Direct product representations

We then dehne an internal coordinate <() such that <() = 0 2ti denotes a a path that has described one complete loop around the Cl in the nuclear branching space. Other than this, we need specify no further details about < ). We do not even need to specify whether the complete set of nuclear coordinates give a direct product representation of the space. It is sufficient that closed loop has wound around the CL Using this definition of ((), we can express the effect of the GP on the... 

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... 

This technique of decomposing a direct product representation will be of great use in the next section. 

To what irreducible representations can the following direct product representations be reduced for the specified point group ... 

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. 

Before showing further applications of direct-product representations to quantum mechanics, we quote without proof a theorem we will need. Let rij a and rkip be the elements of the matrices corresponding to the symmetry operation R in the two different nonequivalent irreducible representations Ta and T it can be shown that... 

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... 

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