Vibrational Selection Rules
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The gross selection rule for vibrational transitions is that the electric dipole moment of the molecule must change in the course of the vibrational motion.

e.g. homonuclear diatomics are infrared inactive - stretching of the bond does not alter the dipole moment of the molecule, it remains at zero. However, heteronuclear diatomics may be infrared active, as bond stretching increases the distance between the positive and negative ends of the molecule, increasing its dipole moment.

Note that this selection rule does not require a molecule to possess a permanent electric dipole moment, the change may be from a state without a dipole moment to a state with one.

The specific vibrational selection rule in terms of the quantum number ν is:

The gaps between vibrational energy levels are larger than those between rotational energy levels, to the extent that at room temperature molecules are almost always in their vibrational ground state, ν = 0. Thus a vibrational spectrum commonly consists of only one line, corresponding to the transition from ν = 0 to ν = 1. This is called the fundamental transition. Other transitions are only weakly observed, due to the tiny populations of the lower states in these transitions.

Note that as a result of our approximations, the implication is that the energy terms of allowed vibrational transition are a constant value:

which further indicates that the spectrum should always consist of a single line anyway.

However, the approximation of the potential energy curve to a parabola is an approximation, and the breakdown of this approximation leads to the transitions lying at slightly different frequencies in reality. Thus several lines may be observed in the spectrum, particularly if the molecule is formed in some vibrationally excited states (ν > 0) as transitions from these states will then have greater intensities than would normally be the case.

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