Representations, completely reduced irreducible

If each of the blocks in the matrices comprising the matrix system A cannot be reduced ftirther, the matrix system has been reduced completely and each of the matrix systems A1, A2, . .. in the direct sum is said to be irreducible. Matrix systems that are isomorphous to a group G are called matrix representations (Chapter 4). Irreducible representations (IRs) are of great importance in applications of group theory in physics and chemistry. A matrix representation in which the matrices are unitary matrices is called a unitary representation. Matrix representations are not necessarily unitary, but any representation of a finite group that consists of non-singular matrices is equivalent to a unitary representation, as will be demonstrated in Section A1.5. 

The submatrices are also representations, i.e., so-called irreducible representations. The completely reduced representation can be regarded as the direct sum (represented by the symbol Q)) of n, multiples of irreducible representations /, where u, is a positive number or zero ... 

K 0p V, where K is an arbir trary semisimple Lie algebra (real or complex), p an arbitrary linear representation (reducible or irreducible), dim V being greater than dim K, and in the decomposition of the representation p into irreducible components trivial summands are absent. Brailov, Pevtsova [106], [111]. All the functions of these complete involutive sets are linear. 

Pig. 5-10. Form of the completely reduced representation for the plane equilateral triangular model of the carbonate ion, CO3 , an example of the group Bsj. The symbols indicate the irreducible representation corresponding to each nonmixing block. 

No change of coordinates can reduce such a representation any further it is therefore called an irreducible representationd A completely reduced representation is evidently made up of a number of irreducible representations, each of its noncombining sets of coordinates forming the basis for one irreducible representation. 

One of these R = methods given in Chap. 6. By inserting these values of xr id (3), together with values of xk taken from Table 5-2, the result is obtained (as may readily be verified) that the reduced representation contains the irreducible representation A[ once, A 2 once, twice, E three times, and E" once (see Fig. 5-10, Sec. 5-7). But since it has already been stated that the normal coordinates form the basis of a completely reduced representation, there must be one normal coordinate which transforms like A[, one like A, three pairs like E, etc. Referring to Fig. 5-9 and Table 5-1, one sees that Qi is of symmetry A[, Q2 of symmetry A, and the pairs Q3, Qi, and Qr>, Qe are of symmetry E. The other six coordinates represent translation and rotation. 

If a transformation can be found which will put all the matrices of a given representation into this general form, the representation is said to be reducible. If no transformation can further diagonalize all submatrices such as c,/, and i in Eq. (3.16) then the set of matrices of a given representation is said to be completely reduced and the sets of submatrices are called the irreducible representations. The fact that the submatrices are also representations can be clarified by referring to Eq. (3.16). If the group multiplication table requires that AB = C then, following the rules for matrix multiplication, it is clear that ad = g be = h, and cf = i so that the set of submatrices a, d, and g, for example, form part of a representation which as described above cannot be further reduced. 

If this matrix characterizing C2 is compared with that obtained for the C2 operation in cartesian displacement coordinates [Eq. (3.10)], it can be seen that the character is — 1 as before but now there are no off-diagonal elements. We have only one-by-one matrices on the diagonal which obviously cannot be further reduced. Since the matrices for the other operations will be in this same form, the representation for the water molecule using normal coordinates as a basis is a completely reduced one. Each of the nine one-by-one matrices on the diagonal of the large matrix in Eq. (3.17) represents the C2 operation in one of the four possible irreducible representations illustrated in Fig. 3.7 and tabulated in Table 3.5. 

Notice here that the six>dimensionaI reducible representation is completely reduced to the direct sum of twice the one-dimensional irreducible representation and twice the two-dimensional irreducible representation. Or the number of times the one-dimensional irreducible representation appears in the reducible one is two and the two-dimensional irreducible representation appears twice. Concerning the character of the reducible representation, one can write that Xd(R) = 2xdi(R)+ 2xd2(R)i for every R of the point group C3.. Another result from group theory is that a necessary and sufficient condition for a representation to be irreducible is... 

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 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... 

For systems with high symmetry, in particular for atoms, symmetry properties can be used to reduce the matrix of the //-electron Hamiltonian to separate noninteracting blocks characterized by global symmetry quantum numbers. A particular method will be outlined here [263], to complete the discussion of basis-set expansions. A symmetry-adapted function is defined by 0 = 04>, where O is an Hermitian projection operator (O2 = O) that characterizes a particular irreducible representation of the symmetry group of the electronic Hamiltonian. Thus H commutes with O. This implies the turnover rule (0 > II 0 >) = (), which removes the projection operator from one side of the matrix element. Since the expansion of OT may run to many individual terms, this can greatly simplify formulas and computing algorithms. Matrix elements (0/x H ) simplify to (4 H v) or... 

R2. Another mle of the character table constraction prescribes that all the symmetry operations from a class have the same character in a representation (irreducible or reducible), a self-satisfied mle of table constmction for the group above. Next are following the rules for completing the characters table. 

Because we have assembled the reducible representations for complete sets of orbitals, the character totals obtained are independent of the choice of symmetry elements or operations from each class in the point group. We can now proceed to using the reduction formula to find the irreducible labels for p- and d-orbitals in O. For the p-orbitals, inspection of the standard character table from Appendix 12 shows that... 

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