## 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 |*S*Σ⟩ and the rotational wave functions
|*J*Ω⟩ invariant. The behavior of the electronic orbital wave
functions under *i* is indicated by the subscripts *g* and *u*.
For example,

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 *C*_{2}

The symmetry operation *C*_{2} 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 *C*_{2}(*y*) ≡ σ_{υ}(*xz*) ·
*i*. The effects of σ_{υ}(*xz*) are given
in eqs (2.11). The effects of *i*
are given in sect. 2.6. We thus conclude that
the effect of *C*_{2}(*y*) on the various basis set functions
is given by the following equations:

where (2-16c) could be written more precisely as *C*_{2}
=
(-1)^{J-Ω} . 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:

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 *C*_{2}(*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
^{1}Π_{u} State

The complete basis set functions for a ^{1}Π_{u}
state can be written as sums and differences of functions of the type
:

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
| ± 1_{u} 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.