Mathematical Methods for Physics and Engineering - Matematica.NET

29.11 PHYSICAL APPLICATIONS OF GROUP THEORYIt follows thatn λn µ∑χ prod [(X) = D prod (X) ] kkk=1n λ n µ∑ ∑ [= D (λ) (X) ] [ii D (µ) (X) ] jj=i=1 j=1{nλ∑ [D (λ) (X) ] iii=1}{ nµ∑ [D (µ) (X) ] }jjj=1= χ (λ) (X) χ (µ) (X). (29.23)This proves the theorem, and a similar argument leads to the corresponding resultfor integrands in the form of a product of three or more factors.An immediate corollary is that an integral whose integrand is the product oftwo functions transforming according to two different irreps is necessarily zero. Tosee this, we use (29.18) to determine whether irrep A 1 appears in the productcharacter set χ prod (X):m A1= 1 ∑ [χ(A 1) (X) ] ∗χ prod (X) = 1 ∑χ prod (X) = 1 ∑χ (λ) (X)χ (µ) (X).gg gXXXWe have used the fact that χ (A1) (X) =1forallX but now note that, by virtue of(29.14), the expression on the right of this equation is equal to zero unless λ = µ.Any complications due to non-real characters have been ignored – in practice,they are handled automatically as it is usually Ψ ∗ φ, rather than Ψφ, that appearsin integrands, though many functions are real in any case, and nearly all charactersare.Equation (29.23) is a general result for integrands but, specifically in the contextof chemical bonding, it implies that for the possibility of bonding to exist, thetwo quantum wavefunctions must transform according to the same irrep. This isdiscussed further in the next section.29.11 Physical applications of group theoryAs we indicated at the start of chapter 28 and discussed in a little more detail atthe beginning of the present chapter, some physical systems possess symmetriesthat allow the results of the present chapter to be used in their analysis. Weconsider now some of the more common sorts of problem in which these resultsfind ready application.1105