Λ-type doubling in a ^{1}Π state arises from perturbations of
the rotational levels of the ^{1}Π state by the rotational levels of
a distant ^{1}Σ state [25]. In
sect. 4.3 we considered a
^{1}Π - ^{1}Σ^{+} 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 ^{1}Σ 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

(4.9) |

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.

(4.10) |

The Λ-doubled rotational levels of the ^{1}Π state are often
represented as symmetrically split about some mean position. Such a
representation causes no difficulty when the ^{1}Π and
^{1}Σ 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 ^{1}Π rotational levels
are perturbed by interaction with the ^{1}Σ^{+} state.
The other half of the levels are completely unaffected.