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

A coordinate $Q_s$ of an $A_{1g}$ 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 $A_{1g}$, including odd values of $n$. Consequentlyfor any number of quanta in that totally symmetric coordinate. The same electric-dipole permission survives every $A_{1g}$ 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.