# Vibration – Rotation Spectra

If the vibrational spectrum of a gas-phase heteronuclear diatomic molecule is obtained at high enough resolution, it is found that each line of the spectrum actually consists of a large number of closely spaced components.

These components arise from the rotational transitions that accompany vibrational transitions – since the gaps between vibrational levels are so much greater than those between rotational levels, the lines representing rotational transitions are much more closely spaced than those representing vibrational transitions. Thus the rotational lines are only observable as components within the broader lines representing vibrational transitions. (We would expect rotational transitions to accompany vibrational transitions, as the vibrational transition leads to an instantaneous change in the bond length of the molecule, which in turn alters its moment of inertia and thus brings about a change in the rotational energy level of the molecule.) The appearance of a vibration-rotation spectrum is as follows:

Analysis of the quantum mechanics of this situation shows that for a diatomic molecule undergoing a vibrational transition with Δν = 1 , the rotational quantum number J can change by ±1. For molecules with an angular momentum about their axis, Δν = 0 is also allowed. (e.g. NO , which has an electronic angular momentum about its axis, arising from an unpaired electron in the molecule.)

The appearance of the spectrum can best be discussed using the combined rotation-vibration terms, S, which depend upon both quantum numbers J and ν. The simplest expression that can be obtained, ignoring both anharmonicity and centrifugal distortion, is as follows:

When we consider the vibrational transition ν + 1 ¬ ν , we can see that the rotational transitions will lead to the formation of two or three branches within the vibrational line.

The transitions for which ΔJ = -1 form the P-branch of the spectrum:

The P-branch thus consists of lines at lower frequency than , the fundamental frequency. They are predicted to be equally spaced, with a distance of 2B between them. (Note however, that the effects of anharmonicity and centrifugal distortion mean that this not true in practice.)

The transitions for which ΔJ = +1 form the R-branch of the spectrum:

The R-branch thus consists of lines at higher frequency than the fundamental frequency, again predicted to be equally spaced by 2B, but again this is not true in practice.

The transitions for which ΔJ = 0 , if they are permitted, form the Q-branch of the spectrum:

This is predicted to be a single line at the fundamental frequency (the frequency corresponding to the pure vibrational transition). However, the effects of anharmonicity and centrifugal distortion mean that it actually consists of a cluster of very closely spaced lines around.