The vibrational selection rule Av = ±1 for electric dipole transitions is only absolute for a pure simple harmonic oscillator. The binding force between real molecules, while proportional to displacement for small oscillations, has non-negligible higher orders, or anharmonic elements for larger oscillations. These anharmonic elements relax the selection rules to Av = ±1, ±2, ±3,... giving rise to "overtone" bands, for
V] symmetric stretch V': bend (A) v; bend (B) v.i asymmetric stretch which > 1. Overtone bands are always far less intense than fundamental bands, for which A^ = ±1.
The population of vibrational energy states is also covered by a Boltzmann distribution, and thus at low temperatures, most molecules are in their lowest vibrational state and the absorption spectra of low-temperature molecules are dominated by transitions from this ground state. However, as the temperature rises, so does the population of the higher vibrational states giving rise to so-called "hot bands".
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