NBS Mono115: Lab-Fixed Components of Elec. Dipole Moment Oper.

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 $|\langle f|$ µ_Z $|i\rangle|^2$ . 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 $$|\langle f|$ µ_X + i µ_Y $|i\rangle|^2$ or $$|\langle f|$ µ_X - i µ_Y $|i\rangle|^2$ .

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 C₂(y)

eq 3.02 (3.2)

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 $\pm \leftrightarrow \mp$ . 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 · C₂.