Interactions - Group Theory Applied to Chemistry - page 128

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 first-
order, 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 degen-
erate irrep of the electronic state, minus the totally-symmetric modes, since these
cannot lower the symmetry:

∈ [ Γ 0

Γ Γ a × Γ a ]− (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

Ωω | Ωω | Ωω | ΓγΓ γ

H
∂Q Γγ ∂Q Γ γ

(6.62)

The second-order matrix element then becomes:
Γ a γ a

Γ a γ b

∂ 2 H

∂Q Γγ ∂Q Γ γ

0

Ωω

Γ a Ωω ΓγΓ γ = Γ a Ω | Γ a γ a | ΩωΓ a γ b (6.63)

The second-order elements thus are related to a product of two 3 Γ symbols. 5 A spe-
cial element arises when Ω is totally symmetric. In this case, the coupling coeffi-
cients 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 6 Γ symbol, by analogy with
the 6 j coupling coefficients in atomic spectroscopy.