(2.21) |

By applying the symmetry operation σ to the integrand below, we obtain

(2.22) |

The integral given in (2.22) is therefore nonvanishing only if
*n*_{1} + *n*_{2} + *n*_{3} is
an even integer. We have thus obtained a selection rule.

Equations (2.21) lead to a selection rule because Ψ_{1},
Ψ_{2}, and *L* all transform into some constant times
themselves. The quantities Ψ_{1}, Ψ_{2}, and *L*
might transform into some constant times Ψ_{3}, Ψ_{4},
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 σ_{υ}. Thus, by applying
σ_{υ} to both
wave functions and to in the matrix element , and by using the transformation equations
(2.11), we obtain the following equality

(2.23) |

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 σ_{υ}. However, the individual
operators occurring in can be shown to transform as follows
[15,16]:

(2.24) |

Transformation equations for *L*_{±}, *L _{z}*,

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 σ_{υ} to the wave functions and operators on the left side of
the two equalities below, we obtain

(2.25) |

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

(2.26) |

Equation (2.26) rests ultimately on the assumption that the
nonrotating-molecule basis set functions can be written as the
product |Λ⟩ |*S*Σ⟩ of an orbital function
|&Lambda⟩ and a spin function |*S*Σ⟩, and on the further
assumption that the same orbital function is associated with all
2*S* + 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.