Fundamental frequencies, molecular

Molecules vibrate at fundamental frequencies that are usually in the mid-infrared. Some overtone and combination transitions occur at shorter wavelengths. Because infrared photons have enough energy to excite rotational motions also, the ir spectmm of a gas consists of rovibrational bands in which each vibrational transition is accompanied by numerous simultaneous rotational transitions. In condensed phases the rotational stmcture is suppressed, but the vibrational frequencies remain highly specific, and information on the molecular environment can often be deduced from hnewidths, frequency shifts, and additional spectral stmcture owing to phonon (thermal acoustic mode) and lattice effects. 

Entropies can be calculated or estimated, and hence enthalpies can be derived from equilibrium measurements. Gaseous entropies are calculated by statistical mechanics using experimental or estimated molecular dimensions and fundamental frequencies (93). For solids, numerous methods based on additivity rules, or regularities in series of compounds, are available. Khriplovich and Paukov (140), for example, list 20 such relationships and were able to estimate entropies to about 1%. Empirical equations are also available for ion entropies (59). 

The fundamental frequencies 9t (t = 1, 2,... 3tf—6) are related to and since Xt are the roots of det B—XE) — 0, r, are related to the matrix B and to the molecular force constants Bif. Hence the vibrational energy levels for a non-linear polyatomic molecule in the harmonic oscillator approximation are given by... 

The anharmonicity constant vexe is small compared to ve, but its effect increases as v increases, and the overtones deviate more and more from simple multiples of the fundamental frequency with increasing vSee Fig. 4.7. The infrared region extends from 10 to 14,000 cm-1 (7000 A). Molecular vibrational frequencies run from 100 to 4000 cm-1, so that the fundamental and lower overtones lie in the infrared region. 

However, what unite all applications of NIRS for PAC are the unique features of the NIR spectrum. The NIR is in effect the chemical spectroscopy of the hydrogen atom in its various molecular manifestations. The frequency range of the NIR from about 4000 cm-1 up to 12 500 cm-1 (800-2500 nm) covers mainly overtones and combinations of the lower-energy fundamental molecular vibrations that include at least one X—H bond vibration. These are characteristically significantly weaker in absorption cross-section, compared with the fundamental vibrational bands from which they originate. They are faint echoes of these mid-IR absorptions. Thus, for example, NIR absorption bands formed as combinations of mid-IR fundamental frequencies (for example v + u2), typically have intensities ten times weaker than the weaker of the two original mid-IR bands. For NIR overtone absorptions (for example 2v, 2v2) the decrease in intensity can be 20-100 times that of the original band. 

Molecular Models. After selecting the most important fundamental frequencies from which the infrared spectrum is built, it becomes of interest to attribute to each a definite bond or type of vibration. Although such relations are quite speculative, there is now sufficient evidence to establish some of them with a fair degree of confidence. 

The molecule is pyramidal, having C3v symmetry with the nitrogen atom at the apex. The molecular dimensions have been determined by electron diffraction (266) and by microwave spectroscopy (161,271). The molecule with this symmetry will have four fundamental vibrations allowed, both in the infrared (IR) and the Raman spectra. The fundamental frequency assignments in the IR spectrum are 1031, vt 642, v2 (A ) 907, v3 (E) and 497 cm-1, v4 (E). The corresponding vibrations in the Raman spectrum appear at 1050, 667, 905, and 515 cm-1, respectively (8, 223, 293). The vacuum ultraviolet spectrum has also been studied (177). The 19F NMR spectrum of NF3 shows a triplet at 145 + 1 ppm relative to CC13F with JNF = 155 Hz (146, 216, 220,249, 280). 

Estimates of the eleven fundamental frequencies of Ll2SO are based on comparisons with Na2S0 (g) and K2S0 (g) (7), data on crystalline Li2S0 (9), and on a normal coordinate analysis of Devlin (1 ). Uncertainties in the estimates of the molecular... 

The formal requirement for these transitions to occur is that the molecular vibration should produce a change in dipole moment. Thus, in a molecule such as carbon dioxide, the symmetrical stretching vibration (vj) will be infrared inactive and the bending (V2) and asymmetric stretch (V3) modes will be active. The fundamental frequency at which a particular vibration will occur is given by the classical formula for a diatomic harmonic oscillator ... 

The seeding of molecular harmonics is in some sense a throwback to experiments where second harmonics were first observed in condensed-matter isotropic systems. For example, in glass fibers it was observed that a harmonic was produced after long exposures to fundamental frequency laser light [99,100]. It was later found that, by introduction of a low intensity seeding beam at the harmonic frequency, the onset of the harmonic in the glass was essentially instantaneous [101]. It was at this time that the proposal of a SWM mechanism... 

With both types of vibrational spectroscopy, distinctive spectra and facility in interpretation are possible because only vibrational transitions corresponding to changes in the vibrational quantum number of+1 are allowed by the spectral selection rules. That is, An = 1, where n is the vibrational quantum number. Due to this, the frequencies observed are usually the fundamental frequencies. In addition, because of analogies between the mathematical descriptions of classical and quantum mechanical vibrating molecular systems, it is possible to rationalize many spectral observations by analogy with classical vibrating systems that possess characteristic force constants and reduced masses. This rationalization has become the basis for systematizing much of the structural and chemical information derived from vibrational spectra. 

Fundamental frequency of the resonator Correlation function for surface roughness Root mean square height of a roughness Wave vector of shear waves in quartz, (Uy pq//rq Correlation length of surface roughness Thickness of the liquid film Thickness of interfacial layer Molecular dynamics Pressure in a liquid Quartz crystal microbalance Hydrodynamic roughness factor Electrochemical roughness factor Coordinates (normal and lateral)... 

The inclusion of molecular vibrations will effect the partition function which, in addition to the electronic contribution, will contain the vibration part [28]. Each molecule is assumed to behave as a set of (3 - 6) independent harmonic oscillators with the fundamental frequencies vf so that the vibration partition functions are... 

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