Fermi resonance

{{Short description|Shift of energy in physics}} A '''Fermi resonance''' is the shifting of the energies and intensities of absorption bands in an [[infrared spectrum|infrared]] or [[Raman spectroscopy|Raman spectrum]]. It is a consequence of quantum-mechanical wavefunction mixing.<ref>[[Kazuo Nakamoto]], "Infrared and Raman Spectra of Inorganic and Coordination Compounds: Theory and Applications in Inorganic Chemistry (Volume A)", John Wiley, 1997, {{ISBN|0-471-16394-5}}.</ref> The phenomenon was first explained by the Italian physicist [[Enrico Fermi]].<ref>{{Cite journal |last=Fermi |first=E. |date=1931-03-01 |title=Über den Ramaneffekt des Kohlendioxyds |url=https://link.springer.com/article/10.1007/BF01341712 |journal=Zeitschrift für Physik |language=de |volume=71 |issue=3 |pages=250–259 |doi=10.1007/BF01341712 |issn=0044-3328|url-access=subscription }}</ref>

==Selection rules and occurrence== [[File:FermiResScheme.png|thumb|360px|right|Idealized appearance of a normal mode and an overtone before and after Fermi resonance. Beneath the idealized spectra are idealized energy-level schemes.]] Two conditions must be satisfied for the occurrence of Fermi resonance: * The two [[molecular vibration|vibrational modes]] of a molecule transform according to the same [[irreducible representation]] in their molecular [[molecular symmetry|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.{{cn|date=March 2020}}

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 [[ketone]]s 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.<ref>Robert M. Silverstein, Francis X. Webster, David Kiemle, "Spectrometric Identification of Organic Compounds", 7th ed., John Wiley & Sons, 2005, {{ISBN|0-471-39362-2}}.</ref>

===CO₂=== In CO₂, the bending vibration ν₂ (667&nbsp;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&nbsp;cm⁻¹, and, ignoring anharmonicity, the frequency of 02⁰0 is 1334&nbsp;cm⁻¹, twice the 667&nbsp;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.

==References== <references/>

{{DEFAULTSORT:Fermi Resonance}} [[Category:Quantum chemistry]] [[Category:Computational chemistry]] [[Category:Vibrational spectroscopy]]

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