Fundamental vibrational

For a nonlinear molecule composed of N atoms, 3N—6 eigenvalues provide the normal or fundamental vibrational frequencies of the vibration and and the associated eigenvectors, called normal modes give the directions and relative amplitudes of the atomic displacements in each mode. 

To get the frequency v in centimeters-, the nonstandard notation favored by spectioscopists, one divides the frequency in hertz by the speed of light in a vacuum, c = 2.998 x lO " cm s-, to obtain a reciprocal wavelength, in this case, 4120 cm-. This relationship arises because the speed of any running wave is its frequency times its wavelength, c = vX in the case of electromagnetic radiation. The Raman spectral line for the fundamental vibration of H2 is 4162 cm-. .., not a bad comparison for a simple model. 

Infrared absorption properties of 2-aminothiazole were reported with those of 52 other thiazoles (113). N-Deuterated 2-aminothiazole and 2-amino-4-methylthiazo e were submitted to intensive infrared investigations. All the assignments were performed using gas-phase studies of the shape of the vibration-rotation bands, dichroism, isotopic substitution, and separation of frequencies related to H-bonded and free species (115). With its ten atoms, this compound has 24 fundamental vibrations 18 for the skeleton and 6 for NHo. For the skeleton (Cj symmetry) 13 in-plane vibrations of A symmetry (2v(- h, 26c-h- Irc-N- and 7o)r .cieu.J and... 

The planar structure of thiazole (159) implies for the molecule a Cj-type symmetry (Fig. 1-8) and means that all the 18 fundamental vibrations are active in infrared and in Raman spectroscopy. Table 1-22 lists the predictions made on the basis of this symmetry for thiazole. 

The determination of specific heats (159) has led to the conclusion that thiazole is associated intermoiecularly. The measurements can be carried out by adiabatic calorimetry (159) or by using the observed fundamental vibration frequencies and molecidar parameters (160, 161). 

One type of single point calculation, that of calculating vibrational properties, is distinguished as a vibrations calculation in HyperChem. A vibrations calculation predicts fundamental vibrational frequencies, infrared absorption intensities, and normal modes for a geometry optimized molecular structure. 

The symmetry properties of a fundamental vibrational wave function are the same as those of the corresponding normal coordinate Q. For example, when the C3 operation is carried out on Qi, the normal coordinate for Vj, it is transformed into Q[, where... 

Table 6.4 Fundamental vibration wavenumbers of crotonaldehyde obtained from the infrared and Raman spectra...

The CO2 laser is a near-infrared gas laser capable of very high power and with an efficiency of about 20 per cent. CO2 has three normal modes of vibration Vj, the symmetric stretch, V2, the bending vibration, and V3, the antisymmetric stretch, with symmetry species (t+, ti , and (7+, and fundamental vibration wavenumbers of 1354, 673, and 2396 cm, respectively. Figure 9.16 shows some of the vibrational levels, the numbering of which is explained in footnote 4 of Chapter 4 (page 93), which are involved in the laser action. This occurs principally in the 3q22 transition, at about 10.6 pm, but may also be induced in the 3oli transition, at about 9.6 pm. 

Color from Vibrations and Rotations. Vibrational excitation states occur in H2O molecules in water. The three fundamental frequencies occur in the infrared at more than 2500 nm, but combinations and overtones of these extend with very weak intensities just into the red end of the visible and cause the blue color of water and of ice when viewed in bulk (any green component present derives from algae, etc). This phenomenon is normally seen only in H2O, where the lightest atom H and very strong hydrogen bonding combine to move the fundamental vibrations closer to the visible than in any other material. 

Infrared Spectrophotometry. The isotope effect on the vibrational spectmm of D2O makes infrared spectrophotometry the method of choice for deuterium analysis. It is as rapid as mass spectrometry, does not suffer from memory effects, and requites less expensive laboratory equipment. Measurement at either the O—H fundamental vibration at 2.94 p.m (O—H) or 3.82 p.m (O—D) can be used. This method is equally appticable to low concentrations of D2O in H2O, or the reverse (86,87). Absorption in the near infrared can also be used (88,89) and this procedure is particularly useful (see Infrared and raman spectroscopy Spectroscopy). The D/H ratio in the nonexchangeable positions in organic compounds can be determined by a combination of exchange and spectrophotometric methods (90). 

Table 20 Fundamental Vibrational Frequencies (in cm of Parent Heterocycles...

In cases where information about atomic arrangements cannot be obtained by X-ray crystallography owing to the insolubility or instability of a compound, vibrational spectroscopy may provide valuable insights. For example, the explosive and insoluble black solid SesNaCla was shown to contain the five-membered cyclic cation [SesNaCl]" by comparing the calculated fundamental vibrations with the experimental IR spectrum. ... 

Kwiatkowski and Lesczcynski and (2) Nowak, Adamowicz, Smets, and Maes. Within the harmonie approximation, ab initio methods yield very aeeurate frequeneies for the fundamental vibrations (normal eoor-dinate ealeulations) although in most eases the values need to be sealed (sealing faetor 0.9 to 0.98 depending on the theoretieal method used). The eomparison with the experimental speetrum suffers for the following reasons (1) most tautomerie eompounds are studied in solution while the ealeulated speetrum eorresponds to the gas phase (2) eombination, overtone, and Fermi resonanee bands are not eomputed and (3) ealeulations are mueh less aeeurate for absolute intensities than for frequeneies. This last problem ean be partially overeome by reeording the eomple-mentary Raman speetrum. Some representative publications are shown in Table V. 

There are, at present, two overriding reasons an experimentalist would choose to employ laser Raman spectroscopy as a means of studying adsorbed molecules on oxide surfaces. Firstly, the weakness of the typical oxide spectrum permits the adsorbate spectrum to be obtained over the complete fundamental vibrational region (200 to 4000 cm-1). Secondly, the technique of laser Raman spectroscopy is an inherently sensitive method for studying the vibrations of symmetrical molecules. In the following sections, we will discuss spectra of pyridine on silica and other surfaces to illustrate an application of the first type and spectra of various symmetrical adsorbate molecules to illustrate the second. 

In Chapter 10, we will make quantitative calculations of U- U0 and the other thermodynamic properties for a gas, based on the molecular parameters of the molecules such as mass, bond angles, bond lengths, fundamental vibrational frequencies, and electronic energy levels and degeneracies. 

For a polyatomic molecule, the complex vibrational motion of the atoms can be resolved into a set of fundamental vibrations. Each fundamental vibration, called a normal mode, describes how the atoms move relative to each other. Every normal mode has its own set of energy levels that can be represented by equation (10.11). A linear molecule has (hr) - 5) such fundamental vibrations, where r) is the number of atoms in the molecule. For a nonlinear molecule, the number of fundamental vibrations is (3-q — 6). 

As an example, CO2 (a linear molecule) has four fundamental vibrations while H2O (a nonlinear molecule) has three fundamental vibrations (normal... 

Each type of vibration (normal mode) has associated with it a fundamental vibrational frequency and a set of energy levels. 

Table 10.2 Fundamental vibrational frequencies of some common molecules.

Tables 10.1, 10.2, and 10.3e summarize moments of inertia (rotational constants), fundamental vibrational frequencies (vibrational constants), and differences in energy between electronic energy levels for a number of common molecules or atoms/The values given in these tables can be used to calculate the rotational, vibrational, and electronic energy levels. They will be useful as we calculate the thermodynamic properties of the ideal gas.

We have seen earlier that for a linear polyatomic molecule, the vibrational motions can be divided into (3rj — 5) fundamentals, where rj is the number of atoms. For a nonlinear molecule (3rj - 6) fundamentals are present. In either case, each fundamental vibration can be treated as a harmonic oscillator with a partition function given by equations (10.100) and (10.101). Thus. 

Calculation of Thermodynamic Properties We note that the translational contributions to the thermodynamic properties depend on the mass or molecular weight of the molecule, the rotational contributions on the moments of inertia, the vibrational contributions on the fundamental vibrational frequencies, and the electronic contributions on the energies and statistical weight factors for the electronic states. With the aid of this information, as summarized in Tables 10.1 to 10.3 for a number of molecules, and the thermodynamic relationships summarized in Table 10.4, we can calculate a... 

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