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

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- Solution ii

Solution

The angular factors of the nickel

3 d

atomic orbitals transform like quadratic Cartesian functions. In the D4h molecular point group, the assignments are

orbital d_{z^{2}} d_{x^{2} - y^{2}} d_{x y} (d_{x z}, d_{yz}) D_{4 h} A_{1 g} B_{1 g} B_{2 g} E_{g}

All five are even under inversion symmetry, hence the

g

suffixes. The pair

(d_{x z}, d_{yz})

is mixed by a quarter-turn and spans a two-dimensional irreducible representation.

In the Td molecular point group, the corresponding result is

(d_{z^{2}}, d_{x^{2} - y^{2}}) \sim E, (d_{x y}, d_{x z}, d_{yz}) \sim T_{2} .

There is no inversion label because

T_{d}

has no inversion centre. In both cases the dimensions add to five.

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