Chemistry Reference
In-Depth Information
7.7 Application: Spin Hamiltonian for the Octahedral Quartet
State
The octahedral double group contains a four-dimensional spin representation, which
is commonly denoted as the
Γ
8
quartet, or
U
in Griffith's notation. The direct square
of this irrep is given by
Γ
8
×
Γ
8
=[
A
2
+
2
T
1
+
T
2
]+{
A
1
+
+
T
2
}
E
(7.67)
According to the time-reversal selection rules, time-odd interactions with a magnetic
field will be based on the symmetrized square. The spin-operator of the Zeeman
Hamiltonian transforms as
T
1
g
, which is indeed included in the symmetrized square.
The present case is, however, special since the
T
1
g
irrep occurs twice in the product.
The multiplicity separation cannot be achieved on the basis of symmetrization since
both
T
1
g
irreps appear in the symmetrized part. One way to distinguish the two
products is through the subduction process from spherical symmetry. The addition
rules of angular momenta give rise to
3
/
2
×
3
/
2
=[
p
+
f
]+{
s
+
d
}
(7.68)
On these grounds the two
T
1
g
interactions can be distinguished on the basis of a
different spherical parentage corresponding to
p
or
f
coupling. We shall return to
this point in a moment. For a systematic treatment of this problem, we start by set-
ting up a suitable function space. The spherical
S
3
/
2 spin-quartet level subduces
directly the octahedral
Γ
8
. We can thus use the quartet spin functions as symmetry
bases. The components of
S
=
=
3
/
2 can be obtained by a fully symmetrized product
of the basic spinor:
|
3
/
2
+
3
/
2
=
α
1
α
2
α
3
1
√
3
(α
1
α
2
β
3
+
|
3
/
2
+
1
/
2
=
α
1
β
2
α
3
+
β
1
α
2
α
3
)
(7.69)
1
√
3
|
3
/
2
−
1
/
2
=
(α
1
β
2
β
3
+
β
1
α
2
β
3
+
β
1
β
2
α
3
)
|
3
/
2
−
3
/
2
=
β
1
β
2
β
3
In this expression the fundamental
spinor resembles a quark state: three
quarks are coupled together to form the quartet result. The use of the quartet spin
bases does not mean that our
Γ
8
really corresponds to a quartet spin. It only means
that we can introduce a fictitious spin operator,
S
, which acts on the
Γ
8
components
in the same way as the real spin momentum would act on the components of a
spin quartet. The transformations of the
Γ
8
spinor under the elements of the group
O
∗
may be obtained by combining the transformation matrices for the fundamental
(
{|
α
,
|
β
}
)
spins with the quartet coupling scheme in Eq. (
7.69
). In the group
O
∗
,this
irrep is denoted as
Γ
6
. In Table
7.7
the results are shown for two generators of the
octahedral group. These matrices can be taken as the canonical basis relationships
|
α
|
β
