Overtone frequency

Some sample calculations are displayed in Fig. 10. As may be seen, the spectral density (124) involves two sub-bands in the vs (X-H) frequency region, like it is observed within the exchange approximation. Note that other submaxima appear at overtone frequencies (near 2oo0, 3co0,. ..) with a much lower intensity (less than 0.1% of the doublet intensity) and will not be studied here. 

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). 

Child, M. S., and Halonen, L. O. (1984), Overtone Frequencies and Intensities in the Local Mode Picture, Adv. Chem. Phys. 57, 1. 

The strong lowering of the overtone frequency of the surface OH groups amounting to 8-23% and accompanied by a marked peak intensity decrease, when basic compounds containing 0 and N atoms are adsorbed, points to the formation of strong hydrogen bonds between the participants. 

The anharmonicity, 2u)eXe, which is defined experimentally as the difference between twice the fundamental frequency, woi and the overtone frequency, ujq2>... 

In summary, none of the three types of data available—deuteration shift, overtone frequencies, or overtone intensities—provides a basis for generalization that the effect of the H bond is to make v, more an-harmonic. To the contrary, each type of data can be used to support the proposal that v, is of normal anharmonicity or is possibly slightly more harmonic after H bond formation. 

The MD calculated DOS is, to first-order, independent of temperature and only small anharmonic effects appear at non-zero temperatures. These anharmonic effects arise from the fact that most potential functions are not parabolic. As the displacements increase in magnitude the molecules explore non-parabolic regions of the potential and the overtone frequencies with perfect integers of the base frequency C0o> such as 2coo, 3c0o..., due to the Fourier expansion of the non-parabolic potential function. 

The bound state of two phonons for the overtone frequency region was evidently first identified by Ron and Hornig (34). In this investigation they measured the absorption spectrum of the HC1 crystal in the region of overtone frequen-... 

Biphonons in the overtone frequency region of an intramolecular vibrations qualitative consideration... 

Upon Fermi resonance in an isolated molecule, the frequency of one of the molecular vibrations turns out to be close to the overtone frequency, or the sum frequency of some other vibrations. We may say that in this case the mentioned resonance occurs between two excited states of the molecule (for simplicity we assume here that all vibrations are nondegenerate). If the anharmonicity is absent these states do not interact and as the overtones usually have a small oscillator strength in the considered approximation we can expect in the spectrum, for example, only one absorption line. However, owing to the anharmonicity of intramolecular vibrations, the resonance can lead to the characteristic doublet of comparable intensity in the absorption spectra or the RSL spectra or (depending upon the symmetry of the molecule and the type of vibration) in both. If degenerate vibrations also participate in the Fermi resonance, the number of lines in these spectra can even be larger (43). 

Along with the functions g and G a number of other Green s functions must be known in order to calculate the dielectric tensor of a crystal in the overtone frequency region, as well as the RSL cross-section and the cross-section of nonlinear optical processes. Among these others we required the two-particle Green s function G 4k/ (t), which is determined by the relation... 

FlG. 6.8. Dependence of the dielectric function on the frequency in the region of overtone frequencies E b and E b are the energies of the longitudinal and transverse biphonons, respectively m n and emax are the minimum and maximum energy values in the band of two-particle states. 

Fig. 6.8 Dependence of the dielectric function on the frequency in the region of overtone frequencies 195...

Case 2. Av 1. For the anharmonic oscillator, the selection rule requiring that Ai = 1 is no longer a rigid requirement. There is a small probability of transitions with Av = 2 and an even smaller probability of transitions with Av = 3. If we insert these conditions in Eq. (25.19), that is, v = 2, v = 0, we can develop expressions analogous to Eqs. (25.21) and (25.22) for the R and P branches of a band centered on the first overtone frequency, 2vq(1 — 3Xg), or approximately twice the fundamental vibrational frequency. This overtone band is much weaker than the fundamental band. If i = 3,v = 0, there is a second overtone band that is much weaker than the first overtone band. The requirement, AJ = 1, still applies. 

Tables 11-16 and Il-lc lists the observed frequencies of a number of diatomic molecules, ions, and radicals as reponed recently. The matrix isolation technique was employed extensively to observe the spectra of unstable and reactive species (Sec. 1-22). Resonance Raman spectra of colored diatomic species exhibit a series of overtone frequencies which can be used to calculate anharmonicity constants (Sec. 1-21).

Table 2.27 summarizes literature data on the overtone frequencies of selected surface hydroxyl groups. 

The basic difficulty in the Bohr model arises from the use of classical mechanics to describe the electronic motions in atoms. The evidence of atomic spectra, which show discrete frequencies, indicates that only certain energies of motion are allowed the electronic energy is quantized. However, classical mechanics allows a continuous range of energies. Quantization does occur in wave motion for example, the fundamental and overtone frequencies of a violin string. Hence Louis de Broglie suggested... 

Fig. 3.14 Level diagram for the generation of higher-order Stokes sidebands, which differ from the vibrational overtone frequencies...

The lowest frequency occurs when n = I and is called the fundamental. Doubling the frequency corresponds to raising the pitch by an octave. Those solutions having values of n > I are known as the overtones. As mentioned previously, one important property of waves is the concept of superposition. Mathematically, it can be shown that any periodic function that is subject to the same boundary conditions can be represented by some linear combination of the fundamental and its overtone frequencies, as shown in Figure 3.8. In fact, this type of mathematical analysis is known as a Fourier series. Thus, while the note middle-A on a clarinet, violin, and piano all have the same fundamental frequency of 440 Hz, the sound (or timbre) that the different instruments produce will be distinct, as shown in Figure 3.9. 

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