Selected character tables

TABLE 6.3 Selected character tables for point groups of non-linear molecules. More tables appear in the Appendix. 

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. 

Linear molecules belong to either the (with an inversion centre) or the (without an inversion centre) point group. Using the vibrational selection rule in Equation (6.56) and the (Table A. 3 7 in Appendix A) or (Table A. 16 in Appendix A) character table we can... 

For a symmetric rotor molecule such as methyl fluoride, a prolate symmetric rotor belonging to the C3 point group, in the zero-point level the vibrational selection mle in Equation (6.56) and the character table (Table A. 12 in Appendix A) show that only... 

As we proceed to molecules of higher symmetry the vibrational selection rules become more restrictive. A glance at the character table for the point group (Table A.41 in Appendix A) together with Equation (6.56) shows that, for regular tetrahedral molecules such as CH4, the only type of allowed infrared vibrational transition is... 

Appendix B Character Tables for Selected Point Groups... 

When deriving selection rules from character tables it is noted that vibrations are usually excited from the ground state which is totally symmetric. The excited state has the symmetry of the vibration being excited. Hence A vibration will be spectroscopically active if the vibration has the same symmetry species as the relevant operator. 

In the ideal case of free Eu + ions, we first must observe that the components of the electric dipole moment, e x, y, z), belong to the irreducible representation in the full rotation group. This can be seen, for instance, from the character table of group 0 (Table 7.4), where the dipole moment operator transforms as the T representation, which corresponds to in the full rotation group (Table 7.5). Since Z)° x Z) = Z) only the Dq -> Fi transition would be allowed at electric dipole order. This is, of course, the well known selection rule A.I = 0, 1 (except for / = 0 / = 0) from quantum mechanics. Thus, the emission spectrum of free Eu + ions would consist of a single Dq Ei transition, as indicated by an arrow in Figure 7.7 and sketched in Figure 7.8. 

Electric dipole n polarized emissions are those in which the electric field of the emitted light is parallel to z. Thus the selection rule for n emissions from level A1 are those defined by the direct product A1 x A2. By an inspection of the character table of group 1)3 (Table 7.6), we can easily prove that Ai x A2 = A2, so that only the A A2 emission is allowed by n polarized radiation (as shown in Figures 7.7 and 7.8). 

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. 

We also see that if Tj occurs at all it will occur only once. It is very easy to check this theorem by using the character tables in Appendix IIA, as the following illustrative example shows. Many other examples may be selected by the reader to develop familiarity with the manipulation of direct products. 

We may illustrate these rules, using the carbonate ion. We see in the D h character table that (x, y) form a basis for the E representation and z for the Ay representation. For the polarizability tensor components we see that one or more of these belong to the A, , and E" representations. Thus, for any molecule of Dv, symmetry, we have the following selection rules ... 

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. 

The selection rules for the fundamentals of these modes are obtained immediately from the right columns of the character table and are... 

Reduction of fr (Eq. 3.1) shows that it is composed of ee, and bin. The character table reveals that no orbitals transform as but that p. belongs to while df. and dn belong to eg. That these three orbitals on platinum are allowed by symmetry to participate in out-of-plane it bonding is reasonable since they are all oriented perpendicular to the plane of the ion (the xy plane). Selection of orbitals on platinum suitable for in-plane it bonds is left as an exercise. (Hint In choosing vectors to represent the suitable atomic orbitals, remember that the in-plane and out-of-p[ane ir bonds will be perpendicular to each other and that the regions of overlap lor the former will be on eadi side of a hording axis. Thus the in-plane vectors should be positioned perpendicular to the bonding axes.)30... 

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