Point group character tables

Next, we discuss representations and character tables. A group of order h can be represented by h matrices, each of dimensions hxh. Flowever, there exist so-called irreducible representations for each group, which are block-diagonal submatrices, which span the space. Table 7.6 presents the character tables for all 32 crystallographic point groups, and some other groups as well. Table 7.7 is an abbreviated form of Table 7.6. 

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. 

SymApps converts 2D structures From the ChemWindow drawing program into 3D representations with the help of a modified MM2 force field (see Section 7.2). Besides basic visualization tools such as display styles, perspective views, and light source adjustments, the module additionally provides calculations of bond lengths, angles, etc, Moreover, point groups and character tables can be determined. Animations of spinning movements and symmetry operations can also he created and saved as movie files (. avi). 

Molecular point-group symmetry can often be used to determine whether a particular transition s dipole matrix element will vanish and, as a result, the electronic transition will be "forbidden" and thus predicted to have zero intensity. If the direct product of the symmetries of the initial and final electronic states /ei and /ef do not match the symmetry of the electric dipole operator (which has the symmetry of its x, y, and z components these symmetries can be read off the right most column of the character tables given in Appendix E), the matrix element will vanish. 

The character tables for all important point groups, degenerate and non-degenerate, are given in Appendix A. 

Inspection of this character table, given in Table A. 12 in Appendix A, shows two obvious differences from a character table for any non-degenerate point group. The first is the grouping together of all elements of the same class, namely C3 and C as 2C3, and (t , and 0-" as 3o- . 

The III character table is given in Table A.46 in Appendix A. The very high symmetry of this point group results in symmetry species with degeneracies of up to five, as in and... 

Then, if we look through all the point group character tables in Appendix A to see if any of the translational symmetry species is totally symmetric, it is apparent that molecules belonging to only the following point groups have a permanent dipole moment ... 

Assign the allene molecule to a point group and use the character table to form the direct products A-2 x5i,5i X 82,82 xE and E X E. Show how the symmetry species of the point group to which 1,1-dilluoroallene belongs correlate with those of allene. 

Each of these can be assigned to one of the symmetry species of the point group to which the molecule belongs. These assignments are indicated in the right-hand column of each character table given in Appendix A and will be required when we consider vibrational Raman spectra in Section 6.2.3.2. 

Acetylene (HC=CH) belongs to the point group whose character table is given in Table A.37 in Appendix A, and its vibrations are illustrated in Figure 6.20. Since V3 is a vibration and T T ) = 2"+, the 3q transition is allowed and the transition moment is polarized along the z axis. Similarly, since Vj is a vibration, the 5q transition is allowed with the transition moment in the xy plane. 

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... 

This result is similar to that for e x e, in Equation (4.29), in the point group and can be verified using the Tig , character table in Table A.36 in Appendix A. As the two electrons (or one electron and one vacancy) in the partially occupied orbitals may have parallel (S = 0) or antiparallel (5=1) spins there are six states arising from the configuration in Equation... 

J. A. SalthouSe and M. J. Ware, Point Group Character Tables and Related Data, p. 29, Cambridge University Press, 1972. 

Vibrations of the symmetry class Ai are totally symmetrical, that means all symmetry elements are conserved during the vibrational motion of the atoms. Vibrations of type B are anti-symmetrical with respect to the principal axis. The species of symmetry E are symmetrical with respect to the two in-plane molecular C2 axes and, therefore, two-fold degenerate. In consequence, the free molecule should have 11 observable vibrations. From the character table of the point group 04a the activity of the vibrations is as follows modes of Ai, E2, and 3 symmetry are Raman active, modes of B2 and El are infrared active, and Bi modes are inactive in the free molecule therefore, the number of observable vibrations is reduced to 10. 

A planar molecule of point group 03b is shown in Fig. 5. The sigma orbitals i, <72 and (73 represented there will be taken as the basis set Application of the method developed in Section 8.9 yields the characters of the reducible representation given in Table 14. With the use of the magic formula (Eq. (37)] the structure of the reduced representation is of the form Ta — A, ... 

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