As a further aid in understanding the geometric symmetry operation συ, let us consider its effect on several electronic wave functions. In particular, let us consider diatomic-molecule wave functions derived from atomic wave functions. The one-electron atomic configuration np gives rise to an orbital P state (L = 1), which in turn gives rise to diatomic-molecule orbital Σ(Λ = 0) and Π(Λ = ±1) states. The wave functions for these diatomic-molecule states are
if we use the phase conventions of Condon and Shortley  (p. 52). These functions transform as follows under the operation συ(xz) (see table 3)
which can be written as
Equations (2.4) and (2.5) can be
used to illustrate a fundamental point concerning the relationship between the
choice of phase factors for a set of wave functions and their transformation
properties under symmetry operations. It can be seen that the first of the
transformation equations (2.5) is unaltered if |p Σ〉 in
(2.4) is defined to be
It turns out that all Σ electronic wave functions are characterized by an intrinsic transformation property under the operation συ. Furthermore, this transformation property is customarily indicated by a superscript attached to the Σ label; i.e., we define Σ+ and Σ- states, such that
It further turns out that all doubly-degenerate orbital electronic states,
i.e., those with |Λ| > 0, are not characterized by
an intrinsic transformation property under the operation
Consider now the two-election atomic configuration np n′p. This configuration also gives rise to an orbital P state, which in turn gives rise to diatomic-molecule orbital Σ and Π states. The wave functions for these diatomic-molecule states are
if we use the phase conventions of Condon and Shortley
 (p. 76). These wave functions transform as
follows under the symmetry operation
which can be written as
We note in passing that the Σ state of (2.4) is a Σ+ state, while that of (2.8) is a Σ- state.
It turns out that electronic orbital wave functions |LΛ〉 having phase factors consistent with those of Condon and Shortley , and arising from atomic states of even parity, all transform according to (2.10), while wave functions having such phase factors, and arising from atomic states of odd parity, all transform according to (2.6). (The parity of an atomic state is determined by its behavior under the inversion operation i in table 3.)
There are several complications which arise in deciding whether to use (2.6) or (2.10). First, wave functions of the type |LΛ〉 are most often used in discussing Rydberg states. Under these circumstances, many of the electrons in the molecule are assigned to the "core" and are not considered explicitly. Since the core plays the role of the atomic nucleus, one must consider the parity of the atomic state which corresponds to the diatomic-molecule electronic wave function involving only electrons outside the core. Second, L is often not a good quantum number, so that no particular value of L suggests itself for use in (2.6) or (2.10). Under these circumstances one can often obtain consistent and correct results for states with |Λ| > 0 by arbitrarily giving L some value and arbitrarily choosing one of the two relations (2.6) or (2.10) to represent the transformation properties. This apparently casual choice of signs actually causes no difficulty. Of course, it does require a particular phase choice for the wave functions, which must be consistent with the phase choice implicit in any matrix element expressions used. However, when L is not a good quantum number, matrix elements involving the electronic orbital part of the wave function are not evaluated explicitly (they are left as adjustable parameters). Hence, contradictions between matrix elements and transformation properties are not introduced (see sect. 2.8). Finally, the transformation properties of Σ± states are determined by eq (2.7). Sometimes, however, it is convenient to incorporate a Σ state into a transformation scheme utilizing eq (2.6) or (2.10). It is then necessary to correlate the choice of sign and the choice of L to obtain the proper Σ state transformation properties.
Since the transformation properties of the electronic spin basis set functions |SΣ〉 and of the rotational basis set functions |JΩ〉 are more difficult to illustrate with simple examples than are the transformation properties of the electronic basis set functions |LΛ〉, we shall merely state the final results here: The functions |SΣ〉 and |JΩ〉, when chosen to have phase factors consistent with those of Condon and Shortley , transform like functions |LΛ〉 of even parity [l5,16]. We can thus summarize the effect of the operation συ on the various basis set functions described in chapter 1 by the following equations:
where (2.11c) could be written more precisely as
It was pointed out in sect. 2.2 that when the symmetry operation συ(xz) 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 all the particles in the molecule are replaced by their negatives. States of definite parity transform into themselves or into their negatives under this operation. It can easily be seen by application of (2.12) that functions of the form ± have a definite parity. We next consider two examples of the determination of the parity of rotational energy levels.