E 6C 4 3C 2 ¼ C
2
4
À
Á
8C 3 6C 2
v T 5 À1
1
À1
1
X T is decomposed in T 2 + E, in the case of O h , T 2g + E g .
The results obtained only by the symmetry considerations are the same as those
deriving from the application of the perturbation operator, that is that the d orbitals
split in two sets of orbitals, one three times degenerate (T 2g ) and the other two times
(E g ). The symmetry considerations allow to predict the number and type of energy
levels, but do not give any quantitative energy measure. See the prevision in
Table 3.3.
Let us consider that the correlation between the type of orbital (l value) and the
irreducible representations is valid also for the states (L values) and the irreducible
representations of the states.
The prevision of the states, both in the case of monoelectronic and
multi-electronic states, can be transferred from the octahedral symmetry to the
different ones. In fact, the same wave functions, in different point groups, are base
of different irreducible representation (Table 3.4).
The crystal field perturbation can have different weight on the energy levels, that
is it can be stronger (strong field) or weaker (weak field) than the interelectronic
perturbation. If we suppose the d
n electronic configurations in a strong crystal field
perturbation, for the d
1 configuration, the two perturbed terms are
2 T 2g and
2 E g . In
the case of d
2 , we have t
2
2g , t
1
2g e
1
g , e
2
g orbital arrangements, and the terms are given
by the direct product of the single arrangements (direct product of the irreducible
representations).
t 2g  t 2g ¼ T 1g þ T 2g þ E g þ A 1g
t
1
2g  e
1
g
¼ T 1g þ T 2g
e g  e g ¼ E g þ A 1g þ A 2g
However, while the terms are easy to be obtained by the group theory, the related
spin states are not. In fact, the d
2 configuration allows two spin states, the singlet
and the triplet, and their attribution to the terms is not immediate. We will use the
Table 3.3 Association
between orbitals and their
symmetry properties in O h
Types of orbital
l value
Irreducible representation
s
0
a 1g
p
1
t 1u
d
2
e g + l 2g
f
3
a 2u þ t 1u þ t 2u
g
4
a 1g þ e g þ t 1g þ t 2g
h
5
e u þ 2t 1u þ t 2u
i
6
a 1g þ a 2g þ e g þ t 1g þ 2t 2g
3.3 Crystal Field Perturbation
49
2
4
À
Á
8C 3 6C 2
v T 5 À1
1
À1
1
X T is decomposed in T 2 + E, in the case of O h , T 2g + E g .
The results obtained only by the symmetry considerations are the same as those
deriving from the application of the perturbation operator, that is that the d orbitals
split in two sets of orbitals, one three times degenerate (T 2g ) and the other two times
(E g ). The symmetry considerations allow to predict the number and type of energy
levels, but do not give any quantitative energy measure. See the prevision in
Table 3.3.
Let us consider that the correlation between the type of orbital (l value) and the
irreducible representations is valid also for the states (L values) and the irreducible
representations of the states.
The prevision of the states, both in the case of monoelectronic and
multi-electronic states, can be transferred from the octahedral symmetry to the
different ones. In fact, the same wave functions, in different point groups, are base
of different irreducible representation (Table 3.4).
The crystal field perturbation can have different weight on the energy levels, that
is it can be stronger (strong field) or weaker (weak field) than the interelectronic
perturbation. If we suppose the d
n electronic configurations in a strong crystal field
perturbation, for the d
1 configuration, the two perturbed terms are
2 T 2g and
2 E g . In
the case of d
2 , we have t
2
2g , t
1
2g e
1
g , e
2
g orbital arrangements, and the terms are given
by the direct product of the single arrangements (direct product of the irreducible
representations).
t 2g  t 2g ¼ T 1g þ T 2g þ E g þ A 1g
t
1
2g  e
1
g
¼ T 1g þ T 2g
e g  e g ¼ E g þ A 1g þ A 2g
However, while the terms are easy to be obtained by the group theory, the related
spin states are not. In fact, the d
2 configuration allows two spin states, the singlet
and the triplet, and their attribution to the terms is not immediate. We will use the
Table 3.3 Association
between orbitals and their
symmetry properties in O h
Types of orbital
l value
Irreducible representation
s
0
a 1g
p
1
t 1u
d
2
e g + l 2g
f
3
a 2u þ t 1u þ t 2u
g
4
a 1g þ e g þ t 1g þ t 2g
h
5
e u þ 2t 1u þ t 2u
i
6
a 1g þ a 2g þ e g þ t 1g þ 2t 2g
3.3 Crystal Field Perturbation
49