The rotational energy levels of a ^{3}Σ state have been discussed
in sect. 1.10; the symmetry properties of the
rotational levels of a ^{3}Σ^{+} state have been
discussed in sect. 2.5. It is easy to
show, by arguments analogous to those of sect. 2.5, that the parities of
the rotational levels of a ^{3}Σ^{-} state are just the
opposite of those of a ^{3}Σ^{+} state, i.e., states of
even *N* are of odd parity and states of odd *N* are of even
parity.

The wave functions for the rotational levels of a ^{3}Σ state
were not determined in sect. 1.10. These wave
functions can be determined, however, by finding the eigenvectors of the sum of
the matrices given in (1.24) and
(1.27). We consider a
^{3}Σ^{-} state which is very near case (b); for the
purposes of calculating intensities, we thus set λ = 0. The
three normalized eigenfunctions of given *J* then become:

(3.22a) |

(3.22b) |

(3.22c) |

These three functions are eigenfunctions of the matrix sum
(1.24) plus
(1.27) when λ = 0, and
belong to the eigenvalues *E* + *B* ⟨⟩ + *BN*(*N* + 1), where
*N* = *J* + 1, *N* = *J*, and
*N* = *J*-1, respectively.

We must next calculate all matrix elements of *µ _{Z}* allowed
by the selection rules on

(3.23) |

Furthermore, by applying the symmetry operation
σ_{υ} (see
sect. 2.9), we find that the first and second
terms in (3.23) are equal.

The next step is to replace *µ _{Z}* by the right-hand side of
(3.12). Before doing this, however, we
examine the matrix elements of the molecule-fixed components of the dipole
moment operator in the nonrotating-molecule basis set under consideration. The
nonrotating-molecule basis set for the

(3.24) |

The one remaining point to check involves the matrix element between the two
states having Ω = 0. Selection rules require that all matrix
elements of *µ _{z}* vanish between 0

(3.25) |

Consequently, both the ^{1}Σ^{+} and the
^{3}Σ^{-} states give rise in the strong spin-orbit
coupling limit to 0^{+} states, and the second matrix element of (3.24)
is allowed by symmetry.

Spectroscopists sometimes speak of a doubly forbidden transition. Such a label
is useful, if it is employed carefully. The degree of multiple forbiddenness is
best defined to be the number of first-order perturbations which must be
carried out in succession before a given transition is made allowed. Thus, in
the particular case of a
^{3}Σ^{-} - ^{1}Σ^{+}
transition, a single first-order spin-orbit perturbation (satisfying the
selection rules Δ*S* = 0, ±1;
ΔΩ = 0) suffices to make the transition allowed (e.g.,
the mixing of ^{3}Σ^{-} and ^{1}Π), so that
this transition is only singly forbidden. On the other hand, a
^{5}Σ^{-} - ^{1}Σ^{+}
transition is made allowed only after two successive first-order spin-orbit
perturbations, and it is therefore doubly forbidden.

Taking into account the fact that the first and second terms in
(3.23) are identical, the fact that the only nonvanishing
matrix elements of the molecule-fixed components of the dipole moment operator
in the basis set under consideration are given in (3.24),
and the fact that *µ _{Z}* can be expanded as given in
(3.12), we can rewrite (3.23) in the form

(3.26) |

For simplicity we define two quantities *µ*_{||} and
*µ*_{⊥}

(3.27) |

which can both be made real as follows. Since the two wave functions
**|**0^{-} 1 0⟩ and
**|**0^{+} 0 0⟩ both have only zero values for the
angular momentum projection quantum numbers, their phases can be chosen such
that they transform into themselves under the time inversion operation θ
(see sect. 2.11)

(3.28) |

Applying the time inversion operation to all quantities in the first equation of (3.27) we obtain

(3.29) |

Clearly, the quantity *µ*_{||} is real under these conditions.
Applying the time inversion operation in a similar manner to the second
equation in (3.27), and using transformation properties
for the wave function **|**0^{-} 1 1&rang obtained from
eq (2.38), we find

(3.30) |

Applying the symmetry operation σ_{υ} to the matrix element on the righthand side of (3.30)
allows one to conclude that

(3.31) |

Thus, the quantity *µ*_{⊥} is also real. (Note that the time
inversion operation θ was used together with the reflection operation
σ_{υ} in
demonstrating that *µ*_{⊥} is real. The use of both
θ and σ_{υ}
will generally be necessary when the matrix elements under consideration
involve functions with nonzero values for angular momentum projection quantum
numbers.)

If we now substitute from table 6 and
eqs (3.27) in (3.26), we
obtain for this matrix element of *µ _{Z}*

(3.32) |

where Ω has been given its value of
zero. The intensity is proportional to the square of this quantity summed over
*M*. Thus,

(3.33) |

In a similar fashion.

(3.34) |

*µ*_{0} (his
notation) = +*µ*_{||} (this notation), but that
*µ*_{1}*µ*_{⊥}
(this notation).