Reducible and Irreducible Representations

Molecular Symmetry David J. Willock 2009 John Wiley Sons, Ltd. ISBN 978-0 70-85347  

In Section 4.8 we found how the matrix representation of a set of basis vector transformations for a point group can sometimes be made simpler. For example, the 2 x 2 matrix representation for x, y at O in H2O can be reduced to two T x 1 matrices (i.e. a simple number for each operation) for x and y. This process of simplifying a representation to a set of irreducible standard representations is central to the application of symmetry in chemistry and corresponds to finding the fundamental modes of vibration that underlie molecular motion. 

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Prior to interpreting the character table, it is necessary to explain the terms reducible and irreducible representations. We can illustrate these concepts using the NH3 molecule as an example. Ammonia belongs to the point group C3V and has six elements of symmetry. These are E (identity), two C3 axes (threefold axes of rotation) and three crv planes (vertical planes of symmetry) as shown in Fig. 1-22. If one performs operations corresponding to these symmetry elements on the three equivalent NH bonds, the results can be expressed mathematically by using 3x3 matrices. ... 

The next step is to separate this representation into its component irreducible representations. This requires another property of groups. The number of times that any irreducible representation appears in a reducible representation is equal to the sum of the products of the characters of the reducible and irreducible representations taken one operation at a time, divided by the order of the group. This may be expressed in equation form, with the sum taken over all symmetry operations of the group. ... 

As an example, we will consider a trigonal-bipyramidal complex ML5, in which L is a CT donor only. The point group is and the reducible and irreducible representations are shown here ... 

The relationships among symmetry operations, matrix representations, reducible and irreducible representations, and character tables are conveniently illustrated in a flowchart, as shown for C2v symmetry in Table 4.8. 

Equation 4.39 follows by inspection of the reduced matrix above. We will avoid confusion of the character symbols for reducible and irreducible representations by always using a superscript (j) for the jth irreducible representation. No superscript appears on a reducible matrix character. Multiplying relations in Eq. 4.39 on both sides by and summing over all group elements R yields... 

Thus far, we have defined representations and shown how they may be generated from basis functions. We have distinguished between reducible and irreducible representations and have indicated that there is an unlimited number of equivalent representations corresponding to any given two- or higher-dimensional representation. An example of a pair of equivalent, reducible, two-dimensional representations, derived in Section 13-11, is given in Table 13-17. Equivalent representations are related through unitary transformations, which are a special kind of similarity transformation (see Chapter 9), and two matrices that differ only by a similarity transformation have the same... 

In general, it will be possible to simplify the total representation for a basis of our choosing into a sum of the standard irreducible representations from the point group character table. The reducible and irreducible representations are linked by the fact that the sum of characters of the irreducible representations for a basis must give the characters of their reducible representation ... 

In Chapter 5 we will exploit this relationship between the reducible and irreducible representations further and find a general formula for obtaining the , values that control the composition of F for any basis and any point group. This chapter finishes with a few more examples of reducible and irreducible representations. 

Section 4.11 used the matrix representation to deal with a set of three basis vectors jc, y, z on the central atom of a square planar D41, complex. It was shown that this basis can be reduced to + A2U by inspection of the matrices for the operations in the Ah group. The characters for the reducible and irreducible representations are shown in Table 5.2. 

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