Nonlinear molecules spectra

A nonlinear molecule of N atoms with 3N degrees of freedom possesses 3N — 6 normal vibrational modes, which not all are active. The prediction of the number of (absorption or emission) bands to be observed in the IR spectrum of a molecule on the basis of its molecular structure, and hence symmetry, is the domain of group theory [82]. Polymer molecules contain a very high number of atoms, yet their IR spectra are relatively simple. This can be explained by the fact that the polymer consists of identical monomeric units (except for the end-groups). 

As the molecule vibrates it can also rotate and each vibrational level has associated rotational levels, each of which can be populated. A well-resolved ro - vibrational spectrum can show transitions between the lower ro-vibrational to the upper vibrational level in the laboratory and this can be performed for small molecules astronomically. The problem occurs as the size of the molecule increases and the increasing moment of inertia allows more and more levels to be present within each vibrational band, 3N — 6 vibrational bands in a nonlinear molecule rapidly becomes a big number for even reasonable size molecules and the vibrational bands become only unresolved profiles. Consider the water molecule where N = 3 so that there are three modes of vibration a rather modest number and superficially a tractable problem. Glycine, however, has 10 atoms and so 24 vibrational modes an altogether more challenging problem. Analysis of vibrational spectra is then reduced to identifying functional groups associated... 

The number of fundamental vibrational modes of a molecule is equal to the number of degrees of vibrational freedom. For a nonlinear molecule of N atoms, 3N - 6 degrees of vibrational freedom exist. Hence, 3N - 6 fundamental vibrational modes. Six degrees of freedom are subtracted from a nonlinear molecule since (1) three coordinates are required to locate the molecule in space, and (2) an additional three coordinates are required to describe the orientation of the molecule based upon the three coordinates defining the position of the molecule in space. For a linear molecule, 3N - 5 fundamental vibrational modes are possible since only two degrees of rotational freedom exist. Thus, in a total vibrational analysis of a molecule by complementary IR and Raman techniques, 31V - 6 or 3N - 5 vibrational frequencies should be observed. It must be kept in mind that the fundamental modes of vibration of a molecule are described as transitions from one vibration state (energy level) to another (n = 1 in Eq. (2), Fig. 2). Sometimes, additional vibrational frequencies are detected in an IR and/or Raman spectrum. These additional absorption bands are due to forbidden transitions that occur and are described in the section on near-IR theory. Additionally, not all vibrational bands may be observed since some fundamental vibrations may be too weak to observe or give rise to overtone and/or combination bands (discussed later in the chapter). 

The rotation-vibration interaction of Section 4.32 produces different effects in nonlinear molecules than those discussed in the previous section. In nonlinear molecules the quantum numbers are vavhvcKJM >. The connection between the group quantum numbers Ico , co2> xi > 2 -A 3/ > and the usual quantum numbers is given by Eq. (4.85). The different effect can be traced to the different nature of the rotational spectrum. In lowest order, the spectrum of a bent molecule is given by Eq. (4.107) and Figure 4.21. The rotation-vibration interaction introduces terms with selection rules... 

All nonlinear molecules have 3n — 6 vibrational modes, where n is the number of atoms. Some of these modes arc active in the infrared spectrum, some are active in the Raman spectrum, and others do not give directly observable transitions. Analyses of these spectra usually make use of isotopically substituted molecules to provide additional experimental data, and in recent years, theoretical calculations of vibrational spectra have aided both in making assignments of the observed bands, and in providing initial estimates of force constants.97 Standard methods are available for relating the experimental data to the force constants for the vibrational modes from which they are derived.98... 

The H3+ molecule ion is the simplest nonlinear molecule, of course, and its potential surface has been calculated in detail and with relatively high accuracy490 (see Figs. 63-65). This ab initio surface, as well as that for Li+-H2,491 have been employed in calculations of cross sections for vibrational excitation in nonreactive scattering collisions492 (see also Section II.B.2.b). An ab initio calculation of the vibrational spectrum of H3+ has also been carried out,207 and it has been suggested that vibrational chemiluminescence should be observable from the reaction H2+ (H2, H)H3+. 

A nonlinear molecule with n atoms generally has 3n — 6 fundamental vibrational modes. Water (3 atoms) has 3(3) -6 = 3 fundamental modes, as shown in the preceding figure. Methanol has 3(6) - 6 = 12 fundamental modes, and ethanol has 3(9) - 6 = 21 fundamental modes. We also observe combinations and multiples (overtones) of these simple fundamental vibrational modes. As you can see, the number of absorptions in an infrared spectrum can be quite large, even for simple molecules. 

Jahn-Teller effect - An interaction of vibrational and electronic motions in a nonlinear molecule which removes the degeneracy of certain electronic energy levels. It can influence the spectrum, crystal structure, and magnetic properties of the substance. 

Because the central atom is not in tine with the other two, the symmetric stretching vibration produces a change in dipole momeni and is thus IR active. For example, stretching peaks at 3657 and 3766 cm (2.74 and 2.66 pm) appear in the IR spectrum for the symmetric and asymmet ric stretching vibrations of the water molecule. I here is only one component to the scissoring vibration for this nonlinear molecule because motion in the plane of the molecule constitutes a rotational degree of freedom. For water, the bending vibration causes absorption at 1595 cm (6.27 pm). The difference in behavior oflinear and nonlinear triaiomic molecules with two and three absorption bands, respectively. illustrates how IR absorption spectroscopy can sometimes be used to deduce molecular shapes. 

The same rules for number of bands in a spectrum apply to Raman spectra as well as IR spectra 3N—6 for nonlinear molecules and 3N—5 for linear molecules. There may be fewer bands than theoretically predicted due to degeneracy and nonactive modes. Raman spectra do not usually show overtone or combination bands they are much weaker than in IR. A rule of thumb that is often tme is that a band that is strong in IR is weak in Raman and vice versa. A molecule with a center of symmetry, such as CO2, obeys another rule if a band is present in the IR spectrum, it will not be present in the Raman spectrum. The reverse is also true. The detailed explanation for this is outside the scope of this text, but the rule explains why the symmetric stretch in carbon dioxide is seen in the Raman spectrum, but not in the IR spectrum, while the asymmetric stretch appears in the IR spectrum but not in the Raman spectmm. 

The essential signature of a molecule is tiiat it vibrates. For a molecule composed of N atoms, there are 3N mechanical degrees of freedom associated with the motions of the system. Three degrees of freedom are determined by the translational motions of the center of mass, and for a nonlinear molecule there are three degrees of freedom connected with the overall rotational motion of the molecule. For a macromolecule, some of the remaining 3N - 6 degrees of freedom are associated with isomerizations of the chain backbone and the side chains. Finally, there exists a set of quantized vibrational states for the molecule. If the frequencies of the vibrational states depend on the conformational state of the molecule, the measurement of the vibrational spectrum can be used to infer the conformational composition of the ensemble of macromolecules. The frequencies of quantized molecular vibrations greatly exceed the frequencies associated with isomerization of the chain backbone. 

As we ll discuss in Chap. 6, a nonlinear molecule with N atoms has 3N 6 vibrational modes, each involving movements of at least two, and sometimes many atoms. The overall vibrational wavefunction can be written as a product of wavefunctions for these individual modes, and the overall Franck-Condon factor for a given vibronic transition is the product of the Franck-Condon factors for all the modes. When the molecule is raised to an excited electronic state some of its vibrational modes will be affected but others may not be. The coupling factor (S) provides a measure of these effects. Vibrational modes for which S is large are strongly coupled to the excitation, and ladders of lines corresponding to vibronic transitions of each of these modes will feature most prominently in the absorption spectrum. 

Suppose that three conformations are proposed for the nonlinear molecule H2O2 (8,9, and 10). The infrared absorption spectrum of gaseous HjOj has bands at 870,1370,2869, and... 

Electronically Excited State A A.,. Theoretical studies on the effects of orbital angular momentum in nonlinear molecules [23, 24] showed NH2 to be nonlinear at equilibrium (otg = 144° 5°), with a small barrier to linearity (see p. 171) [24]. An empirical fit of the bending potential function confirmed the slightly bent nuclear arrangement yielding otg = 144.2° and rg = 1.007 A [25]. Ab initio treatments established this structure [10, 16]. These results corrected the earlier conclusion of a linear structure indicated by an analysis of the X<-A absorption spectrum [2]. 

Linear molecules have two equal principal moments of inertia, corresponding to rotation about the center of mass about two mutually perpendicular axes, with the third principal moment equal to zero. Nonlinear molecules usually have three different moments of inertia. In this case, the vibration-rotation spectrum can be very complex, even for a simple molecule such as water. The rotational fine stmcture of the H-O-Ti bending mode of water is shown in Figure 1.3. 

For nonlinear molecules, three rotational constants are used in the energy level expressions. They are associated with rotations about three orthogonal axes in the nonlinear molecule called principal axes. One or more of the rotational constants can be measured from a microwave spectrum, and isotopic substitution studies can be employed to extract geometrical parameters (i.e., bond lengths and angles). 

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