Fermi resonance

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A Fermi resonance is a splitting of lines in an infrared spectrum or Raman spectrum the theory of which is one of the many contributions of the Italian / American physicist Enrico Fermi. It occurs when two different vibrational states of a molecule transform according to the same irreducible representation of the molecular point group and have almost the same energy. This can occur for triatomic and larger molecules and the spectra of CO₂ provide a good example.

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 ν₁, one quantum of excitation in ν₂ with angular momentum about the molecular axis equal to ±1, no excitation in ν₃) and clearly transforms according to the irreducible representation Π_u. Putting two quanta into ν₂ 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) as, and a very similar energy to, state 100 (a single excitation in the ν₁ symmetric stretch (calculated unperturbed frequency 1337 cm^-1), no excitation in ν₂, no excitation in ν₃). The similarity in energy may be thought of as an accidental near-degeneracy. The states 02⁰0 and 100 can therefore mix together, 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.