Fundamental vibration selection rule

Vibrational spectra and symmetry 8.5.1 Fundamental vibrational selection rule... 

These signals correspond precisely to the characteristic vibrations of the molecule. If v = 0 they are called fundamental bands. As levels with v > 0 are normally not significantly populated, these bands account for most of the intensity in an absorption spectrum. The IR absorption spectrum and the Stokes Raman scattering spectrum both correspond to v —> v + 1 transitions, so the fundamental vibrational selection rule is normally stated as Av = +1. We should remember that Av = — 1 is also permitted, but this corresponds to the emission of radiation in the IR or to the anti-Stokes lines in the Raman spectrum, which will be weak, as the population of molecules with v > 0 will be small. 

The vibrational selection rules treated in Sec. 2.7 are strictly valid in the gas phase, because intermolecular interactions are mostly absent. As an example we present the rotation-vibration infrared and Raman spectra of benzene CgHg in Fig. 4.3-1 on a common scale. According to the rule of mutual exclusion (see Sec. 2.7.3.4), none of the fundamentals should coincide in the two spectra. Of the 20 normal vibrations of QHf, four are infrared active (1A2 , 3 i ), seven Raman active (24 E g, and nine... 

The fundamental principles upon which the calculation of selection rules are based have been given in Secs. 3-4, 3-5, and 3-6. In this chapter these principles will be applied to the problem of determining the vibrational selection rules for symmetrical molecules. It will be found that certain transitions are forbidden merely because of the symmetry properties of the molecule. Other transitions are found not to be forbidden by symmetry considerations such transitions may nevertheless be missed experimentally because of low intensity due to other causes. On the other hand, transitions forbidden by symmetry sometimes seem to appear in the spectra of liquids, presumably due to the distortion of the symmetry by the neighboring molecules. However, in spite of the fact that so-called forbidden transitions may occur weakly in liquids and so-called allowed transitions are quite frequently not observed, the selection rules given by symmetry considerations are of very great importance as a guide in the interpretation of molecular spectra. 

The vibrational selection rule, rule (7) above, which is applicable again here, implies that only one vibrational mode can change, and by at most one unit, in any allowed transition. Transitions in which this is obeyed are called fundamentals and they indeed tend to represent the strongest transitions. As a result of harmonicities however both overtone Au > 1 for a single mode) and combination bands Av 0 for several modes) may appear, and are quite important. A discussion of which particular overtone and combination bands will be active in infrared and Raman transitions, a property which stems from more fundamental selection rules governed by the symmetry of the molecule, is beyond our purposes here and can be found in, for example, [Ref. 2.2, Sect. III.3]. 

The case in which all A w s vanish corresponds to a microwave spectrum. According to Eq. (23.5-la), Aw = 1 for only one normal mode at a time. Transitions obeying this selection rule produce fundamental bands. There is a fundamental band in the infrared region for each normal mode that modulates the molecule s dipole moment. The vibrational selection rules are less well obeyed than the rotational selection rules. There are overtone bands in which A w = 2 (and sometimes 3), and combination bands, in which two (or more) normal modes change their quantum numbers at once. These forbidden bands are usually less intense than the fundamental bands. 

The vibrational selection rule for the harmonic oscillator, Au = 1, applies to polyatomic molecules just as it did to diatomic molecules. Vibrational energy can, therefore, change in units of hcoi/ln. Transitions in which one of the three normal modes of energy changes by Au = - -1 (for example Ui = 0 1, U2 = U3 = 0 or 1 = 1) 2 = 3, i>3 = 2 3) result from absorption of a photon having one of three fundamental frequencies of the molecule. In the actual case, anharmonicities also allow transitions with Au, = 2, 3,... so that, for example, weak absorption also occurs at 2coi, 3(Ui, etc. and at coi + coj, 2vibrational transitions often play major roles in planetary spectroscopy. 

Raman vibrational selection rules are related to the symmetry of the polarizability components. Considering only the fundamental vibrations... 

In this expression, cr, p = x, y, or z, where a p is one of the polarizability components and (i and f) are the initial and final vibrational wave functions. Each of the factors in Eq. (80) belongs to an irreducible representation T, and according to group theory, in order for the integral in Eq. (80) to be nonvanishing, the direct product TiXT xTf must contain the totally symmetric representation. Thus, for a fundamental vibration to be Raman active, YfYf must transform under symmetry operations in the same manner as one of the a p components. This is the basis of the normal Raman vibrational selection rule. 

The surface selection rule operates in addition to the normal IR selection rules in determining which vibrational modes are observed. As a result of the SSR the relative intensities of the fundamental IR adsorption bands of an adsorbed species can be used to give information on the orientation of the species with respect to the surface. Both S- and P-polarised light interact equally with the randomly oriented solution species. 

For a fundamental vibrational mode to be IR-active, a change in the molecular dipole must take place during the molecular vibration. This is described as the IR selection rule. Atoms that possess different electronegativity and are chemically bonded change the net dipole of a molecule during normal molecular vibrations. Typically, antisymmetric vibrational modes and vibrations due to polar groups are more likely to exhibit prominent IR absorption bands. 

In a case where the transition of an energy state is from 0 to 1 in any one of the vibrational states (vi,v2,v3,. ..), the transition is considered as fundamental and is allowed by selection rules. When a transition is from the ground state to v — 2,3,. .., and all others are zero, it is known as an overtone. Transitions from the ground state to a state for which Vj = 1 and vj = 1 simultaneously are known as combination bands. Other combinations, such as v — 1, Vj = 1, v = 1, or v, — 2, v7 — 1, etc., are also possible. In the strictest form, overtones and combinations are not allowed, however they do appear (weaker than fundamentals) due to anharmonicity or Fermi resonance. 

The model fundamental to all analyses of vibrational motion requires that the atoms in the system oscillate with small amplitude about some defined set of equilibrium positions. The Hamiltonian describing this motion is customarily taken to be quadratic in the atomic displacements, hence in principle a set of normal modes can be found in terms of these normal modes both the kinetic energy and the potential energy of the system are diagonal. The interaction of the system with electromagnetic radiation, i.e. excitation of specific normal modes of vibration, is then governed by selection rules which depend on features of the microscopic symmetry. It is well known that this model can be worked out in detail for small molecules and for crystalline solids. In some very favorable simple cases the effects of anharmonicity can be accounted for, provided they are not too large. 

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... 

Anharmonicity leads to deviations of two kinds. At higher quantum numbers, AE becomes smaller, and the selection rule is not rigorously followed as a result, transitions of A 2 or 3 are observed. Such transformations are responsible for the appearance of overtone lines at frequencies approximately two or three times that of the fundamental line the intensity of overtone absorption is frequently low, and the peaks may not be observed. Vibrational spectra are further comphcated by the fact that two different vibrations in a molecule can interact to give absorption peaks with frequencies that are approximately the sums or differences of their fundamental frequencies. Again, the intensities of combination and difference peaks are generally low. 

An alternative experiment that measures the same vibrational fundamentals subject to different selection rules is Raman spectroscopy. Raman intensities, however, are more difficult to compute than IR intensities, as a mixed third derivative is required to approximate the change in the molecular polarizability with respect to the vibration that is measured by the experiment. The sensitivity of Raman intensities to basis set and correlation is even larger than it is for IR intensities. However, Halls, Velkovski, and Schlegel (2001) have reported good results from use of the large polarized valence-triple-f basis set of Sadlej (1992) and... 

This result is tremendously useful, it not only leads to selection rules for vibrational spectroscopy but also, as was the case with electronic wavefunctions (see 8-2), allows us to predict from inspection of the character table the degeneracies and symmetries which are allowed for the fundamental vibrational wavefunctions of any particular molecule. 

Selection Rules. The character table shows that A and E vibrations are both IR and Raman active. Thus, all four fundamental modes for such a molecule should be observable in both the IR and the Raman spectra. 

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