Chemistry Reference
In-Depth Information
tween the complex and real triplet basis, as given in Eq. (
7.39
). One ob-
tains:
√
2
−
c)
1
√
2
1
0
|
H
Ze
|+
1
=−
A
1
|
H
|
E
x
+
iE
y
=
a
+
d
+
i(
−
b
−
√
2
a
c)
1
√
2
1
0
|
H
Ze
|−
1
=
A
1
|
H
|
E
x
−
iE
y
=
+
d
+
i(b
−
2
i
=±
1
±
1
|
H
Ze
|±
1
=
x
|
H
|
x
+
y
|
H
|
y
±
x
|
H
|
y
−
y
|
H
|
x
f
From these equations the parameters may be identified as follows:
a
=
0
=−
b
g
⊥
B
y
=
c
0
d
=
g
⊥
B
x
0
f
=
g
||
B
z
The Zeeman Hamiltonian does not include the zero-field splitting between the
A
1
and
E
states. This can be rendered by a second-order spin operator, which
transforms as the octahedral
E
g
θ
quadrupole component:
e
=
3
S
2
2
2
S
z
−
S
x
−
S
y
=
D
3
D
1
S
z
−
H
ZF
=
2
One then obtains
=
3
7.5 The action of the components of the fictitious spin operator on the
Γ
8
basis is
dictated by the general expressions for the action of the spin operators on the
S
=
D
3
2
basis functions. It is verified that the spin-Hamiltonian that generates the
J
p
part of the matrix precisely corresponds to
·
S
H
p
=
J
p
B
The fictitious spin operator indeed transforms as a
T
1
operator and has the
tensorial rank of a
p
-orbital. However, as we have shown, the full Hamil-
tonian also includes a
J
f
part, which involves an
f
-like operator. To mimic
this part by a spin Hamiltonian, one thus will need a symmetrized triple prod-
uct of the fictitious spin, which will embody an
f
-tensor, transforming in the
octahedral symmetry as the
T
1
irrep. These
f
-functions can be found in Ta-
ble
7.1
and are of type
z(
5
z
2
3
r
2
)
. But beware! To find the correspond-
ing spin operator, it is not sufficient simply to substitute the Cartesian vari-
ables by the corresponding spinor components, i.e.,
z
by
S
z
, etc.; indeed,
while products of
x,y
, and
z
are commutative, the products of the corre-
sponding operators are not. Hence, when constructing the octupolar product
−
