It is important to recall that the molecule-fixed inversion operation i is not equivalent to the laboratory-fixed inversion operation I (see sect. 2.2).
where (2-16c) could be written more precisely as
The transformation properties of the complete wave functions can be determined from the transformation properties of the individual parts as in eq (2.12) above.
It was pointed out in sect. 2.2 that when the symmetry operation C2(y) acts on a complete basis set function (corresponding to both the nonrotational and the rotational part of the problem), then its net effect is equivalent to that obtained when the laboratory-fixed coordinates of the two identical nuclei are permuted. Rotational states characterized as s transform into themselves under this operation; states characterized as a transform into their negatives. We next consider an example of the determination of the parity and of the s, a character of rotational energy levels.
Making use of (2.11) with an arbitrarily chosen value of L = 2, we find
Making use of (2.16) with the same arbitrarily chosen value of L = 2, we find
We therefore conclude that the sum function in (2.18) is - s for even J and + a for odd J, and that the difference function is + a for even J and - s for odd J.
Had we arbitrarily chosen a value of L = 1 in making use of
(2.11) and (2.16),
we would have concluded that the sum function was + a for
even J and - s for odd J, and that the
difference function was - s for even J and
+ a for odd J. For both values of L, we thus
conclude that the rotational levels occur in pairs for given J, one
member of the pair being + a, the other being - s.
Whether the sum or difference function in (2.18) is + a for given
J actually depends on our choice of phases for the two functions