4. Perturbations

4.1. General Remarks

Perhaps the first question to consider in this chapter on perturbations in diatomic molecules [1] (pp. 280-298) is the following: What is a perturbation and what is not? From an experimental point of view, a perturbation occurs when some energy level of the molecule is found in an unexpected position, or when a transition between some pair of energy levels is observed with an unexpected intensity. From a theoretical point of view, a perturbation occurs when calculated quantities disagree with experiment because an important interaction was neglected in the calculation. Both of the preceding statements indicate that what is considered to be a perturbation and what is not depends largely on what behavior for experimentally observable quantities is initially expected, or on what interaction terms in the theoretical formalism are initially included.

For example, one sometimes speaks of the rotational levels of a ¹Π state being perturbed by the rotational levels of a nearby ¹Σ⁺ state. Such an interaction leads to Λ-type doubling in the Π state and to a change in the effective B value for the Σ state. Calling this interaction a perturbation implies that our experimental expectations were based on the behavior of an isolated ¹Π state and an isolated ¹Σ⁺ state, and that our theoretical calculations did not initially include interactions between these two states.

On the other hand, one sometimes considers the rotational levels of a ¹Π and a ¹Σ⁺ state simultaneously, as part of a p-complex (L = 1; Λ = 0, ± 1). This point of view implies that our experimental expectations are not based on the behavior of an isolated ¹Π; and an isolated ¹Σ⁺ state, and that our theoretical calculations already include interactions between the Σ state and Π state of the p-complex, including those which give rise to Λ-type doubling in the Π state and to a change in the effective B value of the Σ state. The p-complex can still be perturbed, but only by states external to itself, e.g., by a ¹Δ state or by a ³Π state.

Perturbations are sometimes a nuisance and sometimes a source of valuable information. If their origins cannot be understood, they are most often simply a nuisance. However, if a perturbation can be dealt with theoretically, it frequently yields information which cannot easily be obtained in other ways.

Many techniques have been developed for dealing theoretically with perturbation problems in molecular spectroscopy. In this chapter we do not consider all of them. Rather, we concentrate on a two-step procedure of general applicability and widespread use. The two steps consist of: (i) setting up a Hamiltonian matrix which correctly contains the effects of the perturbation(s) to be considered, and (ii) diagonalizing this matrix on a computer. It is hoped that the explanatory material and the three examples described below illustrate this procedure well enough to allow a diatomic-molecule spectroscopist to embark on his own perturbation calculations with relatively little help from a professional theoretician.

The Hamiltonian matrix for a perturbation calculation is set up by a procedure similar to that described in chapter 1, except that more electronic states must be included in the basis set. For example, if a ³Δ state were perturbed by a ¹Π state, the basis set for the perturbation calculation would contain the six wave functions of the ³Δ state and the two wave functions of the ¹Π; state. In the absence of the perturbation, one could (and would) consider the six wave functions of the ³Δ state by themselves, and the two wave functions of the ¹Π; state by themselves.

Intensity matrices for perturbed states can be set up by a procedure similar to that described in chapter 3, except that again a larger basis set must be used for the perturbed state(s).