Representation of the 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... 

Thus, the matrices will have a multiplication table with the same structure as the multiplication table of the synnnetry group and hence will fonn an /-dimensional representation of the group. 

In faet, one finds that the six matriees, Df4)(R), when multiplied together in all 36 possible ways obey the same multiplieation table as did the six symmetry operations. We say the matriees form a representation of the group beeause the matriees have all the properties of the group. 

These six matrices form another representation of the group. In this basis, each character is equal to unity. The representation formed by allowing the six symmetry operations to act on the Is N-atom orbital is clearly not the same as that formed when the same six operations acted on the (8]s[,S 1,82,83) basis. We now need to learn how to further analyze the information content of a specific representation of the group formed when the symmetry operations act on any specific set of objects. 

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. 

Another one-dimensional representation of the group ean be obtained by taking rotation about the Z-axis (the C3 axis) as the objeet on whieh the symmetry operations aet ... 

These one-dimensional matriees ean be shown to multiply together just like the symmetry operations of the C3V group. They form an irredueible representation of the group (beeause it is one-dimensional, it ean not be further redueed). Note that this one-dimensional representation is not identieal to that found above for the Is N-atom orbital, or the Ti funetion. 

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. 

Each element of K effects an interchange of the s points and thus a permutation of the configurations (0j,. .., 0 ) of content These permutations form a representation of the group... 

To make these notions precise, the transformation properties of the wavefunction x under spatial and time translations as well as under spatial rotations and pure Lorentz transformations must be specified and it must be shown that the generators of these transformations form a unitary representation of the group of translations and proper Lorentz transformations. This can in fact be shown5 but will not be here. 

It is important to distinguish between mmetiy properties of wave functions on one hand and those of density matrices and densities on the other. The symmetry properties of wave functions are derived from those of the Hamiltonian. The "normal" situation is that the Hamiltonian commutes with a set of symmetry operations which form a group. The eigenfunctions of that Hamiltonian must then transform according to the irreducible representations of the group. Approximate wave functions with the same symmetry properties can be constructed, and they make it possible to simplify the calculations. 

Thus, the values of the operations all being +1 satisfies the laws of the C2 group. This set of four numbers (all +1) provides one representation of the group. Another is given by the relationships... 

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. 

Fig. 23. Schematic representation of the group IV donor-hydrogen complex with hydrogen in AB site. The black spheres represent the group V atoms (As), the large white ones the group III atoms (Ga), the small white one the hydrogen atom and the dotted sphere the impurity. The lone pair on the threefold coordinated group V atom is not represented.

Fig. 27. Schematic representation of the group III vacancy having one of its dangling bonds saturated by hydrogen. The black spheres represent the group V atoms and the white one the hydrogen atom. 

In an n-dimensional space L, the linear operators of the representation can be described by their matrix representatives. This procedure produces a homomorphic mapping of the group G on a group of n x n matrices D(G), i.e., a matrix representation of the group G. From equations (6) it follows that the matrices are non-singular, and that... 

By changing the basis in the n-dimensional space L, the matrices D(R) will be replaced by their transforms by some matrix C. The matrices D (R) = CD R)C l also provide a representation of the group G, which is equivalent to the representation D(R). It should be clear that equivalent representations have the same structure, even though the matrices look different. What is needed to avoid any possible ambiguity are appropriate aspects of D(R) which remain invariant under a change of coordinate axes. One such invariant is easily defined in terms of the diagonal elements of the matrix, as... 

Consider the vector space Ln which is used to generate a representation of the group G. For every element A of G and every vector L, A(f> also belongs to Ln. The vector space is said to be invariant under the transformations of G. 

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 transformations of a cartesian coordinate system (x, y) in the plane of the circle can be used to generate a representation of the group. The... 

As remarked above (cf. Eq. (9)), a representation of the group leads to a representation of the algebra in particular, each of the projection operators will be associated with a matrix. If the representation is the irreducible one I M, we find, using (9) and (2),... 

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. 

In a crystal lattice where each atom contributes one atomic orbital, and where these orbitals are related to each other by the translations characteristic of the lattice, the molecular orbitals must belong to irreducible representations of the group of these translations and hence form so-called Bloch orbitals. 46)... 

This set of representations is usually known as a representation of the group. Obviously, if we choose anotiier space of basis functions., anotiier representation of the group can be constructed, and so an infinite number of representations is possible for a given symmetry group. 

If we wanted to generate the representation matrices of 4, we would find the three elementary transpositions and the identity and then generate the other 20 matrices. On the other hand, we could find the representations of the group S2 in 54. This consists of two matrices, U[ E)] and C/[(12)]. We could then use C/[(23)] and f/[(13)] to generate all of the six elements in S3. We could then use C/[(34)j, f/[(14)], and f/[(24)] to generate the rest of 54. While this may seem at first more time-consuming, it is much more easily automated than the brute force approach. 

To an energy eigenvalue or state of 3C, there will generally correspond several independent eigenvectors or state functions i,. . . , n is the degeneracy of the state. These functions must form a basis for a representation of the group G if is invariant under G. If i2 is an element of G... 

Under the action of the electric field of the 0 environment, the state of the ( ) 3d configuration will split up into two states. The orbital degeneracy of a D state is 2 X 2 -f- 1 = 5. From the above discussion each of the resulting states must belong to one of the irreducible representations of Oh given in Table I. The state W corresponds to an irreducible representation of the group of symmetry operations of a sphere, i.e., the full rotation group R(3). 

We are now ready for the main conclusion of this chapter If all elements of a symmetry group are represented by orthogonal matrices in a consistent coordinate system, the matrices will form a group under the operation of matrix multiplication that is isomorphic to the symmetry group. The set of matrices is said to be a representation of the 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. 

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