Reducible representation of a group

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... 

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. 

Theorem (2) shows that there are only a finite number of nonequivalent irreducible representations of a group of finite order. Any reducible representation must either be the direct sum of two or more irreducible representations or be convertible to such a direct sum by the performance of a similarity transformation on its matrices. In the former case, it is easy to see by inspection what irreducible representations make up the reducible representation in the latter case, this is not obvious, since the matrices... 

The underlaying assumption in the evaluation of both information, the Sij and the representation of a group, is that compounds with similar properties have similar structural features thus producing similar spectra. In view of the fact that no strict rule for quantitative definition of similarity between structures exists it is hard to justify the above assumption. However, many valuable results can be obtained using the correlation between the similarity of properties (structures) and similarity of spectra. If the reduced representation of a spectrum i is written in a vector form Ri as... 

The set of matrices Us Pr) in equation (41) form a reducible representation of the group which is reduced into its irreducible components A by the coefficients a above. If we denote the irreducible representations of iAn and 2 by 17 Is- iVl and Z)(/1>, respectively, then this reduction can be written symbolically as... 

Figure 5-11. Schematic representation of a group of pigments in a photosystem core that harvests a light quantum (hv) and passes the excitation to a special trap chlorophyll. Short straight lines indicate the inducible dipoles of chlorophyll molecules and the wavy lines indicate resonance transfer. In the reaction center an electron (e ) is transferred from the trap chi to some acceptor (A in the reduced form) and is then replaced by another electron coming from a suitable donor (D+ in the oxidized form).

We first assume that the functions have the symmetry of a local site a and thus the wavefunctions xLa realize the irreducible representation of the ath site group, where the index / labels this representation. Let xl.g(R) be the character of a matrix which corresponds to an element R in the /th irreducible representation of the site group, and x g (R) be the character of a matrix corresponding to an element R of a factor group in its th irreducible representation. The character of an reducible representation of a factor group, spanned on functions L (0) and corresponding to an element R, is given by... 

When the direct product of two irreducible matrix representations of a group is reducible, it can be reduced to a direct sum of irreducible representations by cin equivalent transformation with a constant matrix, i.e. the same matrix for all the matrix representatives of the symmetry operators of the group (2). We shall assume the irreducible representations in unitary form then the constant matrix can be chosen as the real orthogonal matrix whose elements are the coupling coefficients occuring in Eq. (5). The orthogonality properties can be expressed as... 

The irreducible representations of a group are generally reducible as representations of a sub-group. In Racah s lemma this statement is considered by introducing the concept of a group which is reduced with respect to a sub-group. By this is meant that the irreducible representa-... 

If we choose to construct molecular orbitals for NH3 using linear combinations of the Is orbitals on the three hydrogen atoms, and the 25,2p, 2py, and 2p orbitals on the nitrogen atom, we can show that the resulting molecular orbitals belong to a reducible representation of the group C v with the characters ... 

As previously, the set of 5 X 5 matrices is a reducible representation of the group C v- In this case the matrices are diagonal and therefore in reduced form. We may therefore immediately write down the irreducible components by taking sets of equivalent diagonal elements. Thus, the irreducible representations 1, 1, —1, —1, 1, —1, 1, —1 each appear once, and 1, 1, 1, 1) appears three times. We are therefore able to express the vector G in terms of the orthogonal vectors in the space of the group elements. This procedure is extraordinarily useful in physical problems in much the same manner that it is useful to know the orthogonal components of an ordinary vector property in Cartesian space. 

It is not always necessary to work with all the synunetry operators and representations of a group. Certain subsets of the group of operators for a molecule may meet the requirements of a group on their own, and using the subset amounts to analyzing a problem in reduced symmetry we then learn about a corresponding subset of the symmetry features of functions and vectors. 

From the relations already developed it is possible to obtain further interesting results. Any matrix representation of a group must be some one of the irreducible representations or some combination of them otherwise it would be an additional irreducible representation, but the number of irreducible representations is limited to the number of classes. Any reducible representation can be reduced to its irreducible representations by a similarity transformation which leaves the character unchanged. Thus we can write for the character of a matrix R of the reducible representation the expression... 

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. 

Figure 4.14 Diagrammatic representation of (a) oxy-radical>mediated S-thioiation and (b) thiol/disulphide-initiated S-thiolation of protein suiphydryl groups. Under both circumstances mixed disuiphides are formed between glutathione and protein thiois iocated on the ion-translocator protein resulting in an alteration of protein structure and function. Both of these mechanisms are completely reversible by the addition of a suitabie reducing agent, such as reduced glutathione, returning the protein to its native form.

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... 

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 . 

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 ... 

---