130
6 Interactions
of the first kind, i.e. can be represented by real functions, JT matrix elements can
thus be chosen to be entirely real, which implies:
Γ a γ a |H
′ |Γ a γ b
=
Γ a γ b |H
′ |Γ a γ a
(6.58)
Combining this result with Eq. (6.57) implies that the coupling coefficients, to firstorder, should obey:
Γ a γ a |ΓγΓ a γ b ==Γ a γ b |ΓγΓ a γ a
(6.59)
In view of Eq. (6.31) this condition can be rewritten as:
Γ a γ a Γ a γ b |Γγ==Γ a γ b Γ a γ a |Γγ
(6.60)
The JT distortion modes are thus restricted to the symmetrized square of the degenerate irrep of the electronic state, minus the totally-symmetric modes, since these
cannot lower the symmetry:
Γ ∈
[Γ a × Γ a ]−Γ 0
(6.61)
Modes that obey this selection rule, are said to be JT active. The evaluation of the
second-order matrix elements requires two steps. One first couples the two distortion
modes to a composite tensor operator: |Ωω|.
∂ 2 H
∂Q Γγ ∂Q Γ ′ γ ′
=
Ωω
|Ωω|
Ωω|ΓγΓ
′ γ
′
(6.62)
The second-order matrix element then becomes:
Γ a γ a
∂ 2 H
∂Q Γγ ∂Q Γ ′ γ ′
Γ a γ b
0
=
Ωω
Γ a Ω Γ a
Ωω|ΓγΓ
′ γ
′
Γ a γ a |ΩωΓ a γ b
(6.63)
The second-order elements thus are related to a product of two 3Γ symbols. 5 A special element arises when Ω is totally symmetric. In this case, the coupling coefficients are given by:
Γ 0 |ΓγΓ
′ γ
′
=
1
√
dim(Γ )
δ ΓΓ ′ δ γγ ′
Γ a γ a |Γ 0 Γ a γ b =δ γ a γ b
(6.64)
5 Such combinations can be cast in a higher-order symbol, known as 6Γ symbol, by analogy with
the 6j coupling coefficients in atomic spectroscopy.