By applying the symmetry operation σ to the integrand below, we obtain
The integral given in (2.22) is therefore nonvanishing only if n1 + n2 + n3 is an even integer. We have thus obtained a selection rule.
Equations (2.21) lead to a selection rule because Ψ1, Ψ2, and L all transform into some constant times themselves. The quantities Ψ1, Ψ2, and L might transform into some constant times Ψ3, Ψ4, and L′, say. When this occurs, we do not obtain a selection rule, but obtain rather a relation between two different matrix elements. For example, , , and of eq (1.1) are all invariant under the symmetry operation συ. Thus, by applying συ to both wave functions and to in the matrix element , and by using the transformation equations (2.11), we obtain the following equality
where the first and second factors, respectively, come from the transformation properties of the first and second wave functions. Just as in (2.11), it is often possible to obtain consistent results, when some of the quantum numbers in (2.23) do not have definite values, by arbitrarily assigning them values. These values must be used throughout all calculations, however; i.e., the values chosen here must agree with those chosen for eqs (2.11) and (2.16).
As mentioned above, is also invariant under συ. However, the individual operators occurring in can be shown to transform as follows [15,16]:
Transformation equations for L±, Lz, L2 and J±, Jz, J2 can be obtained by making the obvious substitutions everywhere in (2.24). By combining the transformation properties (2.24) with those of the wave functions given in (2.11), various matrix elements of the Hamiltonian can be related to each other.
We now show that the matrix elements of in the basis set used to label (1.18) are all equal. By applying the symmetry operation συ to the wave functions and operators on the left side of the two equalities below, we obtain
In addition, if we assume that the matrix elements of the orbital operator do not depend on the spin quantum numbers, then we see that
Equation (2.26) rests ultimately on the assumption that the nonrotating-molecule basis set functions can be written as the product |Λ〉 |SΣ〉 of an orbital function |&Lambda〉 and a spin function |SΣ〉, and on the further assumption that the same orbital function is associated with all 2S + 1 spin functions corresponding to given S. These assumptions will be valid to the extent that Λ, S, and Σ are good quantum numbers. As a consequence of (2.25) and (2.26), the four matrix elements of in the basis set used to label the matrix (1.18) are all equal. The equalities represented by (2.25) are exact; that represented by (2.26) is only approximate.