Irreducible representation characters

Products between the irreducible representation characters within a group will produce representations which are often reducible. A simple calculation can decompose this product to a sum of the irreducible representation characters, as is demonstrated in Table V for two representations from the S3-DP-S2 group. 

Table of symmetry elements, irreducible representations, characters of the individual irreducible representations and assignments of vector and tensor properties for a given point group. 

For the Csv group, the characters for some direct products of bases for irreducible representations are shown in Table 13-25. The direct product has as characters the product of characters of E times itself. These characters (4,0,1) do not agree with any of the irreducible representation character sets, and so E E is reducible. We can tell, in fact, that E E is four-dimensional from the leading character. To resolve EiS>E,we employ the formula (13-7), which gives E E = Ai A2 E, and fits the observation that E E is four-dimensional. The other direct products listed... 

Table 4.10 Subtraction of the irreducible representation characters from F for the y, z basis on the central Ni atom of the complex as [Ni(CN)4p. ...

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]. 

In applications of group theory we often obtain a reducible representation, and we then need to reduce it to its irreducible components. The way that a given representation of a group is reduced to its irreducible components depends only on the characters of the matrices in the representation and on the characters of the matrices in the irreducible representations of the group. Suppose that the reducible representation is F and that the group involved... 

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. 

Whenever a fiinction can be written as a product of two or more fiinctions, each of which belongs to one of the synnnetry classes, the symmetry of the product fiinction is the direct product of the syimnetries of its constituents. This direct product is obtained in non-degenerate cases by taking the product of the characters for each symmetry operation. For example, the fiinction xy will have a symmetry given by the direct product of the syimnetries of v and ofy this direct product is obtained by taking the product of the characters for each synnnetry operation. In this example it may be seen that, for each operation, the product of the characters for Bj and B2 irreducible representations gives the character of the representation, so xy transfonns as A2. 

We have seen that any two of the C2, ( Jxz), (r Jyz) elements may be regarded as generating elements. There are four possible combinations of + 1 or — 1 characters with respect to these generating elements, + 1 and + 1, + 1 and -1,-1 and +1,-1 and —1, with respect to C2 and (tJxz). These combinations are entered in columns 3 and 4 of the C2 character table in Table A.l 1 in Appendix A. The character with respect to / must always be + 1 and, just as (r Jyz) is generated from C2 and (tJxz), the character with respect to (r Jyz) is the product of characters with respect to C2 and (tJxz). Each of the four rows of characters is called an irreducible representation of the group and, for convenience, each is represented by a symmetry species Aj, A2, or B2. The A] species is said to be totally symmetric since all the characters are + 1 the other three species are non-totally symmetric. 

The characters 4,1,0 form a reducible representation in the C3 point group and we require to reduce it to a set of irreducible representations, the sum of whose characters under each operation is equal to that of the reducible representation. We can express this algebraically as... 

In the E irreducible representation, the character of any symmetry operation corresponding to a rotation by an angle is given by... 

Table 12-4 gives the characters, basis functions, and the case a, b, or c to which the irreducible representation of Hu and belong for the point T. The degeneracy in and is the usual Kramers spin degeneracy, which is removed in cases (2) and (4) because of the absence of 6 in these symmetry groups. 

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. 

Only the set of integers 1,1,2 satisfies this relation, the order being arbitrary. In tins group there are two different irreducible representations of order one and one of order two. Thus, tire characters appearing in the column headed... 

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... 

Consider a molecular system of symmetry < 2v. whose character table is given in Table 8-11. The irreducible representations for the components of the dipole moment can be easily established, or even read directly from the table. 

The group developed above to describe the symmetry of the ammonia molecule consisted only of the permutation operations. However, if the triangular pyramid corresponding to this structure is flattened, it becomes planer in me limit. The RF3 molecule shown in Fig. lb is an example of this symmetry. In this case it becomes possible to invert the coordinate perpendicular to the plane of the molecule, the z axis. Obviously, the operation of reflection in the (horizontal) plane of the molecule, <7h> is identical. It is easy, then, to identify the irreducible representations A and A" as symmetric or antisymmetric, respectively, under the coordinate inversion. The group composed of the identity and the inversion of the z axis is then <5 = s> whose character table is of the form of Table 7. 

The internal coordinates for the water molecule are chosen as changes in the structural parameters defined in Fig. 3. The effect of each symmetry operation of the symmetry group ( 2 on these internal coordinates is specified in Table 2. Clearly, the internal coordinate Ace is totally symmetric, as the characters xy(Aa) correspond to those given for the irreducible representation (IR) Ai. On die other hand, the characters x/(Ar), as shown, can not be identified with a specific IR. By inspection of Table 2, however, it is apparent that the direct sum Ai B2 corresponds to the correct symmetry of these coordinates. In more complicated cases the magic formula can always be employed to achieve the correct reduction of the representation in question. 

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 practical applications these relationships lead to a number of important rules about irreducible representations and their characters ... 

A matrix of order l has l2 elements. Each irreducible representation T, must therefore contribute If -dimensional vectors. The orthogonality theorem requires that the total set of Y f vectors must be mutually orthogonal. Since there can be no more than g orthogonal vectors in -dimensional space, the sum Y i cannot exceed g. For a complete set (19) is implied. Since the character of an identity matrix always equals the order of the representation it further follows that... 

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