Point groups character table for

Character tables for point groups are available on the Web. See http //www.mpip-mainz.mpg.de/ gelessus/group.html... 

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. 

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. 

The tetrachloroplatinate(If) anion. PtCI -, was given earlier (Fig. 3.14c) as an example of a molecule belonging to the DAI, point group. The character table for this group is... 

Octahedral complex ML6 has Oh symmetry. However, for simplicity, we may work with the 0 point group, which has only rotations as its symmetry operations and five irreducible representations, A, A2,. ..,T2. The character table for this group is shown in Table 8.4.1. It is seen that the main difference between the Oh and 0 groups is that the former has inversion center i, while the latter does not. As a result, the Oh group has ten symmetry species Aig, Aiu, A2g, A2u, , T2g, r2u. 

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. 

Three of the representations for C2 , labeled Aj, Bj, and B2 below, have been determined so far. The fourth, called A2, can be found by using the properties of a group described in Table 4-7. A complete set of irreducible representations for a point group is called the character table for that group. The character table for each point group is unique character tables for the common point groups are included in Appendix C. 

The tetra-atomic molecules of concern throughout this book belong to one of two point groups Cjv or C. The character tables for these groups are reproduced below ... 

Table A.Ill contains the character table for point Z of Cl. This little group has only one two-dimensional representation and we therefore expect all eigenvalues at that point to be doubly degenerate. Using the transformations summarized in (A. 18) and similar additional relations, we construct the characters of the reducible representations Ztraus...

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... 

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

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