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)].