 Solution

ID: past-exam-of-the-chemistry-a-course-of-the-university-of-cambridge/2024/paper-1/a4/b/iii/solution

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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