146
6 Interactions
Here, the notation refers to a singlet orbital transition, which can be written in determinantal form as:
(χ A → ψ A )
1 =
1
√
2
(χ A α)(ψ A β)
−
(χ A β)(ψ A α)
(6.111)
To first approximation, the metal centre is not taking part in the electronic properties, but merely serves as a structural template which keeps the ligands in place.
Distant interactions between the three transitions can be described by a simple
exciton-coupling model. In this model, the interaction between transitions is approximated by the electrostatic interaction potential between the corresponding transition dipoles. This potential is given by:
V ij =
1
4πǫ 0
µ i · µ j
R 3
ij
−
3(µ i · R ij )(µ j · R ij )
R 5
ij
(6.112)
where R ij is the distance between the dipoles, and R ij = R j − R i . The length of the
distance vector is thus
√
3ρ. The energies of the exciton states are then given by:
1 A 2 |V |
1 A 2
=
(μ ) 2
4πǫ 0 ρ 3
1
6
√
3
1 E|V |
1 E
=−
(μ ) 2
4πǫ 0
√
3ρ 3
1
12
√
3
(6.113)
The 1 A 2 state thus goes up in energy twice as much as the 1 E state goes down,
thus keeping the barycentre energy at the zeroth-order position. Now, in order to
determine the CD strength, we need for the two states both the electric and the
magnetic transition dipoles from the ground state. The electric dipoles are easily
obtained by combining the state vectors:
µ
1 A 1 →
1 A 2
=
1
√
3
µ
A + µ
B + µ
C
=
√
2μ
(0, 0, 1)
µ
1 A 1 →
1 E ǫ
=
1
√
6
2µ
A − µ
B − µ
C
=
1
√
2
μ
(0, 1, 0)
µ
1 A 1 →
1 E θ
=
1
√
2
−µ
B + µ
C
=
1
√
2
μ
(1, 0, 0)
(6.114)
The calculation of the magnetic transition dipoles requires a preamble. The magnetic
moment was already defined in Eq. (4.128) of Chap. 4. By explicitly writing the
angular momentum operator in terms of the linear momentum operator as r × p one
obtains:
m =−
e
2m
l =−
e
2m
r × p
(6.115)
6 Interactions
Here, the notation refers to a singlet orbital transition, which can be written in determinantal form as:
(χ A → ψ A )
1 =
1
√
2
(χ A α)(ψ A β)
−
(χ A β)(ψ A α)
(6.111)
To first approximation, the metal centre is not taking part in the electronic properties, but merely serves as a structural template which keeps the ligands in place.
Distant interactions between the three transitions can be described by a simple
exciton-coupling model. In this model, the interaction between transitions is approximated by the electrostatic interaction potential between the corresponding transition dipoles. This potential is given by:
V ij =
1
4πǫ 0
µ i · µ j
R 3
ij
−
3(µ i · R ij )(µ j · R ij )
R 5
ij
(6.112)
where R ij is the distance between the dipoles, and R ij = R j − R i . The length of the
distance vector is thus
√
3ρ. The energies of the exciton states are then given by:
1 A 2 |V |
1 A 2
=
(μ ) 2
4πǫ 0 ρ 3
1
6
√
3
1 E|V |
1 E
=−
(μ ) 2
4πǫ 0
√
3ρ 3
1
12
√
3
(6.113)
The 1 A 2 state thus goes up in energy twice as much as the 1 E state goes down,
thus keeping the barycentre energy at the zeroth-order position. Now, in order to
determine the CD strength, we need for the two states both the electric and the
magnetic transition dipoles from the ground state. The electric dipoles are easily
obtained by combining the state vectors:
µ
1 A 1 →
1 A 2
=
1
√
3
µ
A + µ
B + µ
C
=
√
2μ
(0, 0, 1)
µ
1 A 1 →
1 E ǫ
=
1
√
6
2µ
A − µ
B − µ
C
=
1
√
2
μ
(0, 1, 0)
µ
1 A 1 →
1 E θ
=
1
√
2
−µ
B + µ
C
=
1
√
2
μ
(1, 0, 0)
(6.114)
The calculation of the magnetic transition dipoles requires a preamble. The magnetic
moment was already defined in Eq. (4.128) of Chap. 4. By explicitly writing the
angular momentum operator in terms of the linear momentum operator as r × p one
obtains:
m =−
e
2m
l =−
e
2m
r × p
(6.115)