7.7 Application: Spin Hamiltonian for the Octahedral Quartet State
185
Table 7.7 Representation matrices for the spinor basis in O ∗
D Γ 6 (C
z
4 ) =
1
√
2
1 − i
0
01 + i
D Γ 6 (C
xyz
3 ) =
1
2
1 − i −1 − i
1 − i 1 + i
D Γ 8 ( ˆ
C
z
4 ) =
1
√
2
⎛
⎜
⎜
⎜
⎜
⎝
−1 − i
00
0
01 − i
00
00
1 + i
0
00
0
−1 + i
⎞
⎟
⎟
⎟
⎟
⎠
D Γ 8 ( ˆ
C
xyz
3 ) =
1
4
⎛
⎜
⎜
⎜
⎜
⎝
−1 − i
√
3(−1 + i)
√
3(1 + i)
1 − i
√
3(−1 − i)
−1 + i
−1 − i
√
3(−1 + i)
√
3(−1 − i)
1 − i
−1 − i
√
3(1 − i)
−1 − i
√
3(1 − i)
√
3(1 + i)
−1 + i
⎞
⎟
⎟
⎟
⎟
⎠
that define the components of the Γ 8 . In Griffith’s notation, these four components
are defined in the following way:
|U
′ κ∼
3
2
+
3
2
|U
′ λ∼
3
2
+
1
2
|U
′ μ∼
3
2
−
1
2
|U
′ ν∼
3
2
−
3
2
(7.70)
Knowing the symmetries of the components, we can now turn to the coupling
coefficients that describe their interactions. The coupling coefficients that we need
are determined by the Zeeman Hamiltonian, which can be written as
H Ze =
μ B
B · (L + 2.0023S)
(7.71)
The electronic part of this operator contains the orbital angular momentum and the
spin operator. In octahedral symmetry the overall electronic operator transforms
as T 1g . Applying the Wigner–Eckart theorem to the interaction elements in this operator yields
μ B
Γ 8 i|(L + 2.0023S)|Γ 8 j
==Γ 8 ||T 1 ||Γ 8 a Γ 8 i|T 1 jΓ 8 k a
++Γ 8 ||T 1 ||Γ 8 b Γ 8 i|T 1 jΓ 8 k b
(7.72)
Here, we have introduced the extra labels a and b in order to distinguish that
there are two coupling channels. The coupling coefficients that are required are
of type Γ 8k |Γ 8i T 1j , while the coefficients, as given in Appendix G, are of type