Selection rules vibrational, polyatomic

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.. 

The selection rule for vibronic states is then straightforward. It is obtained by exactly the same procedure as described above for the electronic selection rules. In particular, the lowest vibrational level of the ground electronic state of most stable polyatomic molecules will be totally synnnetric. Transitions originating in that vibronic level must go to an excited state vibronic level whose synnnetry is the same as one of the coordinates, v, y, or z. 

Polyatomic molecules vibrate in a very complicated way, but, expressed in temis of their normal coordinates, atoms or groups of atoms vibrate sinusoidally in phase, with the same frequency. Each mode of motion functions as an independent hamionic oscillator and, provided certain selection rules are satisfied, contributes a band to the vibrational spectr um. There will be at least as many bands as there are degrees of freedom, but the frequencies of the normal coordinates will dominate the vibrational spectrum for simple molecules. An example is water, which has a pair of infrared absorption maxima centered at about 3780 cm and a single peak at about 1580 cm (nist webbook). 

When applied to linear polyatomic molecules, these same selection rules result if the vibration is of a symmetry (i.e., has k = 0). If, on the other hand, the transition is of n symmetry (i.e., has k = 1), so the transition dipole lies perpendicular to the molecule s axis, one obtains ... 

When exposed to electromagnetic radiation of the appropriate energy, typically in the infrared, a molecule can interact with the radiation and absorb it, exciting the molecule into the next higher vibrational energy level. For the ideal harmonic oscillator, the selection rules are Av = +1 that is, the vibrational energy can only change by one quantum at a time. However, for anharmonic oscillators, weaker overtone transitions due to Av = +2, + 3, etc. may also be observed because of their nonideal behavior. For polyatomic molecules with more than one fundamental vibration, e.g., as seen in Fig. 3.1a for the water molecule, both overtones and... 

Although we have been able to see on inspection which vibrational fundamentals of water and acetylene are infrared active, in general this is not the case. It is also not the case for vibrational overtone and combination tone transitions. To be able to obtain selection rules for all infrared vibrational transitions in any polyatomic molecule we must resort to symmetry arguments. 

As is the case for diatomic molecules, rotational fine structure of electronic spectra of polyatomic molecules is very similar, in principle, to that of their infrared vibrational spectra. For linear, symmetric rotor, spherical rotor and asymmetric rotor molecules the selection rules are the same as those discussed in Sections 6.2.4.1 to 6.2.4.4. The major difference, in practice, is that, as for diatomics, there is likely to be a much larger change of geometry, and therefore of rotational constants, from one electronic state to another than from one vibrational state to another. 

The second problem relates to the inclusion, or otherwise, of molecular symmetry arguments. There is no avoiding the fact that an understanding of molecular symmetry presents a hurdle (although I think it is a low one) which must be surmounted if selection rules in vibrational and electronic spectroscopy of polyatomic molecules are to be understood. This book surmounts the hurdle in Chapter 4, which is devoted to molecular symmetry but which treats the subject in a non-mathematical way. For those lecturers and students who wish to leave out this chapter much of the subsequent material can be understood but, in some areas, in a less satisfying way. 

Subsequently, it has been recognized that other IR techniques (reflection, emission and photothermal techniques) can also be applied, and that IR spectroscopy only gives partial information on the vibrational structure of most polyatomic species. In fact, selection rules apply to IR light absorption phenomena, so that only vibrational modes that are associated with changes of the molecular dipolar moment can be directly excited. 

The analysis of the quanta that are actually absorbed by a polyatomic chemical species and those that are not absorbed (so are transmitted) gives information on the vibrational structure of these species and, consequently, on its chemical and geometric structure. This is the transmission/absorption IR technique. However, selection rules apply to such a phenomenon. They are simplified as follows ... 

Information regarding the normal modes of a polyatomic molecule, which are not IR active, may often be obtained from the Raman spectrum. Raman spectroscopy is an inelastic-scattering technique rather than requiring the absorption or emission of radiation of a particular energy. The selection rule differs from the IR in that it is required that the incident electric field of the radiation can induce a changing dipole moment of the molecule. This results in a different symmetry requirement for the normal modes of vibration to be Raman active, since it now depends on the polarizability of the molecule. 

In Sec. 1.5, the symmetry and the point group allocation of a given molecule were discussed. To understand the symmetry and selection rules of normal vibrations in polyatomic molecules, however, a knowledge of group theory is required. The minimum amount of group theory needed for this purpose is given here. 

In addition to the bands centered on the fundamental frequencies, other bands appear in the spectra of polyatomic molecules. We have mentioned overtone bands in the spectrum of diatomic molecules due to violation of the selection rule, Ap = +1, that is permitted because of anharmonicity. But in polyatomic molecules, combination bands also appear. For example, in the case of water if the absorbed quantum splits to raise from 0 to 1 and V2 from 0- 1, there will be a vibration-rotation band centered on the combination frequency, + V2 This process is relatively less probable than the absorbtion of a single quantum at either fundamental frequency, so the intensity of the band is relatively weak. Nonetheless, combination bands appear with sufficient intensity to be an important feature of the infrared spectra of polyatomic molecules. Even in the case of a simple molecule like water, there are a large number of prominent bands, several of which are listed in Table 25.2. 

According to the selection rule for the harmonic oscillator, any transitions corresponding to An = 1 are allowed (Sec. I-2). Under ordinary conditions, however, only the fundamentah that originate in the transition from u = 0 to u = 1 in the electronic ground state can be observed because of the Maxwell-Boltzmann distribution law. In addition to the selection rule for the harmonic oscillator, another restriction results from the symmetry of the molecule (Sec. 1-9). Thus the number of allowed transitions in polyatomic molecules is greatly reduced. The overtones and combination bands of these fundamentals are forbidden by the selection rule of the harmonic oscillator. However, they are weakly observed in the spectrum because of the anharmonicity of the vibration... 

As noted before, polyatomic molecules have 3 AT-6 or, if linear, 3JV—5 normal vibrations. For any given molecule, however, only vibrations that are permitted by the selection rule for that molecule appear in the infrared and Raman spectra. Since the selection rule is determined by the symmetry of the molecule, this must first be studied. 

The majority of polyatomic molecular rotational spectra arise from J 1 J, / and P branch transitions and for these, even when they follow asymmetric rotor selection rules, the intensities still increase with increasing frequency see for example SO2 (Figure 1.1). A conspicuous exception to this general rule however occurs when g-branch J J vibration-rotation transitions are present in the spectral range under investigation. 

This chapter begins with a classical treatment of vibrational motion, because most of the important concepts that are specific to vibrations in polyatomics carry over naturally from the classical to the quantum mechanical description. In molecules with harmonic potential energy functions, vibrational motion occurs in normal modes that are mutually uncoupled. Coupling between vibrational modes inevitably occurs in the presence of anharmonic potentials (potentials exhibiting cubic and/or higher order terms in the nuclear coordinates). In molecules with sufficient symmetry, the use of group theory simplifies the procedure of obtaining the normal mode frequencies and coordinates. We obtain El selection rules for vibrational transitions in polyatomics, and consider the rotational fine structure of vibrational bands. We finally treat breakdown of the normal mode approximation in real molecules, and discuss the local mode formulation of vibrational motion in polyatomics. 

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