Past exam of the chemistry A course of the University of Cambridge / 2021 / Paper 1 / A4 / a / iii

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- Solution iii

Solution

Use the classes

(E, 2 C_{3}, 3 σ_{v})

of the C3v point group. For the three hydrogen

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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