Fermi resonances systems

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... 

It turns out that the language of normal and local modes that emerged from the bifurcation analysis of the Darling-Dennison Hamiltonian is not sufficient to describe the general Fermi resonance case, because the bifurcations are qualitatively different from the normal-to-local bifurcation in figure Al.2.10. For example, in 2 1 Fermi systems, one type of bifurcation is that in which resonant collective modes are bom [54]. The resonant collective modes are illustrated in figure A12.11 their difference from the local modes of the Darling-Dennison system is evident. Other types of bifurcations are also possible in Fermi resonance systems a detailed treatment of the 2 1 resonance can be found in [44]. 

Figure Al.2.11. Resonant collective modes of the 2 1 Fermi resonance system of a coupled stretch and bend with an approximate 2 1 frequency ratio. Shown is one end of a symmetric triatomic such as H2O. The normal stretch and bend modes are superseded by the horse shoe-shaped modes shown in (a) and (b). These two modes have different frequency, as further illustrated in figure Al.2.12.

Consider a many Fermi resonance system of coupled anharmonic oscillators (10). In an analytical treatment of the energy flow, it is convenient to use an algebraic representation for the Hamiltonian. The Hamiltonian H = // 4- V consists of an unperturbed portion,... 

Thus if M, X, and K are fixed and known and < > is varied, the value of , and the proportionality factor Am/,/ for large can be determined. This has in fact been done for a similar Fermi resonance system as described in the following section. Section V describes the results of calculations designed to test the predictions both of Sections III and IV. 

Here we describe exact quantum calculations on a model many-dimensional Fermi resonance system, using the methods explained in Chapters 1 and 3 (11). As in the previous section of this chapter, the Hamiltonian H = H0 + V consists of an unperturbed portion and a coupling term. As a simple means to model 5 = 6 anharmonic oscillators the normal... 

Logan DE, Wolynes PG. 1990. Quantum localization and energy flow in many-dimensional Fermi resonant systems. J. Chem. Phys. 93 4994—5012. 

See, for example, the following and references contained therein E. L. Sibert 111, W. P. Reinhardt, and J. T. Hynes, /. Chem. Phys., 81, 1115 (1984). Intramolecular Vibrational Relaxation and Spectra of CH and CD Overtones in Benzene and Perdeuterobenzene. S. P. Neshyba and N. De Leon,. Chem. Phys., 86, 6295 (1987). Qassical Resonances, Fermi Resonances, and Canonical Transformations for Three Nonlinearly Coupled Oscillators. S. P. Neshyba and N. De Leon,. Chem. Phys., 91, 7772 (1989). Projection Operator Formalism for the Characterization of Molecular Eigenstates Application to a 3 4 R nant System. G. S. Ezra, ]. Chem. Phys., 104, 26 (1996). Periodic Orbit Analysis of Molecular Vibrational Spectra Spectral Patterns and Dynamical Bifurcations in Fermi Resonant Systems. Also see Ref. 6. 

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. 

The fundamental vibrations have been assigned for the M-H-M backbone of HM COho, M = Cr, Mo, and W. When it is observable, the asymmetric M-H-M stretch occurs around 1700 cm-1 in low temperature ir spectra. One or possibly two deformation modes occur around 850 cm l in conjunction with overtones that are enhanced in intensity by Fermi resonance. The symmetric stretch, which involves predominantly metal motion, is expected below 150 cm l. For the molybdenum and tungsten compounds, this band is obscured by other low frequency features. Vibrational spectroscopic evidence is presented for a bent Cr-H-Cr array in [PPN][(OC)5Cr-H-Cr(CO)5], This structural inference is a good example of the way in which vibrational data can supplement diffraction data in the structural analysis of disordered systems. Implications of the bent Cr-H-Cr array are discussed in terms of a simple bonding model which involves a balance between nuclear repulsion, M-M overlap, and M-H overlap. The literature on M-H -M frequencies is summarized. 

Besides self trapping two alternative explanations, Fermi resonance and conformational substates, have been previously discussed as well [2]. In a recent study [6] we compared the 2D-IR spectrum of ACN with those of two molecular systems, which show the same splitting in the amide I band, and which were chosen as simple representatives of the alternative mechanisms. The three 2D-IR spectra differ completely, albeit in a well understood way. Based on the 2D-IR spectroscopic signature Fermi resonance and conformational sub-states can be definitely excluded as alternative explanations for the anomalous spectra of ACN. The 2D-IR spectrum of the amide I mode in ACN, on the other hand, can be naturally explained by self-trapping, as dicussed above. 

This band often appears as a doublet probably resulting from the accidental degeneracy of an overtone of another low-lying vibration mode (Fermi resonance). 7a, 6 8) The C—H stretching vibrations also undergo a high-frequency shift and appear at 3050 cm-1, a position generally characteristic of the methylene stretch in cyclopropyl systems.67)... 

The ACF of the dipole moment operator of the fast mode may be written in the presence of Fermi resonances by aid of Eq. (10). Besides, the dipole moment operator at time t appearing in this equation is given by a Heisenberg equation involving the full Hamiltonian (225). The thermal average involved in the ACF must be performed on the Boltzmann operator of the system involving the real... 

Here, [G))cnnl (l)] is the ACF of the g part of the system involving Fermi resonances and quantum indirect damping, whereas [G -1 (f)]M are the ACFs corresponding to the u parts involving neither Fermi resonance nor indirect damping that are given by Eq. (276). 

The Fermi resonance Hamiltonian consists of two terms. The first one, Ho, is the Dunham expansion, which characterizes the uncoupled system, while the second term, Hp, is the Fermi resonance coupling, which describes the energy flow between the reactive mode and one perpendicular mode. For the three systems, HCP CPH, HOCl HO - - Cl and HOBr HO + Br, the reactive degree of freedom is the slow component of the Fermi pair and will therefore be labeled s, while the fast component will be labeled /. Thus, the resonance condition writes co/ w 2c0s. More explicitly, for HCP the slow reactive mode is the bend (mode 2) and the fast one is the CP stretch (mode 3), while for HOCl and HOBr the slow mode is the OX stretch (X = Cl,Br) (mode 3) and the fast one is the bend (mode 2). The third, uncoupled mode— that is, the CH stretch (mode 1) for HCP and the OH stretch (mode 1) for HOCl and HOBr—will be labeled u. With these notations, the Dunham expansion writes in the form... 

Let us first neglect the Fermi resonance and analyze the dynamics of the uncoupled systems described by the Dunham expansion alone [Eq. (24)]. Because of the resonance condition co/ 2c0s, quantum states are organized in clumps, or polyads. Each polyad is defined by two quantum numbers, namely the number v of quanta in the uncoupled degree of freedom and the so-called polyad number P ... 

Similar intense bands at approximately the same frequencies are observed when other molecules of comparable proton affinity are adsorbed in HZSM-5 e.g.water [41],dimethylether [38,40,41], acetone and various carboxylic acids [41]. Pelmenschikov et al. [42,43] pointed out that these bands are very similar to the so-called A,B,C triplet of OH bands characteristic of strong molecular hydrogen bonded complexes in liquid or solid phases. The most widely accepted explanation for the A,B,C triplet in hydrogen bonded systems is that due to Claydon and Sheppard [44], who suggested that the A,B,C triplet are in fact pseudobands caused by the superposition onto a very broad single (OH) band of two so-called Evans transmission windows caused by Fermi resonance between the (OH) mode and the first overtones of in-plane ( 2 5(OH) = 2600 cm l) and out of plane ( 2 y(OH) = 1900 cm l) bending... 

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