Selection Rules and Polarizations

In Section 5.3 the general symmetry restrictions on transitions occurring by dipolar interaction with electromagnetic radiation were discussed. Here they will be invoked with particular reference to electronic transitions on metal ions in ligand fields. 

In a complex that possesses a center of symmetry, all states arising from a dn configuration have the g character inherent in the d orbitals. Since the dipole moment vectors belong to odd representations, all of the integrals such as / y/gXi//g dr are identically zero because the direct product of two g functions can never span any u representations. On this basis alone, we would predict that transitions between the various states arising from dn configurations in octahedral environments would have zero absorption intensity. In fact, these transitions do take place but the absorption bands are only —lO 3 times the intensity expected for symmetry allowed electronic transitions. Thus the prediction we have made is substantially correct, but at the same time there is obviously some intensity giving mechanism that has been overlooked. 

This phenomenon of vibronic coupling can be treated very effectively by using group theoretical methods. As will be shown in Chapter 10, the vibrational wave function of a molecule can be written as the product of wave functions for individual modes of vibration called normal modes, of which there will be 3n - 6 for a nonlinear, /i-atomic molecule. That is, we can 

It is further shown in Chapter 10 that, when each of the normal modes is in its ground state, each of the y/j is totally symmetric and hence y/v is totally symmetric. If one of the normal modes is excited by one quantum number, the corresponding it may then belong to one of the irreducible representations other than the totally symmetric one, say T, and thus the entire vibrational wave function f/Y will belong to the representation T,. Simple methods for finding the representations to which the first excited states of the normal modes belong are explained in Chapter 10. In this section we will quote without proof results obtained by these methods. 

To a first approximation, and usually a rather good one, the complete wave function F for a molecule can be written as a product of an electronic wave function y/e, a vibrational wave function y/vn and a rotational wave function 

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Table 8.16. Selection rules and polarizations for the transitions between the Zeeman sublevels of the ground state and those of the T6, T7, and T8 excited states of shallow acceptors in silicon and germanium for B // [001]. The point-group symmetries with...

The simplest situation, as far as symmetry is concerned, arises when O represents a constant of motion, in which case the operator is invariant under the symmetry group of the system. However, O may be any tensorial property and have any S5unmetry, so that selection rules and polarization properties are among the problems of the form of Eq. (1). 

This broad band at 1500 cm was ascribed by Kaufman. Metin, and Saper-stein [10], to an IR observation of the amorphous carbon Raman D and G bands. This is forbidden by the selection rules, and has been attributed to the symmetry breaking introduced by the presence of CN bonds in the amorphous network. As carbon and nitrogen have different electronegativities, the formation of CN bonds gives the necessary charge polarity to allow the IR observation of the collective C=C vibrations in the IR spectrum. This conclusion was stated by the comparison of spectra taken from films deposited from N2 and N2. In the N2-film spectrum, no shift was observed for the 1500-cm band, whereas all other bands shifted as expected from the mass difference of the isotopes. Figure 25 compares... 

Azulene has weak absorption in the visible region (near 7000 A) and more intense band systems in the ultraviolet. The first ultraviolet system, which commences at about 3500 A, has been examined in substitutional solid solution in naphthalene (Sidman and McClure, 1956) and in the vapour state (Hunt and Ross, 1962), and can be observed in fluorescence from the vapour (Hunt and Ross, 1956). Theory predicts that the transition is 1Al<-lAl(C2K), i.e. allowed by the electronic selection rules with polarization parallel to the twofold symmetry axis (see, e.g., Ham, 1960 Mofifitt, 1954 Pariser, 1956b). The vibrational analysis shows that the transition is allowed but does not establish the axis of polarization. The intensity distribution among the vibrational bands indicates a small increase in CC bond distance without change in symmetry. 

Fourier transform infrared microscopes are equipped with a reflection capability that can be used under these circumstances. External reflection spectroscopy (ERS) requires a flat, reflective surface, and the results are sensitive to the polarization of the incident beam as well as the angle of incidence. Additionally, the orientations of the electric dipoles in the films are important to the selection rules and the intensities of the reflected beam. In reflectance measurements, the spectra are a function of the dispersion in the refractive index and the spectra obtained are completely different from that obtained through a transmission measurement that is strongly influenced by the absorption index, k. However, a complex refractive index, n + ik can be determined through a well-known mathematical route, namely, the Kramers-Kronig analysis. 

Single crystal Raman spectra of 123 compounds have defined selection rules and prove that in unpolarized spectra Ag modes dominate. Fig. 4.8-22 shows polarized Raman spectra of these compounds. The strongest spectra are obtained with a (zz) scattering geometry, except for the as-0 2,3) mode at 340 cm, which has a nearly zero Raman tensor component in this direction. It is therefore occasionally referred to as quasi Big. 

Polarization information provides a direct determination of selection rules and allows for rigorous assignments based on group theory allows correlation of spectral features with geometric structure providing detailed insight into electronic structure of metal ion active site... 

For a material to exhibit measurable SHG effects in the solid state it must crystallise in a non-centrosymmetric polar space group. This is a very important "selection rule", and one which is not always easy, or even possible, to "engineer" at the crystal-forming... 

A polarimeter measures the angle that an enantiomer can rotate plane polarized light. Observed rotation (a) is the value recorded directly from a polarimeter. Specific rotation ([a]) is the standardized value for optical rotation, where [a] = a/l c. The absolute configuration (R or S) of an enantiomer is determined by applying the Cahn-Ingold-Prelog selection rules and the steering wheel model. 

For each N2F4 conformer, twelve fundamental vibrations are expected which are distributed among the symmetry species of point groups C2h and C2. They obey the selection rules and exhibit the polarization properties in the Raman spectrum and the types of band contours in the IR spectrum as follows ... 

Following the initial experiments by Kaiser, Axe (1964) extended the theories of Judd (1962) and Ofelt (1962) to calculate selection rules and intensities of two-photon transitions in solids doped with lanthanides. The theoretical predictions with later extensions by Bader and Gold (1968) laid largely untested as an hiatus occurred in the experimental studies in this area, the emphasis having shifted in this period to the observation of (4f) states buried in allowed (5d) or conduction bands. Interest has since returned following the work of Degenais (1981), Downer et al. (1982) and Down and Bivas (1983) on Gd " in LaFj. As noted in fig. 20, these workers observed a number of violations of the selection rules developed by Axe in both the intensity and polarization dependence and were able to trace the descrepancies to various approximations necessary in the... 

As already explained, the probability of photon absorption by a given molecule depends on a number of factors (see the optical selection rules). If polarized light is employed [61], it also depends on the orientation of the absorption transition dipole moment, with respect to the polarization plane of the excitation light (described by the angle (p). Molecules with their absorption dipole moment parallel to the polarization plane of the excitation light are excited preferentially, while those oriented perpendicularly are not excited at all. For a general orientation with angle (j), the dipole moment can be decomposed into parallel and perpendicular components, /ipcos, and /ipsin, respectively, and the excitation probability is proportional to (cos ). ... 

A third molecular property to be considered is the electric dipole moment. The selection rules and intensity of the rotational transitions of asymmetric tops depend on the dipole moment components along the principal inertial axes, that is, on pa> Pb. and pc> which give rise to a-, b-, and c-type spectra, respectively. All conformers possess the same connectivity between the atoms but they differ in the orientation of the functional groups, and this necessarily produces diverse charge distributions reflected in different values of the dipole moment components, as can be seen in Table 1. The microwave power necessary for optimal polarization depends on the dipole moment component involved in a rotational transition. Hence, the difference in the values of the dipole moment components of conformers can be exploited to discriminate between specific conformers just by varying the polarization power. By itself, it cannot be used as a conclusive tool, but it can always corroborate the conformer identification achieved with the previously described molecular properties. 

Mid-IR absorption and Stokes Raman deal with the same vibrations but are subject to different selection rules (and consequently the spectra differ). IR and RS provide complementary images of molecular vibrations. Vibrations which modulate the molecular dipole moment are visible in the IR spectrum, while those which modulate the polarisability appear in the Raman spectrum. Compositions that do not absorb in the IR range generally give a Raman spectrum and strong IR absorbers will produce a weak spectrum by Raman. Examples of silent Raman vibrational modes are specific point groups (e.g. C(, De, Cev, C4h, D, >3h. Den, etc.). Other vibrations may be forbidden in both spectra. Raman spectroscopy complements IR spectroscopy, particularly for the study of non-polar bonds and functional groups e.g. C=C, C—S, S—S, metal-metal bonds). 

To measure the spin coherence time and to estimate the spin-polarized carrier injection efficiency from the electroluminescence data, the selection rules and the valence band structure in ZnO must be understood. The valence band in wurtzite materials is split into three bands (A, B, and C) due to crystal field and spin-orbit coupling as discussed before in Chapter 3. The spin degeneracy of these three bands and the conduction band is lifted in magnetic field resulting in small symmetric Zeeman splittings as shown in Figure 5.6 near the F point [48]. The allowed transitions following the selection rules AI = 1 (for 0 polarization) are indicated... 

A specific example is poly(vinyl chloride) (PVC), which is a monosubstituted vinyl polymer that has a syndiotactic-rich character and a conformation that can be either an extended all-trans structure or a folded syndiotactic structure. The vibrational modes of these conformational models obey different selection rules and have different dichroic properties that can be used to spectroscopically test these structures [7]. The folded syndiotactic model of PVC has the [p,0] classification that requires unique Raman lines (no coincident IR frequency) that are polarized. The extended syndiotactic model has the two unique classifications of [d,0] and [p,(r], which means that the unique Raman lines are depolarized, and the Raman lines that are polarized have perpendicular dichroism in the IR spectrum. In the Raman spectrum of PVC [8], polarized lines are observed at 363, 638, 694, 1172, 1335, 1430 and 1914 cm and IR bands are also observed at each of these frequencies. This result rejects the folded syndiotactic structure, because this structure requires the polarized lines to be unique. In addition, each of these frequencies is perpendicularly dichroic in the IR spectrum, a fact that supports the planar syndiotactic structure. 

The Raman peaks from the Ge QDs sample and the Si substrate obey different selection rules and are observable at different polarization configurations, which can be used to distinguish these two peaks [25]. Figure 12.4 shows the polarized Raman spectra of a Ge QDs sample on Si substrate and that of a reference Si substrate sample. All spectra were taken with the same data accumulation time. Figure 12.4a plots the spectrum recorded from the reference Si substrate in the... 

A very weak peak at 348 mn is the 4 origin. Since the upper state here has two quanta of v, its vibrational syimnetry is A and the vibronic syimnetry is so it is forbidden by electric dipole selection rules. It is actually observed here due to a magnetic dipole transition [21]. By magnetic dipole selection rules the A2- A, electronic transition is allowed for light with its magnetic field polarized in the z direction. It is seen here as having about 1 % of the intensity of the syimnetry-forbidden electric dipole transition made allowed by... 

The polarization dependence of the photon absorbance in metal surface systems also brings about the so-called surface selection rule, which states that only vibrational modes with dynamic moments having components perpendicular to the surface plane can be detected by RAIRS [22, 23 and 24]. This rule may in some instances limit the usefidness of the reflection tecluiique for adsorbate identification because of the reduction in the number of modes visible in the IR spectra, but more often becomes an advantage thanks to the simplification of the data. Furthenuore, the relative intensities of different vibrational modes can be used to estimate the orientation of the surface moieties. This has been particularly useful in the study of self-... 

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