Selection Rules for Optical Transitions

Formal rules, known as selection rules, may be used to decide whether or not an electric dipole transition between two states may take place. Perhaps the most important is the rule governing spin multiplicity spin must not change during an electronic transition. The usual way to write rules of this kind is 

In atoms, for one-electron transitions, we have the selection rules 

For diatomic and linear polyatomic molecules the orbital momentum rule is 

The rules are derived by considering whether or not an interaction with the electric vector of electromagnetic radiation is possible in going from the initial to the proposed final states of the chemical species. Frequently, the symmetry properties of the wavefunctions involved suffice to exclude certain transitions, so that the selection rules are a way of representing the possibilities that remain, although they say nothing about the absolute intensities of the interactions. 

The spin selection rule, AS = 0, might be expected to be of universal applicability, since it does not require the molecule under consideration to have any geometrical symmetry. However, spin-forbidden transitions are also frequently observed. The spin rule is based again on the idea of separability of wavefunctions, this time of the spin and spatial components of the electronic wavefunction. However, the electron experiences a magnetic field as a result of the relative motion of the positive nucleus with respect to it, and this field causes some mixing of spatial and spin components, giving rise to spin-orbit 

In Table 7.5, we show the character (defined as the set of character elements of a representation) of different representations (from / = 1 to 6) of the 0 group. The character elements were obtained from Equation (7.7). These representations, which were irreducible in the full rotation group, are in general reducible in 0, as can be seen by inspecting the character table of 0 (in Table 7.4). Thus, the next step is to decompose them into irreducible representations of 0, as we did in Example 7.1. Table 7.5 also includes this reduction in other words, the irreducible representations of group O into which each representation is decomposed. We will use this table when treating relevant examples in Section 7.6. 

Group theory can also be applied to determine whether an optical transition is allowed in a particular optical center. As we showed in Section 5.3, the probability of a radiative transition between two given states, (initial) and (final), is proportional to 

the band intensity (absorption or emission) related to an optical transition between two states xj/i and f f depends on the value of the matrix element given in 

Equation (7.9). By analyzing this matrix element, we can establish the selection rules for the transition. 

At this point, we should invoke the so-called Wigner-Eckart theorem, whose demonstration is beyond the scope of this book (see, e.g., Tsukerblat, 1994). From this theorem, it is possible to establish the following selection rule  

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Establishing selection rules for optical transitions and determining their polarization character. 

Equation (4.41) contains all selection rules for optical transitions and it contains all transitions between pairs of states involving the same photon energy fko. It thereby models a system with broad bands as being made up of a multitude of 2-level systems. The absorption coefficient can in principle be calculated from theoretical models. Here we will use it as an experimentally determined quantity. 

The experimental techniques most commonly used to measure the phonon distributions are IR absorption, Raman scattering and neutron scattering. The IR and Raman spectra of crystalline silicon reflect the selection rules for optical transitions and are very different from the phonon density of states. The momentum selection rules are relaxed in the amorphous material so that all the phonons contribute to the spectrum. 

By definition of the zero-order basis, selection rules for optical transitions hold rigorously, so that the n states may be divided into subsets of radiant s and nonradiant / levels. These are characterized by transition moments to the vibrationless level of the ground electronic state 0> sueh that (see Section II,E)... 

If, however, we use a set of coordinates (which are not necessarily normal coordinates) that transform under the symmetry operations of the group to which the molecule belongs in the manner indicated by the matrices of the irreducible representations, then all cross products of the type QiQj, where Qi and Qj belong to different irreducible representations, will vanish this choice of coordinates will thus greatly simplify the solution of equation 2-55. We shall return to this point briefly a little later first we wish to derive the selection rules for optical transitions between the various possible vibrational states of polyatomic molecules. 

Diatomic Molecules (Spin Neglected), 258. Symmetry Properties of the Wave Functions, 261. Selection Rules for Optical Transitions in Diatomic Molecules, 262. The Influence of Nuclear Spin, 265. The Vibrational and Rotational Energy Levels of Diatomic Molecules, 268. The Vibrational Spectra of Polyatomic Molecules, 273. 

Because many physical systems possess certain types of symmetry, its adaptation has become an important issue in theoretical studies of molecules. For example, symmetry facilitates the assignment of energy levels and determines selection rules in optical transitions. In direct diagonalization, symmetry adaptation, often performed on a symmetrized basis, significantly reduces the numerical costs in diagonalizing the Hamiltonian matrix because the resulting block-diagonal structure of the Hamiltonian matrix allows for the separate... 

Many Raman scattering Unes are observed for SiC, reflecting zone folding effects in phonon dispersion curves. These lines can be used to identify the polytype of SiC crystals, as mentioned in Sec. III.A. From the shift of the Raman peaks and the discrepancy of the selection rules in optical transitions, information about the internal stress and crystallinity of SiC crystals, respectively, can be obtained, as mentioned in Sec. ni.B. 

In the lowest optically excited state of the molecule, we have one electron (t u) and one hole (/i ), each with spin 1/2 which couple through the Coulomb interaction and can either form a singlet 5 state (5 = 0), or a triplet T state (5 = 1). Since the electric dipole matrix element for optical transitions H em = (ep A)/(me) does not depend on spin, there is a strong spin selection rule (A5 = 0) for optical electric dipole transitions. This strong spin selection rule arises from the very weak spin-orbit interaction for carbon. Thus, to turn on electric dipole transitions, appropriate odd-parity vibrational modes must be admixed with the initial and (or) final electronic states, so that the weak absorption below 2.5 eV involves optical transitions between appropriate vibronic levels. These vibronic levels are energetically favored by virtue... 

Therefore, the selection rule for an optical transition must take into account the direction of the incident photon (hence, that of the created polarization wave) and must be stated as... 

Interband optical transitions in quantum wells are governed by selection rules determined by the symmetry of the wavefimctions. In an ideal one-dimensional potential well, only transitions between levels of identical quantum number would be allowed. In a real quasi-two-dimensional quantum well, band mixing for finite wave vectors within the layer plane leads to a weakening of the selection rule, so that transitions with An 0 may show up in optical spectra. 

In order to understand the charge transfer features of the Blue Copper site, the variable-temperature optical absorption, room-temperature circular dichroism, and magnetic circular dichroism spectra of plastocyanin, stellacyanin, and azurin were studied355. As can be seen for plastocyanin in Fig. 12, the relative intensities (and signs, in the case of CD and MCD) of these transitions vary among the different types of spectra. This is a result of the difference in selection rules for absorption, CD, and MCD spectra, as mentioned in the Introduction. A careful comparison of the three types of spectra and the absorption bandshape temperature dependence (see moment analysis in Ref. 35, pp. 176-177)... 

Several important effects are equilibrium in nature, for example spectral shifts associated with solvent induced changes in solute energy levels are equilibrium properties of the solvent-solute system. Obviously, such observables may themselves be associated with dynamical phenomena, in the example of solvent shifts it is the dynamics of solvation that affects their dynamical evolution (see Chapter 15). Another class of equilibrium effects on radiation-matter interaction includes properties derived from symmetry rules. A solvent can affect a change in the equilibrium configuration of a chromophore solute and consequently the associated selection rules for a given optical transition. Some optical phenomena are sensitive to the symmetry of the environment, for example, surface versus bulk geometry. 

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