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.^[1] 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.^{[citation needed]}

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.^[2]

CO₂

In CO₂, the bending vibration ν₂ (667 cm⁻¹) has symmetry Π_u. The first excited state of ν₂ is denoted 01¹0 (no excitation in the ν₁ mode (symmetric stretch), one quantum of excitation in the ν₂ bending mode with angular momentum about the molecular axis equal to ±1, no excitation in the ν₃ mode (asymmetric stretch)) and clearly transforms according to the irreducible representation Π_u. Putting two quanta into the ν₂ mode leads to a state with components of symmetry (Π_u × Π_u)₊ = Σ⁺_g + Δ _g. These are called 02⁰0 and 02²0 respectively. 02⁰0 has the same symmetry (Σ⁺_g) and a very similar energy to the first excited state of v₁ denoted 100 (one quantum of excitation in the ν₁ symmetric stretch mode, no excitation in the ν₂ mode, no excitation in the ν₃ mode). The calculated unperturbed frequency of 100 is 1337 cm⁻¹, and, ignoring anharmonicity, the frequency of 02⁰0 is 1334 cm⁻¹, twice the 667 cm⁻¹ of 01¹0. The states 02⁰0 and 100 can therefore mix, producing a splitting and also a significant increase in the intensity of the 02⁰0 transition, so that both the 02⁰0 and 100 transitions have similar intensities.