The Selection Rule

This spectrum is called a Raman spectrum and corresponds to the vibrational or rotational changes in the molecule. The selection rules for Raman activity are different from those for i.r. activity and the two types of spectroscopy are complementary in the study of molecular structure. Modern Raman spectrometers use lasers for excitation. In the resonance Raman effect excitation at a frequency corresponding to electronic absorption causes great enhancement of the Raman spectrum. 

We now turn to electronic selection rules for syimnetrical nonlinear molecules. The procedure here is to examme the structure of a molecule to detennine what synnnetry operations exist which will leave the molecular framework in an equivalent configuration. Then one looks at the various possible point groups to see what group would consist of those particular operations. The character table for that group will then pennit one to classify electronic states by symmetry and to work out the selection rules. Character tables for all relevant groups can be found in many books on spectroscopy or group theory. Ftere we will only pick one very sunple point group called 2 and look at some simple examples to illustrate the method. 

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. 

If the experunental technique has sufficient resolution, and if the molecule is fairly light, the vibronic bands discussed above will be found to have a fine structure due to transitions among rotational levels in the two states. Even when the individual rotational lines caimot be resolved, the overall shape of the vibronic band will be related to the rotational structure and its analysis may help in identifying the vibronic symmetry. The analysis of the band appearance depends on calculation of the rotational energy levels and on the selection rules and relative intensity of different rotational transitions. These both come from the fonn of the rotational wavefunctions and are treated by angnlar momentum theory. It is not possible to do more than mention a simple example here. 

Figure Bl.4.9. Top rotation-tunnelling hyperfine structure in one of the flipping inodes of (020)3 near 3 THz. The small splittings seen in the Q-branch transitions are induced by the bound-free hydrogen atom tiiimelling by the water monomers. Bottom the low-frequency torsional mode structure of the water duner spectrum, includmg a detailed comparison of theoretical calculations of the dynamics with those observed experimentally [ ]. The symbols next to the arrows depict the parallel (A k= 0) versus perpendicular (A = 1) nature of the selection rules in the pseudorotation manifold.

Using the selection rule for allowed transitions the relative intensity for the transition from the state Mg) to Mg+l) is given by... 

The transition between levels coupled by the oscillating magnetic field B corresponds to the absorption of the energy required to reorient the electron magnetic moment in a magnetic field. EPR measurements are a study of the transitions between electronic Zeeman levels with A = 1 (the selection rule for EPR). 

The diagonal elements of the matrix [Eqs. (31) and (32)], actually being an effective operator that acts onto the basis functions Ro,i, are diagonal in the quantum number I as well. The factors exp( 2iAct)) [Eqs. (27)] determine the selection rule for the off-diagonal elements of this matrix in the vibrational basis—they couple the basis functions with different I values with one another (i.e., with I — l A). 

The matrix elements (60) represent effective operators that still have to act on the functions of nuclear coordinates. The factors exp( 2iAx) determine the selection rules for the matrix elements involving the nuclear basis functions. 

Thus in the lowest order approximation the angle x is eliminated from the off-diagonal matrix elements of [second and third of Eqs. (60)] it solely determines the selection rules for matrix elements of Hg with respect to nuclear basis functions. 

Qualitatively, the selection rule for IR absorption for a given mode is that the symmetry of qT ) " must he the same as qT ). Qiianii-talivcly, the transition dipole moment is proportion al to tlie dipole derivative with respect to a given normal mode dp/di. ... 

The selection rules for AK depend on the nature of the vibrational transition, in particular, on the component of itrans along the molecule-fixed axes. For the second 3-j symbol to not vanish, one must have... 

The L = L +1 transitions are termed R-branch absorptions and those obeying L = L -1 are called P-branch transitions. Hence, the selection rules... 

For non-linear molecules of the spherical or symmetric top variety, pf j(Rg) (or dpf j/dRa) may be aligned along or perdendicular to a symmetry axis of the molecule. The selection rules that result are... 

In a diatomic or linear polyatomic molecule rotational Raman scattering obeys the selection rule... 

For a symmetric rotor molecule the selection rules for the rotational Raman spectmm are... 

The procedure for determining the selection rules for a particular molecule is ... 

In an E vibrational state there is some splitting of rotational levels, compared with those of Figure 5.6(a), due to Coriolis forces, rather than that found in a If vibrational state, but the main difference in an E — band from an — A band is due to the selection rules... 

The selection rules are the same for oblate symmetric rotors, and parallel bands appear similar to those of a prolate symmetric rotor. However, perpendicular bands of an oblate symmetric rotor show Q branches with AK = - -1 and — 1 on the low and high wavenumber sides, respectively, since the spacing, 2 C — B ), is negative. 

The selection rules governing the promotion of the electron to an excited orbital, and also its falling back from an excited orbital, are... 

So far we have considered only hydrogen, helium, the alkali metals and the alkaline earth metals but the selection rules and general principles encountered can be extended quite straightforwardly to any other atom. 

When you consider the selection rules, which are not particularly restrictive (see Section 7.1.6), governing transitions between these states arising from each configuration, it is not surprising that the electronic spectrum of an atom such as zirconium consists of very many lines. (Remember that the Laporte rule of Equation (7.33) forbids transitions between states arising from the same configuration.)... 

As is the case for vibrational transitions, electronic transitions are mostly of the electric dipole type for which the selection rules are as follows. 

As in atoms, the selection rule breaks down as the nuclear charge increases. For example, triplet-singlet transitions are strictly forbidden in FI2 but in CO the a U — transition is observed weakly. 

The + or — label indicates whether the wave function is symmetric or antisymmetric, respectively, to reflection across any plane containing the intemuclear axis. Whether the + component is below or above the — component for, say, J = 1 depends on the sign of q in Equation (7.94). The selection rules ... 

In non-linear polyatomic molecules the process of deterioration of quantum numbers continues to such an extent that only the total electron spin quantum number S remains. The selection rule... 

For the orbital parts of the electronic wave functions of two electronic states the selection rules depend entirely on symmetry properties. [In fact, the electronic selection rules can also be obtained, from symmetry arguments only, for diatomic molecules and atoms, using the (or and Kf point groups, respectively but it is more... 

Electronic transitions mostly involve interaction between the molecule and the electric component of the electromagnetic radiation (Section 2.1). The selection rules are, therefore. 

Because Raman scattering is also a two-photon process the selection rules for two-photon absorption are the same as for vibrational Raman transitions. For example, for a two-photon electronic transition to be allowed between a lower state j/" and an upper state... 

Both the Ag — X I,g and the b g —X Ig transitions are extremely weak. The selection rules, discussed in Section 7.2.3, show that both transitions violate the A5 = 0 and the g< I >g selection rules. In addition, the a-Xtransition violates the AA = 0, 1 selection rule and the b -X transition the - - — selection rule. Spin-orbit interaction breaks down... 

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