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):
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(3.3) |
with the additional restriction (see below) that µz has no nonvanishing matrix elements between 0+ states and 0- states (all diatomic molecules), and that µx, µy, µz 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
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 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:
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(3.4) |
The transformation properties of µz 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
. 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.,
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(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 for given S. Since the dipole moment operator
is independent of electron spin, we find
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(3.6) |
where µi represents µx,
µy, or µz. The only nonvanishing
spin-allowed matrix elements of the molecule-fixed components of the dipole
moment operator in the nonrotating-molecule basis set thus have the form [7]
(pp. 59-64):
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(3.7) |
with the additional restriction that µz has no nonvanishing spin-allowed matrix elements between Σ+ states and Σ- states (all diatomic molecules), and that µx, µy, µz 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 µz or the components µx and µy. 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
have the form
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(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).
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