Resonance Fermi

Fermi Resonance. Suppose the anharmonic potential V(q) in equation (54) contains a cubic term j rr iq qs, the effect of such a term is to produce an interaction between vibrational states differing by Ayr = +2 and Ars = + 1, since 

Here vr, rs denotes an unperturbed harmonic oscillator function, and we assume neither vibration is degenerate. For example, typical values of f rrs might be of the order 30 cm-1 if the separation between the unperturbed vibrational levels (vr = 2, v, = 0) and (vr = 0, vs = 1) were also about 30 cm-1 the interaction would result in a pushing apart of the energy levels of about 7 cm-1 each way, giving an observed separation of about 44 cm-1. 

A similar interaction would be observed between all Fermi polyads containing sets of vibrational levels related by the selection rule A tv = 2, A tv = +1, and the hamiltonian matrix should be diagonalized for each Fermi polyad without the use of perturbation theory. If, on the other hand, the interaction (63) were smaller, or the separation between the unperturbed levels were larger, the interaction could be treated by perturbation theory it can be shown that, in second-order perturbation theory, equation (63) would contribute a term to the vibrational anharmonic constants 

The Fermi resonance effect usually leads to two bands appearing close together when only one is expected. When an overtone or a combination band have the same or a similar frequency to that of a fundamental, two bands appear, split either side of the expected value, which are of about equal intensity. This effect is greatest when the frequencies match, but it is also present when there is a mismatch of a few tens of wavenumbers. The two bands are referred to as a Fermi doublet. 

The vibrational secular equation deals with the fundamental vibrational modes. The general method for F and G matrices is written as 

If the atomic masses in the G matrix elements are expressed in atomic mass units and the frequencies expressed in wavenumbers, then the equation may be written as 

This may be compared with the equation obtained by treating a 

For a polyatomic molecule, there are 3N - 6 energy levels for which only a single vibrational quantum number is 1 when the rest are zero. These are the fundamental series, where the elevation from the ground state to one of these levels is known as a fundamental. The usual secular equation ignores overtones and combinations, neglecting their effect on any fundamental. 

There are, however, occasions where an overtone or combination band interacts strongly with a fundamental. Often this happens when two excitations give states of the same symmetry. This situation is called Fermi resonance [15] and is a special example of configuration interaction. This phenomenon may occur for electronic transitions as well as vibrational modes. 

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Dubai H-R and Quack M 1984 Tridiagonal Fermi resonance structure in the IR spectrum of the excited CH chromophore in CFgH J. Chem. Phys. 81 3779-91... 

Segall J, Zare R N, Dubai H R, Lewerenz M and Quack M 1987 Tridiagonal Fermi resonance structure in the vibrational spectrum of the CM chromophore in CHFg. II. Visible spectra J. Chem. Phys. 86 634-46... 

Ezra G S 1996 Periodic orbit analysis of molecular vibrational spectra-spectral patterns and dynamical bifurcations in Fermi resonant systems J. Chem. Phys. 104 26... 

IR spectroscopy has been particularly helpful in detecting the presence of keto tautomers of the hydroxy heterocycles discussed in Section 3.01.6. Some typical frequencies for such compounds are indicated in Figure 4. Here again the doublets observed for some of the carbonyl stretching frequencies have been ascribed to Fermi resonance. 

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. 

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]. 

B. Two-Dimensional Quantum Monodromy in Fermi Resonance Polyads... 

The tacit assumption above is that the monodromy matrix is defined with respect to the primitive unit cell, with sides (5v, 8fe) = (0,1) and (1, 0), because the twist angle that determines the monodromy is given by A9 = — (Sv/Sfe)j.. However, situations can arise where other choices are more convenient. For example, the energy levels within a given Fermi resonance polyad are labeled by a counting number v = 0,1,... and an angular momentum that takes only even or only odd values. Thus the convenient elementary cell has sides (8v, 8L) = (0,2) and (1, 0), and the natural basis, say, y, is related to the primitive basis, x, by... 

The effective spectroscopic classical Hamiltonian for Fermi resonance between a nondegenerate mode vi and a doubly degenerate mode V2 takes the form [30, 31]... 

Analysis of the prototypical resonant swing spring model [11-13] shows that Fermi resonance with conserved angular momentum is an intrinsically three-dimensional phenomenon. The form of the 3x3 monodromy matrix was given. 

The standard effective spectroscopic Fermi resonant Hamiltonian allows more complicated types of behavior. The full three-dimensional aspects of the monodromy remain to be worked out, but it was shown, with the help of the Xiao—KeUman [28, 29] catastrophe map, that four main dynamical regimes apply, and that successive polyads of a given molecule may pass from one regime to another. 

The i.r. spectra of two crystal forms of aminomethylphosphonic acid (91) and its and analogues have been studied. A Fermi resonance between vxh and vxd vibrations and certain binary combinations can explain most of the spectra. The related aminophosphinic acid (92) and... 

The Spectral Density of Pure Fermi Resonance Beyond the Exchange Approximation... 

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