216
D Canonical-Basis Relationships
Fig. D.1 Octahedron with x,y,z coordinates in D 4 and D 3 setting
O (D 4 basis)
D(C
z
4 )
D(C
xyz
3 )
|E θ , |E ǫ
10
0 −1
⎛
⎝
−
1
2
−
√
3
2
+
√
3
2
−
1
2
⎞
⎠
|T 1x , |T 1y , |T 1z
⎛
⎝
0 −10
100
001
⎞
⎠
⎛
⎝
001
100
010
⎞
⎠
|T 2ξ , |T 2η , |T 2ζ
⎛
⎝
010
−10 0
00 −1
⎞
⎠
⎛
⎝
001
100
010
⎞
⎠
(See Fig. D.1.) Transformation to trigonal basis set:
|E θ =d z 2 =
1
√
3
(−d x ′2 −y ′2 −
√
2d y ′ z ′ )
|E ǫ =d x 2 −y 2 =
1
√
3
(d x ′ y ′ +
√
2d x ′ z ′ )
|T 1a =
1
√
3
|T 1x +|T 1y +|T 1z
= p z ′
|T 1θ =
1
√
2
|T 1x −|T 1y
= p x ′
|T 1ǫ =
1
√
6
|T 1x +|T 1y −2|T 1z
= p y ′