Representations The Character Table

It is then possible to represent the above-mentioned symmetry operation by the 3x3 matrix of Equation (7.1). In a more general way, we can associate a matrix M with each specific symmetry operation R, acting over the basic functions x, y, and z of the vector (x,y, z). Thus, we can represent the effect of the 48 symmetry operations of group Oh (ABe center) over the functions (x, y, z) by 48 matrices. This set of 48 matrices constitutes a representation, and the basic functions x, y, and z are called basis functions. 

Obviously, we can examine the effect of the Oh symmetry operations over a different set of orthonormal basis functions, so that another set of 48 matrices (another representation) can be constructed. It is then clear that each set of orthonormal basis functions pi generates a representation Fso that, as in Equation (7.1), we can write a transformation equation as follows  

if a suitable space of basis functions is used (a space of basis functions that is closed under the symmetry operations of the group), we can construct a set of representations (each one consisting of 48 matrices) for this space that is particularly useful for our purposes. It is especially relevant that the matrices of each one of these representations can be made equivalent to matrices of lower dimensions. 

The representations that involve the lowest-dimension matrices are called irreducible representations and have a particular relevance in group theory. 

any representation Tcan be expressed as a function of its irreducible representations Pi. This operation is written as P = S a, Pi, where a, indicates the number of times that Pi appears in the reduction. In group theory, it is said that the reducible representation P is reduced into its Pi irreducible representations. The reduction operation is the key point for applying group theory in spectroscopy. To perform a reduction, we need to use the so-called character tables. 

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This means that the four MOs that will be equivalent to the set of four a orbitals must be chosen so as to include one orbital of Ax symmetry and a set of three orbitals belonging to the T2 representation. The character table also tells us that AOs of atom A falling into these categories are as follows ... 

PROBLEM Find the direct product of e 0 e in the point group and decompose it to its irreducible representations. The character table is as follows. 

As we can in principle choose any basis vector we like, we might think that there is an infinite number of possible representations, Fp, each describing the behavior of one basis vector. However, in reality any basis vector we choose must generate either one of the irreducible representations present in the character table, or (as in the last example above) a reducible representation that can itself be reduced to a collection of irreducible representations. The character table therefore covers all possible modes of behavior under the symmetry operations of the point group. The character tables for a selection of point groups most likely to be of interest to chemists are included in the on-Une supplement for Chapter 2. 

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]. 

The rotation-vibration-electronic energy levels of the PH3 molecule (neglecting nuclear spin) can be labelled with the irreducible representation labels of the group The character table of this group is given in table Al.4.10. 

The tables of characters have the general form shown in Table 5. Each colipua represents a class of symmetry operation, while the rows designate the different irreducible representations. The entries in the table are simply the characters (traces) of the corresponding matrices. Two specific properties of the character tables will now be considered. 

A further property of die dieter tables arises from the fact that every symmetry group has an irreducible representation that is invariant under all of die group operations. This irreducible representation is a one-by-one unit matrix (the number one) for every class of operation. Obviously, the characters, are all then equal to one. AS this irreducible representation is by convention taken to be the first row of all Character tables consists solely of ones. The significance of the character tables will become more apparent by consideration of an example. 

An example of the application of Eq. (47) is provided by the group < 3v whose symmetry operations are defined by Eqs. (18). If the same arbitrary function,

symmetry operation can be worked out, as shown in the last column of Table 13. With the use of the projection operator defined by Eq. (47) and the character table (Table 6), it is found (problem 16) that the coordinate z is totally symmetric (representation Ai). However, it is the sum xy + zx that is preserved in the doubly degenerate representation, E. It should not be surprising that the functions xy and zx are projected as the sum, because it was the sum of the diagonal elements (the trace) of the irreducible representation that was employed in each case in the... 

E (for the identity) in Table 6 are accounted for. Furthermore, the totally symmetric representation is r(1) e A the latter notation is dial usually used by speetroscopists The construction of the remainder of the character table is accomplished by application of the orthogonality property of the characters [see Eq. (30) and problem 131. Standard character tables have been derived in this way for the more common groups, as given in Appendix VQI. 

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. 

For readers unfamiliar with these techniques, it might be helpful at this point to work out an example in some detail. We choose that of the allene skeleton, already discussed somewhat in this section, and at first we limit ourselves to achiral ligands, so that G = S4. The character table for S4 is shown in Table 1. In this case, the subgroup is just D2a, and its rotational subgroup is D2. Table 2 shows the classes of T>za, the number of elements in each, the class of S4 and of S4 to which each belongs, and the character of each for the representation, T< >. 

What has been mentioned up to now allows us to infer that the relevant information needed for a representation is given by the characters of its matrices. In fact, the full information for a given group is given by its character table. This table contains the character files of a particular set of representations the irreducible representations. Table 7.2 shows the character table of the Oh point group. A character table, such as Table 7.2, contains the irreducible representations (10 for the Oh group) and their characters, the classes (also 10 for the Oh group), and the set of basis functions. 

Let us consider the character table of the C4 group and the representation F displayed below (see Table 7.3). F is, reducible, because its character does not coincide with any one of the irreducible representations of the Cav group. According to Equation (7.4) and the character table (Table 7.3), we can write... 

Table 7.3 The character table of group C4 . The basis functions are not included for the sake of brevity. A reducible representation, F, is shown below...

Table 7.4 The character tables of group O and its subgroup >4. The irreducible representation T] of group O appears written below as a reducible representation in >4...

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. 

In the ideal case of free Eu + ions, we first must observe that the components of the electric dipole moment, e x, y, z), belong to the irreducible representation in the full rotation group. This can be seen, for instance, from the character table of group 0 (Table 7.4), where the dipole moment operator transforms as the T representation, which corresponds to in the full rotation group (Table 7.5). Since Z)° x Z) = Z) only the Dq -> Fi transition would be allowed at electric dipole order. This is, of course, the well known selection rule A.I = 0, 1 (except for / = 0 / = 0) from quantum mechanics. Thus, the emission spectrum of free Eu + ions would consist of a single Dq Ei transition, as indicated by an arrow in Figure 7.7 and sketched in Figure 7.8. 

The examples used above to illustrate the features of the software were kept deliberately simple. The utility of the symbolic software becomes appreciated when larger problems are attacked. For example, the direct product of S3 (order 6) and S4 (isomorphic to the tetrahedral point group) is of order 144, and has 15 classes and representations. The list of classes and the character table each require nearly a full page of lineprinter printout. When asked for, the correlation tables and decomposition of products of representations are evaluated and displayed on the screen within one or two seconds. Table VII shows the results of decomposing the products of two pairs of representations in this product group. 

The traces, however, are convenient and adequate representatives of the symmetry species. The character table of a group is a listing of the traces of the matrices forming the sets of irreducible representations of the group. 

The sequence of numbers arrived at constitutes the representation of the two Is orbitals with respect to symmetry. Such a combination of numbers is not to be found in the character table it is an example of a reducible representation. Its reduction to a sum of irreducible representations is, in this instance, a matter of realizing that the sum of the a,+ and gu+ characters is the representation of the two Is orbitals ... 

The reducible representation of the six 2p orbitals may be seen, by inspection of the character table and carrying out the following exercise, to be equivalent to the sum of the irreducible representations ... 

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