Next: Lambda doubling
Up: Spectroscopic theory
Previous: Spectroscopic theory
  Contents
An electron in a diatomic molecule has a variety of angular momenta, associated with different electronic states and energy levels. For diatomic molecules, it is quite common to express the angular momenta in terms of projections along the internuclear axis, as these are better quantum numbers than the value of the angular momentum itself. The different types of angular momenta which are important to this study and their projections onto the internuclear axis are given in Table 3.1.
As angular momentum is not a scalar quantity, it is necessary to consider the addition of various angular momenta in terms of vector addition. The coupling of the momenta with the nuclear rotational angular momentum R, gives rise to the different Hund's cases:
- Hund's case (a). Electronic motion is strongly coupled to the internuclear axis, yielding a well defined total electronic angular momentum
which couples with the nuclear rotational angular momentum to give the resultant J. L is strongly coupled to the internuclear axis and S is coupled to L by Spin-Orbit coupling so S is therefore coupled to the internuclear axis (see Figure 3.1).
Figure 3.1:
Coupling of angular momenta for Hund's case (a)
|
- Hund's case (b). Electronic spin angular momentum, S, is not coupled to the internuclear axis, thus
is not defined. Therefore,
cannot be defined, so the electronic orbital angular momentum, L, couples with the nuclear rotation R, to give a resultant N, which couples with S to give the total angular momentum J (see Figure 3.2). Hund's case (b) usually occurs in light molecules when the spin-orbit coupling is small or when
= 0.
Figure 3.2:
Coupling of angular momenta in Hund's case (b)
|
- Hund's case (c). The spin-orbit coupling is strong and the interaction between L and S exceeds that between L and the internuclear axis. The electron spin and orbital angular momentum couple, yielding the resultant Ja which then couples along the internuclear axis to form
.
The nuclear rotation R couples with
to yield the total angular momentum J. See Figure 3.3. The quantum numbers L, S,
and
are no longer good quantum numbers. A description of Hund's case (c) theory can be found in the work of Veseth [70] and examples are given in [71].
- Hund's case (d). Coupling between the electron and the nuclei is extremely weak. The electron orbital angular momentum couples to the nuclear rotational angular momentum, forming a resultant N, which couples with the electron spin angular momentum to give J. This case generally applies to Rydberg states of molecules, where the electron is far removed from the nuclei.
Figure 3.3:
Coupling of angular momenta for Hund's case (c)
|
Due to the large spin-orbit coupling in GeH+ , which arises from the large coupling in the atomic Ge+ of 1,767 cm-1, the molecule should strictly be treated using Hund's case (c). However, for the purposes of this work, it can be treated adequately within a Hund's case (a) formalism, as is the case for the isovalent CH+ and SiH+.
Next: Lambda doubling
Up: Spectroscopic theory
Previous: Spectroscopic theory
  Contents
Tim Gibbon
1999-09-06