Representations of groups

As an example, let us work out a representation of the group C2v, which group consists of the operations , C2, t , g v. Let us say that the C2 axis coincides with the z axis of a Cartesian coordinate system, and let av be the xz plane and 7 be the yz plane. The matrices representing the transformations effected on a general point can easily be seen to be as follows  

It can easily be shown that the matrices multiply together in the same fashion. For example, 

each element in the group C2v is its own inverse, so the same must be true of the matrices. This is easily shown to be so for example, 

We have now, by one procedure, namely, by considering the transformations of a general point, generated a set of matrices which form a representation for the group C2l.. It will be recalled that we have also done the same thing (pages 74-76) for the group T. 

A question that naturally arises at this point is How many representations can be found for any particular group, say C2v, to continue with that as an example The answer is A very large number, limited only by our ingenuity in devising ways to generate them. There are first some very simple ones, obtained by assigning 1 or -1 to each operation, namely, 

One particularly useful way to describe the symmetry operations of a molecule is to write S in terms of a transformation matrix D(S), the 3x3 matrix in equation 4.6, which determines the manner in which a set of basis vectors x are transformed into a new set x as a result of the symmetry operation. 

The two representations Fg and F although equivalent (have the same character) are in fact different in the following sense. Notice that Fg, although a three-dimensional representation (there are three basis vectors), may be written in block diagonal form for all operations S as 

At this point we need to ask what the possible irreducible representations for a given point group are. Just as there is a set of rules that define the formation of a point group, so there is a pair of rules that define the possible set of IRs of a group. 

In short, where the sums are implied whenever there is product of the Fs, equations 4.13 and 4.14 are simplified as in equations 4.15 and 4.16, respectively. 

Each of the irreducible representations could be labeled in a simple way, F, Fi,. . . , F , but there is a notation due to Mulliken, which is very useful. A and 6 refer to one-dimensional representations where A refers to a representation for which the character corresponding to the highest order rotation axis is +1 and B when it is I. and T refer to doubly and triply degenerate representations, respectively. From the definition of these characters, the entry in the character table for the identity operation, y,( ) defines the order of the representation. (I for A, 6 2 for 3 for T, etc.) If there is more than one representation with the same label, then they are distinguished by subscripts 1,2,. . ., and so on. 

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Using the reduction of representations of groups given by Eq. (4.46), one then finds that the rotational spectra of linear molecules contain the angular momenta... 

The same rules for the reduction of representations of groups give the following content of angular momentum in each vibrational band of bent molecules... 

In Table 7.5, we show the character (defined as the set of character elements of a representation) of different representations (from / = 1 to 6) of the 0 group. The character elements were obtained from Equation (7.7). These representations, which were irreducible in the full rotation group, are in general reducible in 0, as can be seen by inspecting the character table of 0 (in Table 7.4). Thus, the next step is to decompose them into irreducible representations of 0, as we did in Example 7.1. Table 7.5 also includes this reduction in other words, the irreducible representations of group O into which each representation is decomposed. We will use this table when treating relevant examples in Section 7.6. 

EXAMPLE 7.4 The direct product between irreducible representations of group O. 

An introduction to the mathematics of group theory for the non-mathematician. If you want to learn formal group theory but are uncomfortable with much of the mathematical literature, this book deserves your consideration. It does not treat matrix representations of groups or character tables in any significant detail, however. 

Many works[5, 6] on group theory describe matrix representations of groups. That is, we have a set of matrices, one for each element of a group, G, that satisfies... 

Any set of algebraic functions or vectors may serve as the basis for a representation of a group. In order to use them for a basis, we consider them to be the components of a vector and then determine the matrices which show how that vector is transformed by each symmetry operation. The resulting matrices, naturally, constitute a representation of the group. We have previously used the coordinates jc, y, and z as a basis for representations of groups C2r (page 78) and T (page 74). In the present case it will be easily seen that the matrices for one operation in each of the three classes are as follows ... 

This table shows how the representations of group Oh are changed or decomposed into those of its subgroups when the symmetry is altered or lowered. This table covers only representations of use in dealing with the more common symmetries of complexes. A rather complete collection of correlation tables will be found as Table X-14 in Molecular Vibrations by E. G. Wilson, Jr., J. C. Decius, and P. C. Cross, McGraw-Hill, New York, 1955. 

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