Reducible representation group

For establishing how many irreducible representations of type i factor group are contained in the above described factor group reducible representation, we have to find the number... 

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... 

These six matrices can be verified to multiply just as the symmetry operations do thus they form another three-dimensional representation of the group. We see that in the Ti basis the matrices are block diagonal. This means that the space spanned by the Tj functions, which is the same space as the Sj span, forms a reducible representation that can be decomposed into a one dimensional space and a two dimensional space (via formation of the Ti functions). Note that the characters (traces) of the matrices are not changed by the change in bases. 

Before considering other concepts and group-theoretical machinery, it should once again be stressed that these same tools can be used in symmetry analysis of the translational, vibrational and rotational motions of a molecule. The twelve motions of NH3 (three translations, three rotations, six vibrations) can be described in terms of combinations of displacements of each of the four atoms in each of three (x,y,z) directions. Hence, unit vectors placed on each atom directed in the x, y, and z directions form a basis for action by the operations S of the point group. In the case of NH3, the characters of the resultant 12x12 representation matrices form a reducible representation... 

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... 

This statement is often taken as a basic theorem of representation theory. It is found that for any symmetry group there is only one set of k integers (zero or positive), the sum of whose squares is equal to g, the order of the group. Hence, from Eq. (29), the number of times that each irreducible representation appears in the reduced representation, as well as its dimension, can be determined for any group. 

This result (problem 14) allows the coefficients to be calculated. Thus, with a knowledge of the symmetry group and the corresponding table of characters, the structure of the reduced representation can be determined. Equation (37) is of such widespread applicability that it is referred to by many students of group theory as the magic formula. ... 

A planar molecule of point group 03b is shown in Fig. 5. The sigma orbitals i, <72 and (73 represented there will be taken as the basis set Application of the method developed in Section 8.9 yields the characters of the reducible representation given in Table 14. With the use of the magic formula (Eq. (37)] the structure of the reduced representation is of the form Ta — A, ... 

When such a complete reduction has been achieved, the component representations rF),r(2 are called the irreducible representations of the group G and the representation T is said to be fully reduced. An irreducible representation may occur more than once in the reduction of a reducible representation T. Symbolically... 

The CMOs transform as irreducible representations of the molecular point group, whereas the LMOs form a basis for a reducible representation. 

In this case, it can be proved that the canonical SCF orbitals, being solutions of Eq. (26), are symmetry orbitals, i.e. that they belong to irreducible representations of the symmetry group. 12) If the number of molecular orbitals is larger than the dimension of the largest irreducible representation of the symmetry group, it must then be concluded that the set of all N molecular orbitals form a reducible representation of the group which is the direct sum of all the irreducible representations spanned by the CMO s. 

Thus, 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. 

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...

Equation (7.7) can be used to determine the characters of the symmetry operations C , where n = In ja, and then, to construct the reducible representations in the group G of the ion in the crystal. Some of the most common character elements are listed below ... 

At this point, we are able to construct the reducible representations D- of a group composed only of rotational elements. For instance, let us consider that the ion in the crystal has a symmetry group G = 0, whose character table (Table 7.4) consists of only rotational symmetry elements classes C . 

These vectors form the basis for a reducible representation. Evaluate the characters for this reducible representation under the symmetry operations of the D h group. 

The character of the reducible representation of the 2p orbitals of E2 molecules may be obtained by writing down, under each symmetry element of the group, the number of such orbitals which are unchanged by each symmetry operation. This produces the representation ... 

The coefficients of the symmetry elements along the top of the above classification (the same as those across the top of the C3v character table), Le. 1,2 and 3, give a total of six which is the order of the point group, denoted by h. The relationship used to test the hypothesis that the reducible representation contains a particular irreducible representation is ... 

When we come to apply the results we have so far discovered to quantum mechanical situations, we will find that the application usually revolves around the reduction of some reducible representation for the point group concerned. We have already seen how to find out which irreducible representations appear in the reduction of a reducible representation, namely if we write... 

Let us now consider the n-dimensional reducible representation r d which is produced from the function space whose basis functions are Qi> 9i> - Qn> d let us assume that in the reduction of I 1 no irreducible representation of the point group occurs more than once. One way of looking at the reduction is to see it as a change of basis functions from gl9 9i> gn to... 

From Table 7-9,2 and using eqn (5-7,2) we can find the diagonal elements of the matrices which represent the 4h point group in the p-orbital basis and in the d-orbital basis. From these elements we get the characters of two reducible representations they are shown in Table 7-9,3, By applying eqn (7-4.2)... 

The regular representation is a reducible representation composed of matrices constructed as follows first write down the group multiplication table in such a way that the order of the rows corresponds to the inverses of the operations heading the columns in this way will appear only along the diagonal of the table. For example, from Table 3 4.2 we would have... 

Given the characters x of a reducible representation T of the indicated point group 9 for the various classes of 9 in the order in which these classes appear in the character table, find the number of times each irreducible representation occurs in T. 

It is always possible to form a new, and in general reducible, representation r of a given point group from any two given representations T and r of the group. This is done by forming a new function space for which the basis functions are all possible products of the basis functions of T and T Let the basis functions of T and t9 be... 

As a second example, let us consider a molecule with the formula AB6 having the symmetry of a trigonal bipyramid Ih. The vector system is shown in Fig. 11-3.2. The set of five hybrid orbitals (or vectors) on A form a basis for a reducible representation of the point group, with the following character ... 

The six necessary hybrid orbitals on the boron atom can also be assigned vectors. If w-bonds are to be formed, these vectors must have the same orientation as the six vectors on the chlorine atoms. If we followed in the footsteps of 11-3, we would now construct the reducible representation Th7b from a consideration of how the six vectors on the boron atom change under the symmetry operations of the B point group. However, it is clear that since the six vectors on the chlorine atoms match the six on the boron atom, exactly the same representation rhyb can be found by using these vectors instead. Since it is less confusing to have three pairs of vectors separated in space than six originating from one point, we will take this latter approach. 

The ten rr-type p-orbitals form the basis for a reducible representation Tr of the 6d point group with the following character ... 

A reducible representation is said to be the direct sum of the irreducible representations of which it is made up. In the standard notation (Section 9.5) for point-group irreducible representations, the 03v representation (9.29) is called A ( and the representation (9.28) is called E. If we denote the reducible representation (9.25) by T, then... 

Since each MO belongs to some irreducible representation of the molecular point group, we must find linear combinations of the AOs that transform according to the irreducible representations group theory enables us to do this. A symmetry operation sends each nucleus either into itself or into an atom of the same type a symmetry operation will thus transform each AO into some linear combination of the AOs (9.63). Therefore (Section 9.6), the AOs form a basis for some representation TAO of the point group of the molecule. This representation (as any reducible representation) will be the direct sum of certain of the irreducible representations r r2,...,r (not necessarily all different) of the molecular point group ... 

The factor lj/h, where h is the order of the group and b 18 the dimension of the y th irreducible representation, has been included in (9.67) for convenience. Application of this procedure to the functions / gives us (unnormalized) symmetry-adapted functions g,. This procedure is applicable to generating sets of functions that form bases for irreducible representations from any set of functions that form a basis for a reducible representation. The proof of the procedure (9.67) for one-dimensional representations is outlined in Problem 9.22 we omit its general proof.5 Symmetry-adapted functions produced by (9.67) that belong to the same irreducible representation are not, in general, orthogonal. 

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