Group character table for

Table B.4. Double group characters table for the Ta point group. The numbers before the symmetry operations correspond to the number of geometrically different axes or symmetry planes. Some of the operations of the double group belong to the same class as those of the original group. When more than one IR is indicated, the first one corresponds to the notation of Mulliken [11], the second one to Koster et al. [9] and the one in parentheses to [3]...

After obtaining the symmetry point group of the metal-adsorbate system, one can determine which symmetry representations of the vibrations of the molecule will be Raman active using group character tables for the given point group. Raman-active modes are those that possess nonzero components of the Raman polarizability tensor. 

The characters of the irreducible representations of a synnnetry group are collected together into a character table and the character table of the group 3 is given in table A1.4.3. The construction of character tables for finite groups is treated in section 4.4 of [2] and section 3-4 of [3]. 

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 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- . 

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... 

In effect, the division by two is the result of the molecular symmetry, as specified by the character table for the group 0. In general it is useful to define a symmetry number a (= 2 in this case), as shown below. The well-known example of the importance of nuclear spin is that of ortho- arid para-hydrogen (see Section 10.9.5). 

Appendix VIII Character Tables for Some of the More Common Point Groups... 

The possible wave functions for the molecular orbitals for molecules are those constructed from the irreducible representations of the groups giving the symmetry of the molecule. These are readily found in the character table for the appropriate point group. For water, which has the point group C2 , the character table (see Table 5.4) shows that only A1 A2, B1 and B2 representations occur for a molecule having C2 symmetry. 

Follow the procedure used in the text in obtaining the character table for the C2 point group and develop the character table for the C3 point group. 

Appendix B Character Tables for Selected Point Groups 823... 

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