2.6. The Symmetry Operation i

The geometric symmetry operation i exists only if the diatomic molecule is homonuclear. Its effect on the various basis set wave functions of chapter 1 is quite simple [15,16]. The operation i leaves the electronic spin functions |SΣ⟩ and the rotational wave functions |JΩ⟩ invariant. The behavior of the electronic orbital wave functions under i is indicated by the subscripts g and u. For example,

 (2.15)

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).

2.7. The Symmetry Operation C2

The symmetry operation C2 also exists only if the diatomic molecule is homonuclear. Its effect on the basis set functions of chapter 1 can be determined most easily by noting that C2(y) ≡ συ(xz) ·  i. The effects of συ(xz) are given in eqs (2.11). The effects of i are given in sect. 2.6. We thus conclude that the effect of C2(y) on the various basis set functions is given by the following equations:

 (2.16a)

 (2.16b)

 (2.16c)

where (2-16c) could be written more precisely as C2 = (-1)J . If L is not a good quantum number, it is again often possible to obtain consistent answers for the symmetry properties by arbitrarily assigning L some value. (The same value of L must be used in (2-16) as in (2.11), of course.) When Λ = 0, the transformation properties of the orbital part of the electronic wave function are completely determined by the state symbol:

 (2.17)

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.

2.8. Example: Symmetry Properties of the Rotational Levels in a 1Πu State

The complete basis set functions for a 1Πu state can be written as sums and differences of functions of the type :

 (2.18)

Making use of (2.11) with an arbitrarily chosen value of L = 2, we find

 (2.19)

Making use of (2.16) with the same arbitrarily chosen value of L = 2, we find

 (2.20)

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 | ± 1u 0 0; ± 1 J M ⟩. In the absence of further computations with these wave functions, neither phase choice introduces contradictions, and no further thought on the matter is necessary.