A ^{3}Δ state cannot easily
interact with a ^{1}Π state via the
operator , since terms of this
operator give rise to nonvanishing matrix elements only if the selection rule
Δ*S* = 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 ^{3}Δ
state and a ^{1}Π state can take
place.

The nonrotating-molecule ^{3}Δ state gives rise to multiplet
components characterized by Ω = ±3, ±2, ±1; the
nonrotating-molecule ^{1}Π 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
^{3}Δ state and the ^{1}Π 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 ^{3}Δ state, set up as described in
chapter 1. (In this block we have written the
spin-orbit energies as *A*ΛΣ, which results in three evenly
spaced components of the ^{3}Δ state in the nonrotating
molecule.) The lower 1 × 1 diagonal block is the Hamiltonian matrix
for a ^{1}Π 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 ^{3}Δ state and for the
^{1}Π 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 ^{3}Δ state and the ^{1}Π 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)].