## 2.4. Example: Parities of the Rotational Levels in a
^{1}**Σ**^{-} State

The complete basis set functions for a
^{1}Σ^{-} state can be written as .
They transform as follows under σ_{υ}(*xz*):

Thus, the rotational levels of even *J* are of odd parity (-), while those
of odd *J* are of even parity (+).

## 2.5. Example: Parities of the Rotational Levels of
^{3}Σ^{+} State

The rotational energy levels of a ^{3}Σ state were calculated in
sect. 1.10. The three basis set functions
used to label the matrix (1.27) transform as
follows when the ^{3}Σ state is a ^{3}Σ^{+}
state.
Consequently the energy levels obtained from the upper
2 × 2 block of
(1.27) are of odd parity when *J* is
even and of even parity when *J* is odd. The energy levels obtained from
the lower right-hand corner of (1.27) are of even parity when *J* is even
and of odd parity when *J* is odd. Since
*J* = *N* ± 1 for the former wave functions and
*J* = *N* for the latter, rotational levels of a
^{3}Σ^{+} state of even *N* are of even parity;
those of odd *N* are of odd parity.

The wave functions in (2.14) all transform into themselves or into their
negatives under the operation σ_{υ}. This is not true in general of the functions
of the original basis set. For this reason, the ^{2}Π
wave functions used to label the matrix (1.18)
cannot be assigned a parity, although appropriate sums and differences of such
wave functions could be.