Anharmonicity Fermi resonances

The spectrum of 4-pyranone-3,5-rf2 allowed analysis of the nature of the splitting observed in the 1667 cm-1 peak of pyran-4-one. The splitting was attributed to a slight anharmonic Fermi resonance involving the out-of-plane deformation mode ascribed to H-3 and H-5 which appears at 851 cm-1 (59CJC2007). This assignment is confirmed since there is no appreciable absorption between 900-800 cm-1 in the spectrum of the deuterated derivative, which also exhibits an unsplit peak at 1648 cm-1 (64JOC2678). 

The first derivative is the gradient g, the second derivative is the force constant (Hessian) H, the third derivative is the anharmonicity K etc. If the Rq geometry is a stationary point (g = 0) the force constant matrix may be used for evaluating harmonic vibrational frequencies and normal coordinates, q, as discussed in Section 13.1. If higher-order terms are included in the expansion, it is possible to determine also anharmonic frequencies and phenomena such as Fermi resonance. 

Bratos and Hadzi have developed another origin of the anharmonicity of the fast mode X-H -Y, the Fermi resonance, which is supported by several experimental studies [1,3,63-70], Widely admitted for strong hydrogen bonds [67], the important perturbation brought to the infrared lineshape by Fermi resonances has also been pointed out in the case of weaker hydrogen bonds [53,71-73]. 

In this section we shall give the connections between the nonadiabatic and damped treatments of Fermi resonances [53,73] within the strong anharmonic coupling framework and the former theory of Witkowski and Wojcik [74] which is adiabatic and undamped, involving implicitly the exchange approximation (approximation later defined in Section IV.C). 

Several studies of Fermi resonances in the absence of H bond have been made [76-80]. We shall account for this situation by simply ignoring the anharmonic coupling between the fast and slow modes (a = 0). The theory then describes the coupling between the fast mode and a bending mode through the potential Htf, with both of these modes being damped in the same way. Because aG = 0, the slow mode does not play any role, so that the total Hamiltonian does not refer to it ... 

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. 

Secondly, and most seriously, the validity even of the harmonic frequencies of Table 1 may be questioned 45). The observed binary and ternary bonds are all of symmetry class T(in thehexacarbonyls) or 41 or (in the case of Mn(CO)5Br), and these symmetry classes are repeated several times both in the fundamental and in the ternary region. Thus we have satisfied the conditions for Fermi resonance. Of course, to show that Fermi resonance is symmetry-allowed is not the same as showing that it occurs, but there is every reason to suspect it in the present case. The physical origin of anharmonicity lies in the existence of direct or crossed cubic and quartic terms in the potential energy expression ). 

Fermi resonance of the vXH vibration with neighbouring overtone and summation frequencies—It has been explained above that Fermi resonance can occur between an anharmonic fundamental vibration such as rXH and other combination (summation) frequencies provided that the latter are of similar frequency to the fundamental and of the same symmetry class. In addition to the frequencies rXH j- nvXH Y that have already been discussed, other interacting summation frequencies might, for example, involve overtones of the SX.H vibration, or combinations of this with rXH Y. Most of the H-bonded systems that can conveniently be studied are part of complex molecules so that many other types of summation bands can often occur in the appropriate region. 

An interaction, through anharmonicity, of the rXH vibration with other overtone and summation bands of similar frequency by Fermi resonance. This factor is more important with complex molecules. 

The large perturbations between near-degenerate vibrational levels produced by anharmonicity were first pointed out by Fermi, and such vibrational perturbations are called Fermi resonance. (The usual warning against the literal interpretation of the word resonance applies.)... 

Figure 6. Pressure dependence of the asymmetric stretching frequency of C02. Experimental frequencies were derived from combination bands and Fermi resonance doublet frequencies. The theoretical line was derived from a mechanical anharmonicity model with force constants from Ref. 73. From Ref. 53 with permission from the American Institute of Physics and the authors.

Here,/ is the anharmonic coupling parameter involved in the Fermi resonance, which is related to the above coupling L through ... 

As a consequence of the above equations, the full Hamiltonian describing the fast mode coupled to the H-bond bridge (via the strong anharmonic coupling theory) and to the bending mode (via the Fermi resonance process) may be written within the tensorial basis (222) according to [24] ... 

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