218
D Canonical-Basis Relationships
I (D 2 basis, Fig. D.2)
D(C 5 )
D(C
xyz
3 )
D(C
z
2 )
|T 1x , |T 1y , |T 1z
1
2
⎛
⎝
1 −φφ −1
φφ −1 −1
φ −1 1
φ
⎞
⎠
⎛
⎝
001
100
010
⎞
⎠
⎛
⎝
−100
0 −10
00 1
⎞
⎠
|T 2x , |T 2y , |T 2z
1
2
⎛
⎝
1 φ −1 −φ
−φ −1 −φ −1
−φ
1 −φ −1
⎞
⎠
⎛
⎝
001
100
010
⎞
⎠
⎛
⎝
−100
0 −10
00 1
⎞
⎠
|G a , |G x , |G y ,
|G z
1
4
⎛
⎜
⎜
⎝
−1 −
√
5
√
5
√
5
√
5 −3 −1 −1
√
51 −13
−
√
5 −1 −31
⎞
⎟
⎟
⎠
⎛
⎜
⎜
⎝
1000
0001
0100
0001
⎞
⎟
⎟
⎠
⎛
⎜
⎜
⎝
10 00
0 −100
00−10
00 01
⎞
⎟
⎟
⎠
I
|Hθ, |Hǫ, |Hξ, |Hη, |Hζ
D(C 5 )
D(C
xyz
3 )
D(C
z
2 )
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
−
1
4 −
√
3
4
1
√
8
1
√
2
−
1
√
8
−
√
3
4
1
4
−
√
3
√
8
0 −
√
3
√
8
−
1
√
8
√
3
√
8
0
1
2
1
2
1
√
2
0
−
1
2
1
2
0
1
√
8
√
3
√
8
1
2
0 −
1
2
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎝
−
1
2 −
√
3
2 000
√
3
2
−
1
2 000
00 0 0 1
00 1 0 0
00 0 1 0
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎠
⎛
⎜
⎜
⎜
⎜
⎜
⎝
10 0 0 0
01 0 0 0
00−100
00 0 −10
00 0 0 1
⎞
⎟
⎟
⎟
⎟
⎟
⎠
It is important to note that in the Boyle and Parker basis the |Hθ and |Hǫ
components do not denote components that transform like the functions d z 2 and
d x 2 −y 2 , but refer to linear combinations of these:
|Hθ=
3
8
d z 2 +
5
8
d x 2 −y 2
|Hǫ=−
5
8
d z 2 +
3
8
d x 2 −y 2
Griffith has presented the subduction of spherical |JM states to point-group
canonical bases for the case of the octahedral group. Similar tables for subduction to the icosahedral canonical basis have been published by Qiu and Ceulemans
[8]. Extensive tables of bases in terms of spherical harmonics for several branching
schemes are also provided by Butler [9].