Terms Fermi resonance

Besides, we have shown elsewhere [22,23,71,72] that the term Fermi resonances is not fully adequate because noticeable perturbations of the vs (X-H Y) bandshape may be obtained in nonresonant cases—that is when the Fermi coupling takes place between the fast mode (of frequency 0)o) and a bending mode whose overtone frequency 2g>s may be very far from 0)o (see Section IV.B.3). 

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. 

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 ). 

In. a number of cases sub-maxima associated with vXH bands have been interpreted in this fashion and in the case of the carboxylic acid dimers this question has been investigated in some detail [4]. A prominent satellite band accompanying the main vOH bands has been assigned to an overtone of the <5QH vibration, and it has been possible to explain formally most of the multiplicity of peaks in the rOH band of formic acid in Fermi resonance terms. Although it is possible that some of these peaks correspond to Stepanov-type sub-bands, no convincing series of this type can be picked out. There seems little doubt that in many cases a considerable number of sub-bands in the rXH region are to be interpreted in terms of Fermi resonance [5, 43,... 

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. 

Now, let us apply the approached model of quantum indirect damping to a situation where a Fermi resonance is possible to occur when the first excited state of the fast mode and the first overtones of some bending modes are close. The basic physical terms are defined in Table VIII. 

Physical Terms Involved in the Fermi Resonance Model... 

If one considers only the first harmonic terms of the bending modes that belong to the g symmetry, then Fermi resonances will only affect the g excited states of the... 

Fermi Resonance. Suppose the anharmonic potential V(q) in equation (54) contains a cubic term 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... 

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... 

The differential Raman scattering cross sections and depolarization ratios in the Fermi resonance region of carbon disulphide CS2 were measured and interpreted in terms of three bond polarizability parameters and the cubic force constant k 22 (Montero et al., 1984). 

The fundamental ground state vibrational frequencies are those from a re-analysis of infra-red and Raman spectra by Guenther and Stoicheff, referenced by Kleman ( ) as a private communication. Several sets of anharmonicity constants and/or rotational constants have been published (2, 4-9). Some are corrected for Fermi resonance to some degree in some manner (6, 7, 9), others are not 2, 4). We adopt the x j, and g22 terms determined from the least-square estimates of the force... 

Both contributions to Tj have been considered simultaneously recently. The importance of resonant exchange dephasing can be tested by isoptopic dilution as it is most important between the same polar modes of neighboring identical molecules, but can also play a role due to quasi-resonance between different modes, such as Fermi resonances. These full theories of vibrational phase relaxation are rather heavy and difficult to apply, mainly because pure and exchange dephasing cannot be treated separately due to the apjjearance of cross terms of the tyjje (Aa)o(t )AcDg(t)) in the calculation of Tj. 

The introduction of Fermi resonances, with a supplementary term in the starting Hamiltonian (eq. (5.A33)), of the form q-qi qgQg, leads to an unchanged value for the centre of the band. It changes its width cr (eq. (5.A42)) into cr with... 

It is less obvious from Eq. (5) that a similar procedure must be applied to the anharmonic term in the case of Fermi resonance (if 20) [13]. 

We shall assume that the conditions for Fermi resonance are satisfied in a free molecule, i.e. that there are two (for the sake of simplicity) nondegenerate vibrations with the frequencies fli and ily, for which, for instance, 2il m In this case, when taking the intramolecular anharmonicity (with the constant T) into account, it is necessary to add to the Hamiltonian (6.11) the sum of two terms Hq(C) and Hp(B,C), where... 

A polyad is a group of near-degenerate vibrational states coupled by one or more low-order anharmonic terms. For example, the very common 2 1 Fermi resonance , which is due to an anharmonic coupling term like ki22QiQ2,... 

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