 Solution

ID: past-exam-of-the-chemistry-a-course-of-the-university-of-cambridge/2021/paper-1/a4/a/iii/solution

Use the classes

(E, 2 C_{3}, 3 σ_{v})

1 s

atomic orbitals, the character of a representation counts orbitals fixed by each symmetry operation. Identity fixes three, a nontrivial threefold rotation fixes none, and a vertical mirror fixes one. Therefore

χ_{H} = (3, 0, 1) .

The character inner product gives multiplicities

n_{A_{1}} = \frac{3 + 3}{6} = 1, n_{A_{2}} = \frac{3 - 3}{6} = 0, n_{E} = \frac{2 \times 3}{6} = 1.

Thus $Γ_{H} = A_{1} \oplus E$ , with the

E

contribution accounting for two independent symmetry orbitals.

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