As a further aid in understanding the geometric symmetry operation
σ_{υ}, 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

(2.4) |

if we use the phase conventions of Condon and Shortley
[7] (p. 52). These functions transform as follows
under the operation σ_{υ}(*xz*) (see
table 3)

(2.5) |

which can be written as

(2.6) |

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
*f* (ρ_{e} ) cosθ_{e}*f* (ρ_{e} ) cosθ_{e}. Thus
the transformation property represented by this first equation is independent
of the choice of phases, and is an intrinsic property of the Σ state
under consideration. On the other hand, the - sign on the right-hand side
of the second of the transformation equations (2.5) can be changed to a
+ sign, simply by defining the functions
| *p* Π_{±} ⟩ in (2.4) to be
*f* (ρ_{e} ) 2^{-l/2}
sinθ_{e}

It turns out that all Σ electronic wave functions are characterized by an
intrinsic transformation property under the operation σ_{υ}. 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

(2.7) |

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 _{υ}.

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

(2.8) |

if we use the phase conventions of Condon and Shortley
[7] (p. 76). These wave functions transform as
follows under the symmetry operation _{υ} (*xz*)

(2.9) |

which can be written as

(2.10) |

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 |*L*Λ⟩
having phase factors consistent with those of Condon and Shortley
[7], 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 |*L*Λ⟩ 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
|*S*Σ⟩ and of the rotational basis set functions
|*J*Ω⟩ are more difficult to illustrate with simple examples
than are the transformation properties of the electronic basis set functions
|*L*Λ⟩, we shall merely state the final results here: The
functions |*S*Σ⟩ and |*J*Ω⟩, when chosen to
have phase factors consistent with those of Condon and Shortley
[7], transform like functions |*L*Λ⟩
of even parity [l5,16]. We can thus summarize the
effect of the operation σ_{υ} on the various basis set functions described in
chapter 1 by the following equations:

(2.11a) |

(2.11b) |

(2.11c) |

where (2.11c) could be written more precisely as _{υ} =^{J-Ω} ,*g* and *u*
diatomic-molecule electronic states correlate with united atom states of even
and odd parity, respectively.) The transformation properties given in (2.11)
are, of course, consistent with the matrix elements given in
(1.13) and
(1.14). The
transformation properties of the complete basis set functions are seen from
eqs (2.11) to be

(2.12) |

It was pointed out in sect. 2.2 that when
the symmetry operation σ_{υ}(*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.