Mathematical Methods for Physics and Engineering : A Comprehensive Guide

29.11 PHYSICAL APPLICATIONS OF GROUP THEORY

It follows that

χprod(X)=

∑nλnμ

k=1

[

Dprod(X)

]

kk

=

∑nλ

i=1

∑nμ

j=1

[

D(λ)(X)

]

ii

[

D(μ)(X)

]

jj

=

{n

∑λ

i=1

[

D(λ)(X)

]

ii

}{nμ

∑

j=1

[

D(μ)(X)

]

jj

}

=χ(λ)(X)χ(μ)(X). (29.23)

This proves the theorem, and a similar argument leads to the corresponding result

for integrands in the form of a product of three or more factors.

An immediate corollary is thatan integral whose integrand is the product of

two functions transforming according to two different irreps is necessarily zero.To

see this, we use (29.18) to determine whether irrep A 1 appears in the product

character setχprod(X):

mA 1 =

1

g

∑

X

[

χ(A^1 )(X)

]∗

χprod(X)=

1

g

∑

X

χprod(X)=

1

g

∑

X

χ(λ)(X)χ(μ)(X).

We have used the fact thatχ(A^1 )(X)=1forallXbut now note that, by virtue of

(29.14), the expression on the right of this equation is equal to zero unlessλ=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 appears

in integrands, though many functions are real in any case, and nearly all characters

are.

Equation (29.23) is a general result for integrands but, specifically in the context

of chemical bonding, it implies that for the possibility of bonding to exist, the

two quantum wavefunctions must transform according to the same irrep. This is

discussed further in the next section.

29.11 Physical applications of group theory

As we indicated at the start of chapter 28 and discussed in a little more detail at

the beginning of the present chapter, some physical systems possess symmetries

that allow the results of the present chapter to be used in their analysis. We

consider now some of the more common sorts of problem in which these results

find ready application.