Fermi resonance
# Fermi resonance

A Fermi resonance is the shifting of the energies and intensities of absorption bands in an infrared or Raman spectrum. It is a consequence of quantum-mechanical wavefunction mixing. The phenomenon was explained by the Italian physicist Enrico Fermi.

## Selection rules and occurrence

Two conditions must be satisfied for the occurrence of Fermi resonance:

* The two vibrational modes of a molecule transform according to the same irreducible representation in their molecular point group. In other words, the two vibrations must have the same symmetries (Mulliken symbols).
* The transitions coincidentally have very similar energies.

Fermi resonance most often occurs between fundamental and overtone excitations, if they are nearly coincident in energy.

Fermi resonance leads to two effects. First, the high-energy mode shifts to higher energy, and the low-energy mode shifts to still lower energy. Second, the weaker mode gains intensity (becomes more allowed), and the more intense band decreases in intensity. The two transitions are describable as a linear combination of the parent modes. Fermi resonance does not lead to additional bands in the spectrum, but rather shifts in bands that would otherwise exist.

## Examples

### Ketones

High-resolution IR spectra of most ketones reveal that the "carbonyl band" is split into a doublet. The peak separation is usually only a few $cm^{−1}.$ This splitting arises from the mixing of $ν_{CO}$ and the overtone of HCH bending modes.

### $CO_2$

In $CO_2,$ the bending vibration $ν_2$ (667 $cm^{−1})$ has symmetry $Π_u.$ The first excited state of $ν_2$ is denoted $01^10$ (no excitation in the $ν_1$ mode (symmetric stretch), one quantum of excitation in the $ν_2$ bending mode with angular momentum about the molecular axis equal to ±1, no excitation in the $ν_3$ mode (asymmetric stretch)) and clearly transforms according to the irreducible representation $Π_u.$ Putting two quanta into the $ν_2$ mode leads to a state with components of symmetry $(Π_u$ × $Π_u)+$ = $Σ^+_g$ + Δ $_g.$ These are called $02^00$ and $02^20$ respectively. $02^00$ has the same symmetry $(Σ^+_g)$ and a very similar energy to the first excited state of $v_1$ denoted 100 (one quantum of excitation in the $ν_1$ symmetric stretch mode, no excitation in the $ν_2$ mode, no excitation in the $ν_3$ mode). The calculated unperturbed frequency of 100 is 1337 $cm^{−1},$ and, ignoring anharmonicity, the frequency of $02^00$ is 1334 $cm^{−1},$ twice the 667 $cm^{−1}$ of $01^10.$ The states $02^00$ and 100 can therefore mix, producing a splitting and also a significant increase in the intensity of the $02^00$ transition, so that both the $02^00$ and 100 transitions have similar intensities.

