Group, characters representation

The coefficients a must be so chosen that 2 0,- transforms in the fashion appropriate to the irreducible representation r. Now the important point is that bases for only certain irreducible representations can be constructed out of linear combinations of the vL- To determine which, one ascertains the group characters associated with the transformation scheme, usually reducible, of the original attached wave functions rpi before linear combinations are taken. This step is easy, as the character xd for a covering operation D is simply equal to q, where q is the number of atoms left invariant by D. This result is true inasmuch as D leaves q of the atoms alone, and completely rearranges the others, so that the diagonal sum involved in the character will contain unity q times, and will have zeros for the other entries. The scheme for evaluating the characters is reminiscent of that in the group... 

The character tables in Appendix A3 include the spinor representations of the common point groups. Double group characters are not given explicitly but, if required, these may be derived very easily. The extra classes in the double group are given by Opechowski s rules. The character of if in these new classes in vector representations is the same as that of R but in spinor representations x(ff) = — (R). The bases of spinor representations will be described in Section 12.8. 

Since a representation—reducible or irreducible—is a set of matrices corresponding to all symmetry operations of a group, the representation can be described by the set of characters of all these matrices. For the simple basis of A/t and Ar2 used before for the HNNH molecule in the C2h point group, the representation consisted of four 2x2 matrices ... 

We list in Table 8.45 the additional characters of the double group D3h and in Table 8.46 the y, M) basis functions for the double group irreducible representations of D3h-... 

The six ligand donor orbitals collectively form a reducible representation E in the point group. This representation can be reduced by the method described in Section 4-4-2 applied to the character table in Table 10-4. This results in r - Aig + Tiu + Eg, shown in the last rows of the table. 

After obtaining the symmetry point group of the metal-adsorbate system, one can determine which symmetry representations of the vibrations of the molecule will be Raman active using group character tables for the given point group. Raman-active modes are those that possess nonzero components of the Raman polarizability tensor. 

In order to construct such sets of orbitals, it is most convenient to naake use of group theory. Each set of equivalent directed valence orbitals has a characteristic symmetry group. If the operations of this group are performed on the orbitals, a representation, which is usually reducible, is generated. By means of the character table of the group this representation, which we shall call the component irreducible representations. The s, p, and d orbitals of the atom also form representations of the group, and can also be divided into sets which form irreducible representations. ... 

D. E. Littlewood, The Theory of Group Characters and Matrix Representations of Groups, Oxford, New York and London, 1940. 

In general, it will be possible to simplify the total representation for a basis of our choosing into a sum of the standard irreducible representations from the point group character table. The reducible and irreducible representations are linked by the fact that the sum of characters of the irreducible representations for a basis must give the characters of their reducible representation ... 

Any collection of basis vectors that complies with the molecular symmetry can generate a character representation of the group, but in most cases it will be a reducible one and so can be simplified. In this section we will show that the simplification of a reducible representation r can be made using the data for the set of irreducible representations available in the standard character tables. 

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... 

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