NBS Monograph 115: 3.2 Molecule-Fixed Components of mu

3.2. Molecule-Fixed Components of μ

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 µ, since such calculations require a knowledge of many-electron molecular wave functions. Instead, we shall treat these matrix elements as parameters, which must be determined from a fit of the calculated rotational intensity expressions to the experimental data. It is still necessary, however, to determine precisely how many such intensity parameters can occur in the rotational intensity expressions for a given electronic transition. Hence, it is necessary to investigate the circumstances under which matrix elements of the molecule-fixed components of µ vanish.

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 J and M, and we need only consider further matrix elements of µ in the various nonrotating-molecule basis sets. The only nonvanishing matrix elements of the molecule-fixed components of µ in the nonrotating-molecule basis set |Ω⟩ have the following form [7] (pp. 59-64):

eq3.03 (3.3)

with the additional restriction (see below) that µ_z has no nonvanishing matrix elements between 0⁺ states and 0^- states (all diatomic molecules), and that µ_x, µ_y, µ_z have no nonvanishing matrix elements between g and g or between u and u electronic states (homonuclear diatomic molecules).

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 $\langle\Omega| \mu_z |\Omega\rangle$ 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 $\langle\Omega| \mu_z |\Omega\rangle$ 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 $\langle\Omega| \mu_z |\Omega\rangle$ 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 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:

eq3.04 (3.4)

The transformation properties of µ_z under σ lead (see sect. 2.9) to the selection rule 0^± ↔ 0^± or Σ^± ↔ Σ^±. The transformation properties of µ under i lead (see sect. 2.9) to the selection rule g ↔ u, u ↔ g.

Consider now the nonvanishing matrix elements of the molecule-fixed components of the dipole moment operator in the nonrotating-molecule basis set $|\Lambda S\Sigma\rangle$ . These matrix elements can be classified as spin-allowed or spin-forbidden. Spin-allowed matrix elements can be obtained by considering the functions $|\Lambda S\Sigma\rangle$ 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 2S + 1 spin functions $|S\Sigma\rangle$ for given S. Since the dipole moment operator is independent of electron spin, we find

(3.6)

where µ_i represents µ_x, µ_y, or µ_z. The only nonvanishing spin-allowed matrix elements of the molecule-fixed components of the dipole moment operator in the nonrotating-molecule basis set $|\Lambda S\Sigma\rangle$ thus have the form [7] (pp. 59-64):

eq3.07 (3.7)

with the additional restriction that µ_z has no nonvanishing spin-allowed matrix elements between Σ⁺ states and Σ^- states (all diatomic molecules), and that µ_x, µ_y, µ_z have no nonvanishing matrix elements between g and g or between u and u electronic states (homonuclear diatomic molecules). Spin-forbidden matrix elements of the dipole moment operator are those forbidden by (3.6) and (3.7), but allowed by (3.3).

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 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 or the components µ_x and µ_y. The former transitions are called parallel transitions because the nonvanishing matrix elements involve the component of the dipole moment parallel to the internuclear axis. The latter transitions are called perpendicular transitions because the nonvanishing matrix elements involve the components of the dipole moment perpendicular to the internuclear axis.

Allowed matrix elements of the dipole moment operator in the nonrotating molecule basis set $|L\Lambda S\Sigma\rangle$ have the form

eq3.08 (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).