Chemistry Reference
In-Depth Information
Multiplication table for the double group
D
3
D
3
E
C
3
C
3
ℵ
C
3
ℵ
C
3
C
2
C
2
C
2
ℵ
C
2
ℵ
C
2
ℵ
C
2
ℵ
E
E
C
3
C
3
ℵ
C
3
ℵ
C
3
C
2
C
2
C
2
ℵ
C
2
ℵ
C
2
ℵ
C
2
ℵ
C
3
C
3
C
3
ℵ
C
3
E
ℵ
C
3
C
2
C
2
ℵ
C
2
ℵ
C
2
ℵ
C
2
C
2
C
3
C
3
ℵ
C
3
E
C
3
ℵ
C
3
ℵ
C
2
C
2
ℵ
C
2
C
2
ℵ
C
2
C
2
ℵ
C
3
ℵ
C
3
ℵ
C
3
E
C
3
C
3
ℵ
C
2
ℵ
C
2
C
2
C
2
C
2
ℵ
C
2
ℵ
ℵ
C
3
ℵ
C
3
E
C
3
ℵ
C
3
C
3
C
2
ℵ
C
2
C
2
ℵ
C
2
C
2
ℵ
C
2
ℵℵ
C
3
ℵ
C
3
C
3
C
3
E
ℵ
C
2
ℵ
C
2
ℵ
C
2
C
2
C
2
C
2
C
2
C
2
C
2
ℵ
C
2
ℵ
C
2
C
2
ℵ
C
2
ℵ
C
3
C
3
E
C
3
ℵ
C
3
C
2
C
2
ℵ
C
2
ℵ
C
2
C
2
C
2
ℵ
C
2
C
3
C
3
ℵ
C
3
E
ℵ
C
3
ℵ
C
2
C
2
C
2
C
2
ℵ
C
2
ℵ
C
2
ℵ
C
2
ℵ
C
3
ℵ
C
3
C
3
C
3
E
ℵ
ℵ
C
2
ℵ
C
2
ℵ
C
2
C
2
C
2
ℵ
C
2
C
2
E
C
3
ℵ
C
3
ℵ
C
3
C
3
ℵ
C
2
ℵ
C
2
C
2
C
2
ℵ
C
2
ℵ
C
2
C
2
ℵ
C
3
E
ℵ
C
3
C
3
C
3
ℵ
ℵ
C
2
ℵ
C
2
ℵ
C
2
ℵ
C
2
C
2
C
2
C
2
C
3
C
3
E
ℵ
C
3
ℵ
C
3
ℵ
7.4 The action of the spin operators on the components of a spin-triplet can
be found by acting on the coupled states, as summarized in Table
7.2
.As
an example, where we have added the electron labels 1 and 2 for clar-
ity:
S
x
|
α
2
=
S
x
|
α
1
|
α
1
S
x
|
α
2
S
x
|+
1
=
α
1
|
α
2
+|
=
2
|
β
2
= √
2
|
β
1
|
α
2
+|
α
1
|
0
√
2
|
i
S
y
|−
=−
1
0
These results can be generalized as follows:
M
S
(S
x
±
iS
y
)
|
M
S
=
(S
∓
M
S
)(S
±
M
s
+
S
z
|
M
S
=
M
S
|
1
)
1
2
The action of the spin Hamiltonian in the fictitious spin basis gives then rise to
the following Hamiltonian matrix (in units of
μ
B
):
|
M
s
±
1
H
Ze
|
0
|+
1
|−
1
1
1
0
|
0
g
√
2
(B
x
+
iB
y
)
g
√
2
(B
x
−
iB
y
)
⊥
⊥
1
+
1
|
g
⊥
√
2
(B
x
−
iB
y
)
g
||
B
z
0
1
−
1
|
g
⊥
√
2
(B
x
+
iB
y
)
0
−
g
||
B
z
We can now identify these expressions with the actual matrix elements in
the basis of the three
D
3
components, keeping in mind the relationship be-
