Molecule-fixed components of ** µ** come into consideration
when matrix elements of the dipole moment operator are to be evaluated, because
molecular wave functions are expressed in terms of molecule-fixed coordinates
and not in terms of laboratory-fixed coordinates. We shall not consider in this
monograph the numerical evaluation of matrix elements of the molecule-fixed
components of

The molecule-fixed components of ** µ** do not involve the rotational
variables. Consequently, matrix elements of these quantities are diagonal in
the rotational quantum numbers

(3.3) |

with the additional restriction (see below) that *µ _{z}* has
no nonvanishing matrix elements between 0

The two wave functions for the nonrotating molecule represented by
**|**Ω⟩ in the third matrix element of (3.3) may correspond to
the *same* state of the molecule or to *two different* states. If
these two wave functions **|**Ω⟩ correspond to the same
vibrational-electronic state of the molecule, then the matrix element
governs the
intensity of pure rotational transitions (which are forbidden in homonuclear
molecules, of course). If the two wave functions **|**Ω⟩
correspond to different vibrational states of the same electronic state, then
the matrix element governs the intensity of a pure
vibrational transition (which is also forbidden in homonuclear molecules). If
the two wave functions **|**Ω⟩ correspond to different electronic
states of the molecule, then the matrix element governs the intensity of an
electronic transition.

Each of the matrix elements in (3.3) can be related to
another matrix element by applying the symmetry operation
σ_{υ} to the two
wave functions and the dipole moment operator involved in the integral. The
transformation properties of the wave functions have been discussed in
chapter 2. The transformation properties of
the molecule-fixed components of the dipole moment operator
[15,16] can be obtained immediately from
table 3:

(3.4) |

The transformation properties of *µ _{z}* under
σ

Consider now the nonvanishing matrix elements of the molecule-fixed components of the dipole moment operator in the nonrotating-molecule basis set . These matrix elements can be classified as spin-allowed or spin-forbidden. Spin-allowed matrix elements can be obtained by considering the functions to be the product of an orbital part and a spin part, i.e.,

(3.5) |

where the quantum numbers Λ, *S*, and Σ are all perfectly
good, and where the same orbital function |Λ⟩ is associated with
all 2*S* + 1 spin functions for given *S*. Since the dipole moment operator
is independent of electron spin, we find

(3.6) |

where *µ _{i}* represents

(3.7) |

with the additional restriction that *µ _{z}* has no
nonvanishing spin-allowed matrix elements between Σ

The matrix elements in (3.7) can be related to other matrix elements in two
ways: first, by assuming that their value is independent of the spin projection
quantum number Σ [which follows from the factorization
(3.5)]; and second, by applying the symmetry operation
σ_{υ} to the two
wave functions and the dipole moment operator involved in the integral (see
sect. 2.10).

In spin-allowed transitions, the intensity of the transition comes from matrix
elements involving *either* the dipole moment component
*µ _{z}*

Allowed matrix elements of the dipole moment operator in the nonrotating molecule basis set have the form

(3.8) |

where the transition *L*′ ↔ *L* must be allowed
in the united atom limit. Forbidden transitions in this basis set are those
forbidden by (3.8), but allowed by (3.7) or
(3.3).