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Selection Rules

Rules governing the probability of transitions between vibrational energy levels, based on symmetry considerations.
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The statement of the theorem

For an infrared transition between vibrational states v′′→v′v'' \to v', the transition is allowed if and only if the transition dipole moment integral ⟨v′∣μ∣v′′⟩\langle v' | \boldsymbol{\mu} | v'' \rangle is non-zero. Mathematically, this requires the derivative of the dipole moment with respect to the normal coordinate QQ: ∂μ∂Q∣Q=0≠0\left. \frac{\partial \boldsymbol{\mu}}{\partial Q} \right|_{Q=0} \neq \mathbf{0}. This condition ensures that the change in dipole moment during the vibration is non-zero.