Chemical Sciences: A Manual for CSIR-UGC National Eligibility Test for Lectureship and JRF/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 mixing. The phenomenon was explained by the Italian-American physicist Enrico Fermi.

Selection rules and occurrence

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

• the two vibrational states of a molecule transform according to the same irreducible representation of the molecular point group.
• The energies of the transitions (accidentally) have almost the same energy.

Since the normal modes of a molecule are generally of disparate energies, they do not mix. Thus, Fermi resonance most often occurs between normal and overtone modes, which are often nearly coincidental 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.

Ideallized appearance of a normal mode and an overtone before and after Fermi resonance. Beneath the idealized spectra are idealized energy level schemes.

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 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 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.