## 3.1. Laboratory-Fixed Components of the Electric Dipole Moment Operator
μ

The intensity of an optical transition between two states often depends on the
polarization of the light passing through the sample in an absorption
experiment, or on the polarization of the light being detected in an emission
experiment. This phenomenon is reflected in the theory as follows.
Theoretical calculations for experiments involving plane polarized light with
the electric vector of the light in the laboratory-fixed *Z* direction
must be performed with the laboratory-fixed *Z* component of the dipole
moment operator [7] (pp. 90-93, 97-100), i.e.,
the intensity in such an experiment for a transition between an initial state
*i* and a final state *f* is proportional to *µ*_{Z}
. Clearly, by
analogy, if the light is plane polarized with the electric vector in the
*X* or *Y* direction, then the theoretical calculations must be
performed with the laboratory-fixed *X* or *Y* component of the
dipole moment operator, respectively.

Theoretical calculations for experiments involving circularly polarized light
traveling in the laboratory-fixed *Z* direction must be performed with the
combinations
*µ*_{X} ± *i µ*_{Y} of the
laboratory-fixed components of the dipole moment operator
[7] (pp. 90-93, 97-100),
i.e., intensities in such experiments are proportional to *µ*_{X} +
*i µ*_{Y} or *µ*_{X} -
*i µ*_{Y}
.

Theoretical results for experiments involving unpolarized (ordinary) light are
obtained by averaging the results of calculations for plane polarized light
having the electric vector in each of the three laboratory-fixed directions.
If, in addition, the emitting or absorbing molecules are in an isotropic
environment, then all directions are equivalent and theoretical results for
experiments using unpolarized light can be obtained by considering plane
polarized light with the electric vector in only one of the laboratory-fixed
directions [7] (pp. 90-93, 97-100).

The laboratory-fixed components of the dipole moment operator transform as
follows [15,16] under the symmetry operations
σ_{υ}(*xz*),
*i*, and *C*_{2}(*y*)

These relations can be proved easily from the results given in
table 5. The first of the transformation
properties in (3.2) leads (see sect. 2.9) to
the overall parity selection rule for electric dipole transitions
. The third of the transformation properties
leads to the selection rule *a* ↔ *a* and
*s* ↔ *s*. The second of the transformation properties
leads to no additional selection rules, since
*i* ≡ σ_{υ} · *C*_{2}.