Interactions - Group Theory Applied to Chemistry - page 138

(CT) transition between metal and ligand gains intensity when the relevant metal
and ligand orbitals interact.
We first calculate the interaction terms between the metal and isolated ligand or-
bitals. The bipy ligand has low-lying unoccupied levels of ψ -character, which form
π -acceptor interactions with the metal t 2 g orbitals. Let H π represent the elementary
interaction between a ligand ψ orbital and a metal t 2 g orbital, directed towards one
ligator. The allowed interactions are then obtained by cyclic permutation:
H π = d xz | H | ψ A

= d xy | H | ψ B =− d xz | H | ψ B
= d yz | H |

ψ A =− d yz | H |

ψ C =− d xy | H | ψ C

(6.88)

In order to apply the model of Day and Sanders, we now consider the CT transition
between the ligand orbital on A and the t 2 g combination that interacts with it. As
showninFig. 6.5 ,the ψ A -acceptor orbital is antisymmetric with respect to the C x
2
axis and antisymmetric in the xy -plane. The only matching t 2 g combination on the
metal is the

| e (t 2 g ) component (see Table 6.4 ). In the local C 2 v symmetry, | ψ A

and | e (t 2 g ) both transform as b 2 (taking the horizontal plane as the local σ 1 ). Their ˆ

interaction element is expressed as:

√ 2 H π

e (t 2 g )

ψ A = √ 2 d xz − ψ A =

1

| H | d yz | H | (6.89)

We now consider the transition dipole moment between these orbitals along the
x direction, with μ x =− e x .In C 2 v symmetry this component transforms as a 1 ,

while μ y and μ z are antisymmetric with respect to the C x 2 axis. According to the
Wigner-Eckart theorem, a transition dipole between two b 2 orbitals must transform
as the direct product b 2 × a 1 ; hence, only the x - component will be dipole-

allowed. In a perturbative approach, which takes into account the symmetry-allowed
interaction between the metal and ligand orbitals, one has:

b 2 =

μ e (t 2 g ) → ψ A = e (t 2 g ) | μ x | ψ A − e (t 2 g ) | H | ψ A

ψ A

| μ x | ψ A (6.90)

E ψ − E t 2 g

In this expression the first term is the contact term between the zeroth-order orbitals.
The second term is the transfer term, arising from the interaction between the donor
and acceptor orbitals. In the simplified model of Day and Sanders this term is the
dominant contribution. The transfer-dipole matrix element in Eq. ( 6.90 ) is approxi-
mated as the dipole length of the transferred charge, which we will represent as μ A .
ψ A

ψ A =− e ψ A ψ A ≈−

x | | μ x | | e | R A |=− e ρ ≡ μ A (6.91)

where R A is the radius vector from the origin to the centre of ligand A , with
length ρ . Since the three ligands are equivalent, we further write:

μ ⊥ μ A = μ B = μ C ≡ (6.92)