 Solution

ID: past-exam-of-the-chemistry-a-course-of-the-university-of-cambridge/2022/paper-1/a5/c/ii/solution

The overall vibrational ground state is invariant under every molecular symmetry operation, so it transforms as

A_{1 g}

. Combining it with the electronic ground state gives

Γ_{initial} = A_{1 g} \otimes A_{1 g} = A_{1 g} .

For a degenerate harmonic normal mode, the zero-quantum wavefunction depends on the invariant sum

Q_{x}^{2} + Q_{y}^{2}

, so the ground vibrational symmetry remains totally symmetric even when excited vibrational states form multiplets.

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