collinearity of each Ni–C–N unit means that including the terminal nitrogen atoms does not lower either point group.

For the square, take the Ni–C directions as $+x,+y,-x,-y$ and $z$ perpendicular to the plane. For the tetrahedron, use the conventional cubic axes with ligand directions proportional to $(1,1,1),(1,-1,-1),(-1,1,-1),(-1,-1,1)$. These choices fix the irreducible representation labels used below.

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.

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.