Combination and overtone absorptions in the near-infrared

So far, the effect of anharmonicity in determining the frequency of overtone (An 1) absorptions, and its effect in relaxing quantum selection rules to allow these transitions to have some absorption intensity have been considered. 

However, in polyatomic molecules transitions to excited states involving two vibrational modes at once (combination bands) are also weakly allowed, and are also affected by the anharmonicity of the potential. The role of combination bands in the NIR can be significant. As has been noted, the only functional groups likely to impact the NIR spectrum directly as overtone absorptions are those containing C—H, N—H, O—H or similar functionalities. However, in combination with these hydride-bond overtone vibrations, contributions from other, lower-frequency fundamental bands such as C=0 and C=C can be involved as overtone-combination bands. The effect may not be dramatic in the rather broad and overcrowded NIR absorption spectrum, but it can still be evident and useful in quantitative analysis. 

Taking the example of the water molecule, a non-linear triatomic, one expects three fundamental vibrations. If a simple harmonic potential is applied, one could write 

For a simple combination band such as the transition from (0,0,0) to (0,1,1) the AE would then be calculated as 

Note that to first order this is simply the sum of the fundamental frequencies, after allowing for anharmonicity. This is an oversimplification, because, in fact, combination bands consist of transitions involving simultaneous excitation of two or more normal modes of a polyatomic molecule, and therefore mixing of vibrational states occurs and 

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