142
6 Interactions
Fig. 6.6 Allowed CT transitions from the t 2g shell to ψ -o rχ -type ligand acceptor orbitals for
tris-chelate complexes with D 3 symmetry
Here, we have made use of the fact that the sum of the three dipole vectors vanishes.
The effective transfer term thus becomes:
µ
e ǫ (t 2g ) → e ǫ (ψ)
=
3
8
κµ A
(6.99)
In the Wigner–Eckart formalism, this matrix element is written as:
e ǫ (t 2g )|μ x ′ |e ǫ (ψ)
==Eǫ|EθEǫ
e(t 2g ) e(μ) e(ψ)
(6.100)
The coupling coefficient in this equation is equal to 1/
√
2. We can thus identify the
reduced matrix element as:
e(t 2g ) e(μ) e(ψ)
=
√
3
2
κμ A
(6.101)
All other e → e transfer terms can then be obtained by simply varying the coupling
coefficients. We give one more example of a transition that requires an operator
which is μ y ′ polarized:
e θ (t 2g )|µ|e ǫ (ψ)
=
1
√
8
κ
ψ
C − ψ
B |µ|2ψ
A − ψ
B − ψ
C
=
1
√
8
κ(µ B − µ C )
(6.102)
The vector µ B − µ C in this expression is directed in the μ y ′ direction, as required
by the selection rule. Moreover, the length of this vector is
√
3μ ⊥ :
(µ B − µ C ) · (µ B − µ C ) = 2
μ
⊥ 2 − 2µ B · µ C = 3
μ
⊥ 2
(6.103)