Appendix D
Canonical-Basis Relationships
The importance of canonical-basis relationships was demonstrated by Griffith in his
monumental work on the theory of transition-metal ions [6]. The icosahedral basis
sets were defined by Boyle and Parker [7].
D 3
D(C
z
3 )
D(C x
2 )
|E x , |E y
⎛
⎝
−
1
2
−
√
3
2
+
√
3
2
−
1
2
⎞
⎠
10
0 −1
D 4
D(C
z
4 )
D(C x
2 )
|E x , |E y
0 −1
10
10
0 −1
D 5
D(C
z
5 )
D(C x
2 )
|E 1x , |E 1y
cos(2π/5) − sin(2π/5)
sin(2π/5) cos(2π/5)
10
0 −1
|E 2x , |E 2y
cos(4π/5) − sin(4π/5)
sin(4π/5) cos(4π/5)
10
0 −1
D 6
D(C
z
6 )
D(C x
2 )
|E 1x , |E 1y
⎛
⎝
1
2
−
√
3
2
+
√
3
2
1
2
⎞
⎠
10
0 −1
|E 2x , |E 2y
⎛
⎝
−
1
2
−
√
3
2
+
√
3
2
−
1
2
⎞
⎠
10
0 −1
A.J. Ceulemans, Group Theory Applied to Chemistry, Theoretical Chemistry and
Computational Modelling, DOI 10.1007/978-94-007-6863-5,
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