First overtone

Designating the frequency of the normal vibrational mode as Vo, we can obtain in process 2 the excited molecules with the energy close to 2hVo, 3/iVo, nhVo. Absorption lines at these frequencies are named overtones (first, second, and higher). The probability of formation in process 2 of the excited molecule with the energy corresponding to the higher overtone decreases rapidly. Therefore, the normal... 

With broad-band pulses, pumping and probing processes become more complicated. With a broad-bandwidth pulse it is easy to drive fundamental and overtone transitions simultaneously, generating a complicated population distribution which depends on details of pulse stmcture [75], Broad-band probe pulses may be unable to distinguish between fundamental and overtone transitions. For example in IR-Raman experiments with broad-band probe pulses, excitation of the first overtone of a transition appears as a fundamental excitation with twice the intensity, and excitation of a combination band Q -t or appears as excitation of the two fundamentals 1761. 

A plot of the spaeing between neighboring energy levels versus vj should be linear for values of vj where the harmonie and first overtone terms dominate. The slope of sueh a plot is expeeted to be -2h(cox)j and the small -vj intereept should be h[cOj - 2(cox)j]. Sueh a plot of experimental data, whieh elearly ean be used to determine the coj and (cox)j parameter of the vibrational mode of study, is shown in the figure below. 

If the vibrational funetions are deseribed within the harmonie oseillator approximation, it ean be shown that the integrals vanish unless vf = vi +1, vi -1 (and that these integrals are proportional to (vi +1)E2 and (vi)i/2 the respeetive eases). Even when Xvf and Xvi are rather non-harmonie, it turns out that sueh Av = 1 transitions have the largest integrals and therefore the highest infrared intensities. For these reasons, transitions that eorrespond to Av = 1 are ealled "fundamental" those with Av = 2 are ealled "first overtone" transitions. 

In multiplying by we use, again, examples of the vibrations of NH3. The result depends on whether we require when (a) one quantum of each of two different e vibrations is excited (i.e. a combination level) or (b) two quanta of the same e vibration are excited (i.e. an overtone level). In case (a), such as for the combination V3 - - V4, the product is written E x E and the result is obtained by first squaring the characters under each operation, giving... 

As chronicled by Dearden [21], the association of compound lipophilicity with membrane penetration was first implied by Overton and Meyer more than a century ago. To enhance this understanding, lipophilicity measurements were initially performed using a variety of lipid phases [22], while the comprehensive review by Hansch et al. [23], with extensive data from literature and their own measurements, lent further support to the now accepted wide use of the octanol-water solvent system. 

The fundamental vibrational frequency is that with n = 1, while the frequen-cies of the harmonics or overtones are obtained with n = 2,3,4. Specifically, n = 2 is called the second harmonic in electronics and the first overtone in musical acoustics. Both terms are employed, often erroneously, in the description of molecular vibrations (see Chapter 9). 

As musicians know, it is the relative intensities of the various members of the overtone series that determine the timbre or tone quality of sound. It is easy to distinguish the sound of a flute from that of the clarinet, although the listener may not know why. The sound of the flute has a relatively intense first overtone, while the boundary conditions imposed on the vibrating air column in the clarinet result in the suppression of all odd overtones. Such phenomena are of course much easier to visualize on a stringed instrument Ask a violinist for a demonstration of the natural harmonics of a given string. 

The D band, the disorder induced mode, normally appears between 1250 and 1450 cm. This band is activated in the first-order scattering process of sp2 carbons by the presence of in-plane substitutional hetero-atoms, vacancies, grain boundaries or other defects and by finite-size effects [134], The G band is the second-order overtone of the D band. 

Applying the exchange approximation and neglecting the zero-point energy terms, we may safely limit the representation of the Hamiltonian Hsf ex within the following reduced base which accounts for the ground states of each mode and for the first (second) overtone of the fast (bending) mode ... 

We may observe that the spectral density (124) is temperature-dependent. However, due to the magnitude of the involved frequencies (first overtone 4500 K), this dependence is irrelevant within the experimental temperature range. 

However, any vibrating system not only has a natural vibration frequency but will also vibrate at twice that frequency, which is known as the first overtone. The first overtone of the vibrations of molecules like water, proteins and fats correspond to a frequency in the near-infrared. Because these frequencies are overtones all of the spectroscopic problems that preclude making quantitative measurements in the mid-infrared are not present in the near-infrared. 

Figure 8.1 The harmonic potential and the Morse potential, together with vibrational energy levels. The harmonic potential is an acceptable approximation for molecular separations close to the equilibrium distance and vibrations up to the first excited level, but fails for higher excitations. The Morse potential is more realistic. Note that the separation between the vibrational levels decreases with increasing quantum number, implying, for example, that the second overtone occurs at a frequency slightly less than twice that of the fundamental vibration.

The ideas of Overton are reflected in the classical solubility-diffusion model for transmembrane transport. In this model [125,126], the cell membrane and other membranes within the cell are considered as homogeneous phases with sharp boundaries. Transport phenomena are described by Fick s first law of diffusion, or, in the case of ion transport and a finite membrane potential, by the Nernst-Planck equation (see Chapter 3 of this volume). The driving force of the flux is the gradient of the (electro)chemical potential across the membrane. In the absence of electric fields, the chemical potential gradient is reduced to a concentration gradient. Since the membrane is assumed to be homogeneous, the... 

Using these factors and a fundamental vibration at 3500 nm, the first overtone would be... 

This equation will give values from 1785 to 1925 nm. In reality, the first overtone would likely be at 3500/2 + several nanometers, usually to a longer wavelength. The anharmonicity gives rise to varying distances between overtones. As a consequence, two overlapping peaks may be separated at a higher overtone. 

Any unknown (isolated) band may be deduced from first principles unfortunately, there is considerable overlap in the NIR region, but this is where Chemometrics will be discussed. An idealized spectrum of combinations and overtones is seen in Fig. 6.3. 

Another potential source of peaks in the NIR is called Fermi resonance. This is where an overtone or combination band interacts strongly with a fundamental band. The math is covered in any good theoretical spectroscopy text, but, in short, the two different-sized, closely located peaks tend to normalize in size and move away from one another. This leads to difficulties in first principle identification of peaks within complex spectra. 

L. England Kretzer and W. A. P. Luck, Band analysis of CH3OH and CH3OD H bond complexes in the first overtone region. J. Mol. Struct. 348, 373 376 (1995). 

When the potential is of the Morse type, the first term in Eq. (2.80) provides an often-used rule of thumb for overtone (v — v > 1) transitions... 

Gryns (1896), Hedin (1897), and especially Overton (1900) looked at the permeability of a wide range of different compounds, particularly non-electrolytes, and showed that rates of penetration of solutes into erythrocytes increased with their lipid solubility. Overton correlated the rate of penetration of the solute with its partition coefficient between water and olive oil, which he took as a model for membrane composition. Some water-soluble molecules, particularly urea, entered erythrocytes faster than could be attributed to their lipid solubility—observations leading to the concept of pores, or discontinuities in the membrane which allowed water-soluble molecules to penetrate. The need to postulate the existence of pores offered the first hint of a mosaic structure for the membrane. Jacobs (1932) and Huber and Orskov (1933) put results from the early permeability studies onto a quantitative basis and concluded molecular size was a factor in the rate of solute translocation. 

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