## 4.4. Example: 3Δ - 1Π Homogeneous Perturbation

A 3Δ state cannot easily interact with a 1Π state via the operator , since terms of this operator give rise to nonvanishing matrix elements only if the selection rule ΔS = 0 is satisfied. To the extent that S is a good quantum number, a heterogeneous perturbation of one of these states by the other is not possible. Actually, of course, S is not a perfectly good quantum number and heterogeneous perturbations between a 3Δ state and a 1Π state can take place.

The nonrotating-molecule 3Δ state gives rise to multiplet components characterized by Ω = ±3, ±2, ±1; the nonrotating-molecule 1Π state gives rise to "multiplet components" characterized by Ω = ±1. Thus a homogeneous perturbation (caused by the spin-orbit interaction term in ) is also possible between the 3Δ state and the 1Π state, corresponding to an interaction between the two multiplet components with Ω = +1 and between the two multiplet components with Ω = -1. We now consider this homogeneous perturbation.

The full Hamiltonian matrix for this problem is of dimension 8 × 8. However, this matrix immediately factors into two identical 4 × 4 diagonal blocks. We consider only one of these below, with rows and columns labeled by the wave functions  = |2 1 1; 3 J M⟩,  |2 1 0; 2 J M⟩, |2 1 -1; 1 J M⟩, and |1 0 0; 1 J M⟩. This Hamiltonian matrix has the following form.

(4.5)

The upper left 3 × 3 diagonal block is just the Hamiltonian matrix for a 3Δ state, set up as described in chapter 1. (In this block we have written the spin-orbit energies as AΛΣ, which results in three evenly spaced components of the 3Δ state in the nonrotating molecule.) The lower 1 × 1 diagonal block is the Hamiltonian matrix for a 1Π state. The off-diagonal element η represents the spin-orbit interaction between the two states with Ω = +1 and is given by (see sect. 1.3)

 (4.6)

where the second line results from the fact that the spin-orbit interaction operator does not involve the rotational variables, and where the third line results from symmetry arguments (see chapt. 2). The second line of (4.6) shows that η is independent of J. The third line represents part of the algebra leading to the factorization of the original 8 × 8 Hamiltonian matrix into two identical 4 × 4 diagonal blocks. The matrix element η can only be evaluated theoretically if the electronic wave functions for the 3Δ state and for the 1Π state are rather well known. Such information is usually not available, so that this matrix element must be treated as an unknown adjustable (real) parameter, to be determined from a fit of calculated results to experimental data. Rotational energy levels are calculated, of course, by diagonalizing (4.5).

The effect of a heterogeneous perturbation (ΔΩ = ±1) between the 3Δ state and the 1Π state can be taken into account by placing the quantity λ[(J - 1)(J + 2)]1/2 in the (2,4) and (4,2) positions of (4.5). This quantity consists of a J-dependent part, determined as in sect. 4.3 from (1.13) and (1.14), and a small J-independent adjustable (real) parameter λ, which is analogous to the parameter B ⟨Π| L+ |Σ⟩ in sect. 4.3. [The fact that η and λ can simultaneously be taken as real must, of course, be proven (see chapter 2 and sect. 3.5)].