The Direct-Product Representation

The direct product representation is usually reducible, unless both component representations are one-dimensional. For instance, in a group such as Dsh, in which no irreducible representation has dimension higher than two, the direct product of Ei and E2 will be four-dimensional, and thus it must be reducible. 

Problem 10-5. In a homonuclear diatomic molecule, taking the molecular axis as z, the pair of LCAO-MO s tpi = 2p A + PxB and tp2 = 2 PyA + 2 PyB forms a basis for a degenerate irreducible representation of D h, as does the pair 3 = 2pxA PxB and 4 = PxA — PxB Identify the symmetry species of these wave functions. Write down the four-by-four matrices for the direct product representation by examining the effect of the group elements on the products 0i 03, 0i 04, V 2 03) and 02 04- Verify that the characters of the direct product representation are the products of the characters of the individual representations. 

In order to apply the direct product representation to the derivation of selection rules, recognize that a matrix element of the form ipi, O lpj) will be equal to zero for symmetry reasons if there is even one symmetry operation that takes the integrand into its negative. The argument follows exactly the course of that of section 10.2. Thus the matrix element will vanish unless the direct product representation is totally symmetric (Ai), or contains A upon reduction. 

Simplification of secular equations. Because the Hamiltonian is totally symmetric - that is, for a molecule of C2v symmetry such as H2O, of symmetry species Ai - the matrix elements Hij = ipi, Ti. ipj) as well as the overlap integrals Sij = (tpi, ipj) will be equal to zero unless the direct product representation r. contains Ai. This is the basis for the assertion that states of different symmetry do not mix. ... 

In general, the direct product representations are reducible and using the formulae of 7-4 we have, if T are irreducible representations... 

If we consider in the direct product representation rH P then since Hitf belong to P, rH P = P and therefore TH — P. Hence, any operator which commutes with all 0M of a point group can be said to belong to the totally symmetric irreducible representation P. 

This is the desired result The character of any symmetry operation in the direct-product representation TC is the product of its characters in the representations TF and Tc. (The direct product of matrices is not, in general, commutative however, A<8>B and B A have equal traces, and thus the corresponding direct-product representations are equivalent to each other.)... 

Let us look at some examples. For S2v, (9.139) and the character table give as the characters of the direct-product representation A2 Bt... 

In order to show how this is done let us take a simple example. For the ion [Co(NH3)6]3+, the ground state y/e transforms as lAlg. There are two excited states with the same spin (S = 0), which belong to the representations Tlg and T2g. In the group Oh the coordinates x, v, z jointly form a basis for the T]u representation. Thus, for the [Alg — lTlg transition the direct product representation of y/ e(x, y, z)y/e is given by... 

This will be true when the direct product representation for the ith and yth normal modes is or contains the irreducible representation for which the kth normal mode forms a basis. These two excited states of the same symmetry will then interact in a way which can be represented by the usual sort of secular equation, namely,... 

We can see immediately that the direct product representation A2 <8> E is the irrep E (direct products with A] are of course trivial). The direct product E(g)E gives the characters (4, 1, 0), a reducible representation that can be reduced to Ax A2 2 . Basis functions for the irreps are also given in the table, as well as the behaviour of (Rs, Ry, Rz), corresponding to rotations about the three coordinate axes. 

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