Fermi resonance

The Fermi resonance (after Enrico Fermi ) is a physical phenomenon of molecular spectroscopy that can be observed in infrared and Raman spectra . In the case of vibrations with random degeneracy, there is a resonance split of the vibration bands.

description

Molecules are put into an excited vibrational state by absorption of infrared radiation . For the investigation of this absorption behavior, this is often shown in the form of an intensity spectrum, for example the absorption versus the radiation energy, which in molecular spectroscopy is often given as a wave number (usual unit: 1 / cm).

Infrared spectrum of a styrene-acrylonitrile copolymer (SAN)

The IR spectrum of an organic molecule (see picture) can be roughly divided into two large areas. In the wave number range below 1500 cm ⁻¹ , the lower energy deformation vibrations are generally found . The stretching vibrations of the functional groups as well as their harmonics and combination vibrations (multiples or combinations of two or more normal vibrations) are usually in the range greater than 1500 cm ⁻¹ . Similar vibrations with the same energy are called degenerate .

Idealized occurrence of normal and harmonics before and after the Fermi resonance. The associated idealized energy level scheme is shown under the idealized spectra.

It is possible that normal, harmonic and combination vibrations happen to have (approximately) the same energies; these vibrations are then called randomly degenerate . This results in a vibration coupling ( Fermi resonance ) in which one vibration draws energy from the other. Since the energy of the harmonic increases and the energy of the normal oscillation decreases, there is an unusual increase in the harmonic. The result is an energetic splitting of the randomly degenerate vibration bands into two bands of similar intensity (Fermi doublet). The intensity now corresponds to that of a fundamental vibration, with which it can easily be confused.

The frequency shift of the individual oscillations corresponds to that of other coupled oscillations, that is, the higher frequency oscillation shifts to an even higher frequency and the lower frequency to an even lower frequency. When analyzing the spectrum, it should be noted that if a harmonic is the higher-frequency oscillation, the frequency can shift to a range above the value calculated from the fundamental frequency. This means that the condition that a harmonic always has a slightly lower value than calculated is no longer given.

However, the Fermi resonance does not occur in all cases in which an upper or combination oscillation has a similar energy to a fundamental oscillation. An important boundary condition is that the upper or combination oscillation corresponds to the same symmetry ( Mulliken symbols ) as the fundamental oscillation.

Examples

Examples of Fermi resonance are the resonance splits:

the symmetrical stretching vibration ν ₁ (≈ 1340 cm ⁻¹ ) and the first harmonic of the deformation vibration ν ₃ of carbon dioxide (CO ₂ ) at 2 × 667 cm ⁻¹ = 1334 cm ⁻¹ , which shifted in two by about ± 50 cm ⁻¹ Bands at 1286 cm ⁻¹ and 1389 cm ⁻¹ result (because of the symmetry only in the Raman spectrum)
the first harmonic of the deformation oscillation of cyanogen chloride (ClCN) (2 × 378 cm ⁻¹ → 753 cm ⁻¹ ) of the C-Cl stretching oscillation at 744 cm ⁻¹ , which results in two bands at 784 cm ⁻¹ and 714 cm ⁻¹

Back calculation

With the help of an approximation formula and the band positions of the Fermi doublet, it is possible to determine the wave numbers of the "undisturbed" vibrations.

{\ displaystyle {\ tilde {\ nu}} _ {\ text {corr.}} = {\ frac {{\ tilde {\ nu}} _ {1} + {\ tilde {\ nu}} _ {2} } {2}} \ pm {\ frac {{\ tilde {\ nu}} _ {1} - {\ tilde {\ nu}} _ {2}} {2}} \ cdot {\ frac {s-1 } {s + 1}}}

For the above-mentioned case of cyanogen chloride, the corrected values of 753 cm ⁻¹ and 744 cm ⁻¹ are obtained for a value of s ≈ 1.45 .

literature

Dana W. Mayo, Foil A. Miller, RW Hannah: Course notes on the interpretation of infrared and Raman spectra . Wiley-Interscience, Hoboken, NJ 2004, ISBN 0-471-24823-1 .

Individual evidence

↑ ^a ^b ^c Helmut Günzler, Hans-Ulrich Gremlich: IR spectroscopy . John Wiley & Sons, 2012, ISBN 978-3-527-66287-6 .
^ ^A ^b ^c ^d ^e Johann Weidlein, Ulrich Müller, Kurt Dehnicke: Schwingungsspektoskopie: An introduction . 2., revised. Edition. Thieme, Stuttgart 1988, ISBN 3-13-625102-4 , p. 37 .

[günzler-1] Helmut Günzler, Hans-Ulrich Gremlich: IR spectroscopy . John Wiley & Sons, 2012, ISBN 978-3-527-66287-6 .

[weidlein-2] A ^b ^c ^d ^e Johann Weidlein, Ulrich Müller, Kurt Dehnicke: Schwingungsspektoskopie: An introduction . 2., revised. Edition. Thieme, Stuttgart 1988, ISBN 3-13-625102-4 , p. 37 .