Polyatomic molecules fundamental vibrational frequencies

Absorption bands that are attributed to overtone and combination vibrations are also observed in the IR spectrum of polyatomic molecules. Overtone vibrations occur at frequencies of approximately integral multiples of the fundamental frequencies. Combination vibrations appear at frequencies that are the sum or the difference of the frequencies of two or more frmdamental vibrations. Overtone and combination bands are much less intense than fundamental bands. 

This works well for the fundamental vibrational frequency of simple diatomic molecules, and is not too far from the average value of a two-atom stretch within a polyatomic molecule. However, this approximation only gives the average or center frequency of the diatomic bond. In addition, one might expect that since the reduced masses of, for example, CH, OH, and NH are 0.85, 0.89, and 0.87, respectively (these constitute the major absorption bands in the near-infrared spectrum), the ideal frequencies of all these pairs would be quite similar. 

The molecular model of the previous section can move as a whole, rotate about its center of mass, and vibrate. The translational motion does not ordinarily give rise to radiation. Classically, this follows because acceleration of charges is required for radiation. The rotational motion causes practically observable radiation if, and only if, the molecule has an electric (dipole) moment. The vibrational motions of the atoms within the molecule may also be associated nuth radiation if these motions alter the electric moment. A diatomic molecule has only one fundameiita] frequency of vibration so that if it has an electric moment its infrared emission spectrum will consist of a series of bands, the lowest of which in frequency corresponds to the distribution of rotational fre-c)uciicies for nonvibrating molecules. The other bands arise from combined rotation and vibration their centers correspond to the fundamental vibration frequency and its overtones. A polyatomic molecule has more than one fundamental frequency of vibration so that its spectrum is correspondingly richer. 

The fundamental vibrational frequencies of several polyatomic molecules are given in Table 8.3, together with local mode descriptions and symmetry labels for the modes. Despite the complications inherent in vibrating polyatomics, several features are predictable from our experience with diatomics and a fundamental knowledge of vibrational modes. For example, we know that the isotopes of H2O, a non-linear triatomic — 3) molecule, should each have... 

TABLE 8.3 Fundamental vibrational frequencies for selected polyatomic molecules. Representations are assigned according to the character tables in Chapter 6, with planar molecules lying in the xz plane. 

So if the bond strength increases or reduced mass decreases, the value of vibrational frequency increases. Polyatomic molecules may exhibit more than one fundamental vibrational absorption bands. The number of these fundamental bands, is related to the degree of freedom in a molecule and the number of degrees of freedom is equal to the number of coordinates necessary to locate all atoms of a molecules in space. 

However, in polyatomic molecules, transitions to excited states involving two vibrational modes at once (combination bands) are also weakly allowed, and are also affected by the anharmonicity of the potential. The role of combination bands in the NIR can be significant. As has been noted, the only functional groups likely to contribute to the NIR spectrum directly as overtone absorptions are those containing C-H, N-H, O-H or similar functionalities. However, in combination with these hydride bond overtone vibrations, contributions from other, lower frequency fundamental bands such as C=0 and C=C can be involved as overtone-combination bands. The effect may not be dramatic in the rather broad and overcrowded NIR absorption spectrum, but it can still be evident and useful in quantitative analysis. 

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... 

Note that to first order this is simply the sum of the fundamental frequencies, after allowing for anharmonicity. This is an oversimplification, because, in fact, combination bands consist of transitions involving simultaneous excitation of two or more normal modes of a polyatomic molecule, and therefore mixing of vibrational states occurs and... 

B. Vibrational Structure of Electronic Transitions 1. Normal vibrations and their symmetry classification An electronic band system belonging to a polyatomic molecule normally contains a large number and variety of transitions in which vibrational quantum changes are superimposed on the electronic jump. The analysis, besides supplementing infrared and Raman evidence of the ground state frequencies, yields values for the fundamental frequencies of the excited state and is one of the principal sources of information as to its structure. 

Frequencies assigned to the fundamental vibrations of the triatomic germylenes, stannylenes and plumbylenes are collected in Table 7 together with corresponding data on triatomic silylenes for comparison. The vibrational frequencies are sensitive to the environment of the molecule. Therefore, the most precise frequency values measured under each type of condition used (in the gas phase and in different low-temperature inert matrices) by different spectroscopic methods are shown in the table for each of the CAs. Nonfundamental frequencies of triatomic and observed frequencies of polyatomic germylenes, stannylenes and plumbylenes are listed in the text. 

Figure 2 Vibrational energy relaxation (VER) mechanisms in polyatomic molecules, (a) A polyatomic molecule loses energy to the bath (phonons). The bath has a characteristic maximum fundamental frequency D. (b) An excited vibration 2 < D decays by exciting a phonon of frequency ph = 2. (c) An excited vibration >d decays via simultaneous emission of several phonons (multiphonon emission), (d) An excited vibration 2 decays via a ladder process, exciting lower energy vibration a> and a small number of phonons, (e) Intramolecular vibrational relaxation (IVR) where 2 simultaneously excites many lower energy vibrations, (f) A vibrational cascade consisting of many steps down the vibrational ladder. The lowest energy doorway vibration decays directly by exciting phonons. (From Ref. 96.)...

Some representative examples of common zero-temperature VER mechanisms are shown in Fig. 2b-f. Figures 2b,c describe the decay of the lone vibration of a diatomic molecule or the lowest energy vibrations in a polyatomic molecule, termed the doorway vibration (63), since it is the doorway from the intramolecular vibrational ladder to the phonon bath. In Fig. 2b, the excited doorway vibration 2 lies below large molecules or macromolecules. In the language of Equation (4), fluctuating forces of fundamental excitations of the bath at frequency 2 are exerted on the molecule, inducing a spontaneous transition to the vibrational ground state plus excitation of a phonon at Fourier transform of the force-force correlation function at frequency 2, denoted C( 2). 

Raman Selection Rules. For polyatomic molecules a number of Stokes Raman bands are observed, each corresponding to an allowed transition between two vibrational energy levels of the molecule. (An allowed transition is one for which the intensity is not uniquely zero owing to symmetry.) As in the case of infrared spectroscopy (see Exp. 38), only the fundamental transitions (corresponding to frequencies v, V2, v, ...) are usually intense enough to be observed, although weak overtone and combination Raman bands are sometimes detected. For molecules with appreciable symmetry, some fundamental transitions may be absent in the Raman and/or infrared spectra. The essential requirement is that the transition moment F (whose square determines the intensity) be nonzero i.e.. 

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