A Fermi resonance is the shifting of the energies and intensities of absorption bands in an infrared or Raman spectrum

Raman spectroscopy

Raman spectroscopy is a spectroscopic technique used to study vibrational, rotational, and other low-frequency modes in a system.It relies on inelastic scattering, or Raman scattering, of monochromatic light, usually from a laser in the visible, near infrared, or near ultraviolet range...

. It is a consequence of quantum mechanical mixing. The phenomenon was explained by the Italian physicist Enrico Fermi

Enrico Fermi

Enrico Fermi was an Italian-born, naturalized American physicist particularly known for his work on the development of the first nuclear reactor, Chicago Pile-1, and for his contributions to the development of quantum theory, nuclear and particle physics, and statistical mechanics...

.

Selection rules and occurrence

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

• the two vibrational state
  Molecular vibration
  A molecular vibration occurs when atoms in a molecule are in periodic motion while the molecule as a whole has constant translational and rotational motion...
  
  s of a molecule transform according to the same irreducible representation
  Group representation
  In the mathematical field of representation theory, group representations describe abstract groups in terms of linear transformations of vector spaces; in particular, they can be used to represent group elements as matrices so that the group operation can be represented by matrix multiplication...
  
    of the molecular point group
  Point group
  In geometry, a point group is a group of geometric symmetries that keep at least one point fixed. Point groups can exist in a Euclidean space with any dimension, and every point group in dimension d is a subgroup of the orthogonal group O...
  
  . In other words the two vibrations must have the same symmetries (Mulliken symbols).
• The transitions (accidentally) have almost the same energy.

Fermi resonance most often occurs between normal and overtone modes, 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 really lead to additional bands in the spectrum.

Ketones

High resolution IR spectra of most ketone

Ketone

In organic chemistry, a ketone is an organic compound with the structure RCR', where R and R' can be a variety of atoms and groups of atoms. It features a carbonyl group bonded to two other carbon atoms. Many ketones are known and many are of great importance in industry and in biology...

s reveal that the "carbonyl band" is split into a doublet. The peak separation is usually only a few wavenumbers. This splitting arises from the mixing of ν_CO and the overtone of HCH bending modes.

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 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 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.