Fermi resonance bands

FIGURE 3.26. Benzoyl chloride. A. Aromatic C—H stretch, 3065 cm-1. B. The C=0 stretch, 1774 cm-1 (see Table 3.3). (Acid chloride C=0 stretch position shows very small dependence on conjugation aroyl chlorides identified by band such as at C.) C. Fermi resonance band (of C=0 stretch and overtone of 8/2 cm1 band), 1730 cm-1. 

FIGURE 2.29. 2-Methyl-l,5-pentanediamine. N—H stretch, hydrogen-bonded, primary amine coupled doublet asymmetric, 3368 cm-1. Symmetric, 3291 cm. (Shoulder at about 3200 cm-1, Fermi resonance band with overtone of band at 1601 cm-1. Aliphatic C—H stretch, 2928,2859 cm-1. N—H bend (scissoring) 1601 cm-1. SsCH2 (scissoring), 1470 cm-1. C—N stretch, 1069 cm-1. N—H wag (neat sample), —900-700 cm-1. 

The next natural step is the discussion of Fermi resonance effects in molecular crystals. Let molecules having Fermi resonance between intramolecular vibrations form a molecular crystal due to weak (van der Waals) forces. Then the individual molecular vibrational excitations discussed above become coupled to each other and form collective Fermi resonance bands. We shall consider here a simple two-layer ID model with intermolecular interaction only between nearest neighbors (see Fig. 9.6). 

The Raman spectrum of quenched products (Fig. 7) consists of the symmetric stretching of excess P-O2 at 1585 cm, two Fermi-resonance bands of CO2 at 1270 and 1400 cm", and three new additional sharp bands at 734, 1079, and 2242 cm". The systematic of the latter three bands are very similar to those of nitrosonium nitrate N0" N03", an ionic dimer of nitrogen dioxide. This similarity suggests that the products also include a species with carbonates and carbosonium, CO COa ". The vibrations of carbonate ions appear at 713 and 1082 cm" in CaCOs [84], and the CO vibration appears about 2150 cm" at 5 GPa [85]. Electronic structure calculations for [86, 87] suggest that there are several low lying states of CO " ", whose vibrational frequencies vary between 1000 and 2000 cm". The yield of... 

In some aromatic acid chlorides one may observe another rather strong band, often on the lower-frequency side of the C=0 band, which makes the C=0 appear as a doublet. This band, which appears in the spectrum of benzoyl chloride (Fig. 2.56) at about 1730 cm is probably a Fermi resonance band originating from an interaction of the C=0 vibration, with an overtone of a strong band for 1-C stretch often appearing in the range from 900 to 800 cm . When a fundamental vibration couples with an overtone or combination band, the coupled vibration is called Fermi resonance. The Fermi resonance band may also appear on the higher-frequency side of the C=0 in many aromatic acid chlorides. This type of interaction can lead to splitting in other carbonyl compounds, as weU. 

Factor group, 295, 296 character tables of, 206ff fc = 0 states, intensities of, 312ff Factor group splitting. 299 Fermi-resonance bands in infrared spectra, 321 Fermions, 257 Fluorene space group, 326... 

An interesting feature of the above discussion is that molecular crystal spectra may be used to identify vibrationally induced components of a spectrum. One can make this identification on energy or polarization grounds alone if a nontotally symmetric frequency is observed in the spectrum. If, however, the perturbing vibration is totally symmetric and the mixed-in state has the same symmetry as the perturbed state, it becomes difficult to distinguish the vibrationally induced bands from those that are allowed. However this distinction becomes clear in the crystal since these bands should show different splittings. Similar arguments could be used to identify Fermi resonance bands in the infrared spectra of crystals. 

From an energetic point of view, the bands at higher wavenumbers can be assigned to the Ss rings. However, the intensities were found as ca. 0.65 1 (pure infected) instead of 2.8 1 which would be expected from the natural abundance of the isotopomers. These discrepancies were solved by applying the mathematical formalism utilized in the treatment of intramolecular Fermi resonance (see, e.g., [132]). Accordingly, in the natural crystal we have to deal with vibrational coupling between isotopomers in the primitive cell of the crystal [109]. 

Then, in complex molecules where many bending modes may participate to a Fermi resonance, we can expect the Vs (X-H Y) bandshape to be perturbed in a very complex way. It is also of importance to note that the perturbation due to a sole Fermi resonance is delocalized on the whole vs (X-H - -Y) band. 

We may finally conclude that, with the purpose of comparison with experiments, one has to be careful and must remember the following (i) Two sub-bands of the same intensity may not be the consequence of a resonant situation 0, (ii) The frequency of each submaxima is governed by the three parameters co0, o>0. and A, and (iii) The frequency of the Evans hole, which appears between the two sub-bands, is given within the exchange approximation by the average frequency j ( 0 + 2o>0), but it is dependent on A within the full treatment of Fermi resonances. 

Figure 12. Hydrogen bond involving a Fermi resonance damping parameters switching the intensities. The lineshapes were computed within the adiabatic and exchange approximations. Intensities balancing between two sub-bands are observed when modifying the damping parameters (a) with y0 =0.1, and y5 = 0.8 (b) with ya = ys = 0.8 (c) with yB — 0.8 and ys — 0.1. Common parameters oto = 1, A = 150cm, 2g)5 = 2850cm-1, and T = 30 K.

In a case where the transition of an energy state is from 0 to 1 in any one of the vibrational states (vi,v2,v3,. ..), the transition is considered as fundamental and is allowed by selection rules. When a transition is from the ground state to v — 2,3,. .., and all others are zero, it is known as an overtone. Transitions from the ground state to a state for which Vj = 1 and vj = 1 simultaneously are known as combination bands. Other combinations, such as v — 1, Vj = 1, v = 1, or v, — 2, v7 — 1, etc., are also possible. In the strictest form, overtones and combinations are not allowed, however they do appear (weaker than fundamentals) due to anharmonicity or Fermi resonance. 

Another potential source of peaks in the NIR is called Fermi resonance. This is where an overtone or combination band interacts strongly with a fundamental band. The math is covered in any good theoretical spectroscopy text, but, in short, the two different-sized, closely located peaks tend to normalize in size and move away from one another. This leads to difficulties in first principle identification of peaks within complex spectra. 

It was noted earlier that the charge density of a narrow resonance band lies within the atoms rather than in the interstitial regions of the crystal in contrast to the main conduction electron density. In this sense it is sometimes said to be localized. However, the charge density from each state in the band is divided among many atoms and it is only when all states up to the Fermi level have contributed that the correct average number of electrons per atom is produced. In a rare earth such as terbium the 8 4f electrons are essentially in atomic 4f states and the number of 4f electrons per atom is fixed without reference to the Fermi level. In this case the f-states are also said to be locaUzed but in a very different sense. Unfortunately the two senses are often confused in literature on the actinides and, in order not to do so here, we shall refer to resonant states and Mott-localized states specifically. 

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