Character Tables for Selected Point Groups

Appendix B Character Tables for Selected Point Groups... 

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... 

Show that the HOMO LUMO transition (highest occupied to lowest unoccupied MO) in the tt-bonding system of cyclopentadienyl must be allowed by electric dipole selection rules. (You do not need the character table for this point group to answer the question.)... 

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. 

An isolated n-atom molecule has 3n degrees of freedom and in—6 vibration degrees of freedom. The collective motions of atoms, moving with the same frequency and which in phase with all other atoms, give rise to normal modes of vibration. In principle, the determination of the form of normal modes for any molecule requires the solution of equation of motion appropriate to the n-symmetry. Methods of group theory are important in deriving the symmetry properties of the normal modes. With the aid of the character tables for point groups and the symmetry properties of the normal modes, the selection rules for Raman and IR activity can be derived. For a molecule with a center of symmetry, e.g. AXe, octahedral molecule, a non-Raman active mode is also IR active, whereas for the BX4 tetrahedral molecule, some modes are simultaneously IR and Raman active. 

The character table allows identifying the active infrared modes. These modes are those with nonzero translation represented by T in the character table. For the C2v point group, the A and B modes are active in the infrared spectroscopy. Adding to this analysis the surface selection rule, only those modes with translation in the direction perpendicular to the surface will be active. Considering that the surface is included in the xy plane, only the Ai modes can be active, since only the Ai have translation in the z direction. The Ai mode is the symmetric species with respect to the rotation about the symmetry axis, as described above. Therefore, for the adsorbed N03 ion, only the symmetric modes will be active in the infrared spectroscopy. 

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

As we can in principle choose any basis vector we like, we might think that there is an infinite number of possible representations, Fp, each describing the behavior of one basis vector. However, in reality any basis vector we choose must generate either one of the irreducible representations present in the character table, or (as in the last example above) a reducible representation that can itself be reduced to a collection of irreducible representations. The character table therefore covers all possible modes of behavior under the symmetry operations of the point group. The character tables for a selection of point groups most likely to be of interest to chemists are included in the on-Une supplement for Chapter 2. 

Table 3.15 lists the character tables and selection rules for important point groups. Polyatomic linear molecules which belong to the and groups are a special case. There are an infinite number of rotations about the axis on which the nuclei lie which yield an equivalent configuration of the molecule. The determination of the selection rules is based on the application of a reduction formula which for nonlinear molecules involves a summa-... 

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... 

Linear molecules belong to either the D h (with an inversion centre) or the (without an inversion centre) point group. Using the vibrational selection rule in Equation (6.56) and the D h (Table A.37 in Appendix A) or (Table A. 16 in Appendix A) character table we can see that the vibrational selection rules for transitions from the zero-point level (LjJ in Di0o/l, A1 in ) allow transitions of the type... 

Symmetry selection rules for Raman spectrum can be derived by using a procedure similar to that for the IR spectrum. One should note, however, that the symmetry property of symmetry species of six components of polarizability are readily found in character tables. In point group C3V, for example, normal vibrations of the NH3 molecule (2A1 and 2E) are Raman-active. More generally, the vibration is Raman-active if the component(s) of the polarizability belong(s) to the same symmetry species as that of the vibration. 

Selecting the Character Table button on the main command bar of the GT Calculator files activates the main display window as shown, again, for the case of the Ih point group, in Figure 1.4. 

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