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 Incoming links: Tables for Group Theory by P. W. Atkins and others

Past exam of the chemistry A course of the University of Cambridge / 2024 / Paper 1 / A4 / b / iii / Solution 2026-10-20

Taking the matrix trace of those matrices gives

class χ_{p} E 3 2 S_{4} - 1 C_{2} - 1 2 C_{2}^{'} - 1 2 σ_{d} 1

The one-dimensional

p_{z}

subspace has characters

(1, - 1, 1, - 1, 1)

, identifying

B_{2}

. The pair

(p_{x}, p_{y})

has characters

(2, 0, - 2, 0, 0)

, identifying

E

. Therefore

Γ_{p_{x}, p_{y}, p_{z}} = B_{2} \oplus E .

The dimensions add to three, as required. The

p_{x}, p_{y}

pair must be retained together: a ninety-degree improper rotation mixes them, so either one alone is not a complete symmetry basis.^[ref]

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Past exam of the chemistry A course of the University of Cambridge / 2024 / Paper 1 / A4 / c / ii / Solution 2026-10-20

The highest expected orbital degeneracy is five. Ideal buckminsterfullerene has Ih molecular point group symmetry. Its irreducible orbital representations include five-dimensional

H_{g}

and

H_{u}

representations; no single-valued orbital irrep has higher dimension.

This counts distinct spatial molecular orbitals. It does not multiply by electron spin, and it excludes accidental coincidences between unrelated irreps. In a real environment, symmetry lowering can split these degeneracies.^[ref]

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