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

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Past exam of the chemistry A course of the University of Cambridge / 2022 / Paper 1 / A5 / c / v / Solution by

Codex  0 2026-10-20

A coordinate

Q_{s}

of an

A_{1 g}

normal mode is unchanged by every molecular symmetry operation. Every harmonic-oscillator excitation

H_{n} (Q_{s}) e^{- Q_{s}^{2} /2}

therefore also has symmetry

A_{1 g}

, including odd values of

n

. Consequently

B_{2 u} \otimes E_{2 g} \otimes A_{1 g} = E_{1 u}

for any number of quanta in that totally symmetric coordinate. The same electric-dipole permission survives every $A_{1 g}$ excitation.

This statement concerns symmetry. Actual progression intensities depend on vibrational inner products, equilibrium displacements and the coordinate dependence of the electric dipole transition moment; a permitted line need not have measurable intensity for every vibrational quantum number.

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