Λ-type doubling in a 1Π state arises from perturbations of the rotational levels of the 1Π state by the rotational levels of a distant 1Σ state . In sect. 4.3 we considered a 1Π - 1Σ+ perturbation problem in which the two electronic states were both included in the matrix to be diagonalized. In this section we consider the same problem, but take into account the effect of the 1Σ state by a Van Vleck transformation.
We thus intend to diagonalize exactly only the upper left 2 × 2 diagonal block of the matrix given in (4.1). This block can be corrected, according to eq (4.8), by adding the quantity
to each of the four matrix elements in the block. Diagonalization of the resulting 2 × 2 matrix yields the desired Λ-doubled rotational energy levels.
The Λ-doubled rotational levels of the 1Π state are often represented as symmetrically split about some mean position. Such a representation causes no difficulty when the 1Π and 1Σ states are well separated. However, if the states are approaching case (d) coupling, it is essential to represent the probelm as indicated here, namely only one-half of the 1Π rotational levels are perturbed by interaction with the 1Σ+ state. The other half of the levels are completely unaffected.