4.2. Homogeneous and Heterogeneous Perturbations

Many perturbations in diatomic molecules can be characterized as either homogeneous or heterogeneous [l] (pp. 284-286). Homogeneous perturbations take place between electronic states satisfying the selection rule ΔΛ = 0 or ΔΩ = 0. Heterogeneous perturbations take place between electronic states satisfying the selection rule ΔΛ = ±1 or ΔΩ = ±1. Ambiguities can clearly arise in this classification scheme. Is, for example, the perturbation between a case (a) ²Π_1/2 state and a case (a) ²Π_3/2 state homogeneous (ΔΛ = 0) or heterogeneous (ΔΩ = ±1)? Some measure of consistency can be achieved by requiring the nomenclature to reflect the selection rules on Ω when, as in cases (a) and (c), the rotational energy levels are given approximately by BJ(J + 1), and to reflect the selection rules on Λ when, as in case (b), the rotational energy levels are given approximately by BN(N + 1). It is interesting to note that the transition from Hund's case (a) to Hund's case (b) as J increases in a ²Π state might thus be described as resulting from a heterogeneous perturbation of the rotational levels of the ²Π_1/2 state by those of the ²Π_3/2 state (or vice versa).

These two types of perturbations can be described in another way. Homogeneous perturbations are those which can occur in the nonrotating molecule, i.e., perturbations caused by . The rigorous selection rule is thus ΔΩ = 0 for nonvanishing homogeneous-perturbation matrix elements in the basis sets used in this monograph, with the approximate selection rules ΔS = 0, ΔΛ = 0, ΔΣ = 0, when S, Λ, Σ are good quantum numbers in the basis set. Heterogeneous perturbations are those which can only occur in the rotating molecule, i.e., perturbations caused by . The rigorous selection rules are thus ΔJ = 0 and ΔΩ = ±1 for nonvanishing heterogeneous-perturbation matrix elements in the basis sets used in this monograph, with the approximate selection rules ΔS = 0, and ΔΛ = ±1, ΔΣ = 0 or ΔΣ = ±1, ΔΛ = 0, when S, &Lambda, Σ are good quantum numbers in the basis set, as well as the approximate rule ΔL = 0 when L is a good quantum number in the basis set. Matrix elements for homogeneous perturbations do not involve the rotational quantum number J; matrix elements for heterogeneous perturbations do involve J. Many of the selection rules stated in this paragraph can be derived by considering the operators and together with appropriate angular momentum commutation relations [7] (pp. 59-64).

Homogeneous perturbations arise most frequently in practice because of spin-orbit interaction (see sect. 4.4), but they may also occur because of configuration interaction. In the latter case, the perturbations are often very large and difficult to treat accurately [22-24]. Heterogeneous perturbations occur because of uncoupling phenomena, i.e., uncoupling of the spin angular momentum (see sect. 1.9) or uncoupling of the orbital angular momentum (see sect. 4.3) from the internuclear axis because of rotation. These uncoupling phenomena can be attributed to Coriolis interactions in the rotating molecule.