6.8 Application: Linear and Circular Dichroism
143
Hence, the transfer-dipole length for this y ′ -polarized transition also measures
√
3/8κ, which is exactly the same as for the x ′ -polarized transition, given in Eq.
(6.99). This is expected since the corresponding coupling coefficients, Eǫ|EθEǫ
and Eθ|EǫEǫ, are equal.
Using the transfer model, we can also express the reduced matrix elements for
the e → a 2 channel. Even though there is no overlap between these orbitals, they do
give rise to a transfer-term intensity. Orbital interaction does indeed delocalize the
e(t 2g ) orbitals over the ligands. The dipole operators, centred on the complex origin,
will then couple the e(ψ) and a 2 (ψ) ligand-centred orbitals. Hence, we write:
µ
e ǫ (t 2g ) → a 2 (ψ)
=−
e ǫ (t 2g )|H|e ǫ (ψ)
E ψ − E t 2g
e ǫ (ψ)|µ|a 2 (ψ)
=
3
2
κ
e ǫ (ψ)|µ|a 2 (ψ)
(6.104)
The dipole matrix element in this expression can easily be evaluated:
e ǫ (ψ)|µ|a 2 (ψ)
=
1
3
√
2
2ψ
A − ψ
B − ψ
C |µ|ψ
A + ψ
B + ψ
C
=
1
3
√
2
(2µ A − µ B − µ C )
=
1
√
2
µ A
(6.105)
The total transfer term is obtained by combining Eqs. (6.104) and (6.105):
µ
e ǫ (t 2g ) → a 2 (ψ)
=
√
3
2
κµ A
(6.106)
A final task is to calculate the transition-moments between the corresponding multielectronic states based on the orbital-transition moments obtained. In the tris-chelate
complex under consideration, a 1 A 1 → 1 E state transition can be associated with
each allowed orbital-transition. The 1 A 1 corresponds to the closed-shell ground
state, based on the (t 2g ) 6 configuration. Both the e → a 2 and e → e transitions will
give rise to a twofold-degenerate 1 E state. As an example, the θ states are written in
determinantal notation as follows, where we write only the orbitals that are singly
occupied:
1 E θ (e → a 2 )
=
1
√
2
e ǫ (t 2g )α
a 2 (ψ)β
−
e ǫ (t 2g )β
a 2 (ψ)α
1 E θ (e → e)
=
1
2
e θ (t 2g )α
e θ (ψ)β
−
e θ (t 2g )β
e θ (ψ)α
−
e ǫ (t 2g )α
e ǫ (ψ)β
+
e ǫ (t 2g )β
e ǫ (ψ)α
(6.107)
143
Hence, the transfer-dipole length for this y ′ -polarized transition also measures
√
3/8κ, which is exactly the same as for the x ′ -polarized transition, given in Eq.
(6.99). This is expected since the corresponding coupling coefficients, Eǫ|EθEǫ
and Eθ|EǫEǫ, are equal.
Using the transfer model, we can also express the reduced matrix elements for
the e → a 2 channel. Even though there is no overlap between these orbitals, they do
give rise to a transfer-term intensity. Orbital interaction does indeed delocalize the
e(t 2g ) orbitals over the ligands. The dipole operators, centred on the complex origin,
will then couple the e(ψ) and a 2 (ψ) ligand-centred orbitals. Hence, we write:
µ
e ǫ (t 2g ) → a 2 (ψ)
=−
e ǫ (t 2g )|H|e ǫ (ψ)
E ψ − E t 2g
e ǫ (ψ)|µ|a 2 (ψ)
=
3
2
κ
e ǫ (ψ)|µ|a 2 (ψ)
(6.104)
The dipole matrix element in this expression can easily be evaluated:
e ǫ (ψ)|µ|a 2 (ψ)
=
1
3
√
2
2ψ
A − ψ
B − ψ
C |µ|ψ
A + ψ
B + ψ
C
=
1
3
√
2
(2µ A − µ B − µ C )
=
1
√
2
µ A
(6.105)
The total transfer term is obtained by combining Eqs. (6.104) and (6.105):
µ
e ǫ (t 2g ) → a 2 (ψ)
=
√
3
2
κµ A
(6.106)
A final task is to calculate the transition-moments between the corresponding multielectronic states based on the orbital-transition moments obtained. In the tris-chelate
complex under consideration, a 1 A 1 → 1 E state transition can be associated with
each allowed orbital-transition. The 1 A 1 corresponds to the closed-shell ground
state, based on the (t 2g ) 6 configuration. Both the e → a 2 and e → e transitions will
give rise to a twofold-degenerate 1 E state. As an example, the θ states are written in
determinantal notation as follows, where we write only the orbitals that are singly
occupied:
1 E θ (e → a 2 )
=
1
√
2
e ǫ (t 2g )α
a 2 (ψ)β
−
e ǫ (t 2g )β
a 2 (ψ)α
1 E θ (e → e)
=
1
2
e θ (t 2g )α
e θ (ψ)β
−
e θ (t 2g )β
e θ (ψ)α
−
e ǫ (t 2g )α
e ǫ (ψ)β
+
e ǫ (t 2g )β
e ǫ (ψ)α
(6.107)