Appendix B
Effect of the ˆ
S Operator and the Matrix
Elements for
1
2 ≤ S ≤
5
2
|S, M S
ˆ
S x |S, M S
ˆ
S y |S, M S
ˆ
S z |S, M S
| 1
2 , 1
2
1
2 | 1
2 , − 1
2
i
2 | 1
2 , − 1
2
1
2 | 1
2 , 1
2
| 1
2 , − 1
2
1
2 | 1
2 , 1
2
−i
2 | 1
2 , 1
2 − 1
2 | 1
2 , − 1
2
|1, 1
√
2
2 |1, 0
−i
√
2
2 |1, 0| 1, 1
|1, 0
√
2
2
|1, 1+|1, −1
i
√
2
2
|1, 1−|1, −1
0
|1, −1
√
2
2 |1, 0
i
√
2
2 |1, 0− | 1, −1
| 3
2 , 3
2
√
3
2 | 3
2 , 1
2
−i
√
3
2 | 3
2 , 1
2
3
2 | 3
2 , 3
2
| 3
2 , 1
2
√
3
2 | 3
2 , 3
2 +| 3
2 , − 1
2
i
√
3
2 | 3
2 , 3
2 −i| 3
2 , − 1
2
1
2 | 3
2 , 1
2
| 3
2 , − 1
2 | 3
2 , 1
2 +
√
3
2 | 3
2 , − 3
2
i| 3
2 , 1
2 − i
√
3
2 | 3
2 , − 3
2 − 1
2 | 3
2 , − 1
2
| 3
2 , − 3
2
√
3
2 | 3
2 , − 1
2
i
√
3
2 | 3
2 , − 1
2 − 3
2 | 3
2 , − 3
2
|2, 2| 2, 1− i|2, 1
2|2, 2
|2, 1| 2, 2+
√
6
2 |2, 0
i|2, 2− i
√
6
2 |2, 0| 2, 1
|2, 0
√
6
2
|2, 1+|2, −1
i
√
6
2
|2, 1−|2, −1
0
|2, −1
√
6
2 |2, 0+|2, −2
i
√
6
2 |2, 0−i|2, −2− | 2, −1
|2, −2| 2, −1− i|2, −1− 2|2, −2
© Springer International Publishing Switzerland 2016
C. Graaf and R. Broer, Magnetic Interactions in Molecules and Solids,
Theoretical Chemistry and Computational Modelling,
DOI 10.1007/978-3-319-22951-5
215