Appendix C
Matrix Representation of the ZFS Model
Hamiltonian
ˆ
SD ˆ
S in an arbitrary axis frame. The simpler form of the Hamiltonian that applies
when the system is oriented along the magnetic axis frame is easily derived by
putting all D ij to zero for i = j, making the trace equal to zero and substituting
D 33 −
1
2 (D 11 + D 22 ) by D and
1
2 (D 11 − D 22 ) by E.
S = 1
|1, 1| 1, 0| 1, −1
1, 1|
1
2 (D 11 + D 22 + D 33 )
−
√
2
2 (D 13 + iD 23 )
1
2 (D 11 − D 22 + 2iD
12
)
1, 0|
−
√
2
2 (D 13 − iD 23 )
D 11 + D 22
√
2
2 (D 13 + iD
23
)
1, −1|
1
2 (D 11 − D 22 − 2iD 12 )
√
2
2 (D 13 − iD 23 )
1
2 (D 11 + D 22 + D 33 )
S = 3
2 | 3
2 , 3
2 | 3
2 , 1
2 | 3
2 , − 1
2 | 3
2 , − 3
2
3
2 , 3
2 | 3
4 (D 11 + D 22 +3D 33 )
−
√
3(D 13 + iD 23 )
√
3
2 (D 11 − D 22 + 2iD 12 )
0
3
2 , 1
2 | −
√
3(D 13 − iD 23 )
1
4
7(D 11 + D 22 ) + D 33
0
√
3
2 (D 11 − D 22 + 2iD 12 )
3
2 , − 1
2 |
√
3
2 (D 11 − D 22 − 2iD 12 )
0
1
4
7(D 11 + D 22 ) + D 33
√
3(D 13 + iD 23 )
3
2 , − 3
2 |
0
√
3
2 (D 11 − D 22 − 2iD 12 )
√
3(D 13 − iD 23 )
3
4 (D 11 + D 22 + 3D 33 )
S = 2
|2, 2| 2, 1| 2, 0
2, 2|
D 11 + D 22 + 4D 33
3(D 13 + iD 23 )
√
6
2 (D 11 − D 22 + 2iD 12 )
2, 1|
3(D 13 − iD 23 )
5
2 (D 11 + D 22 ) + D 33
−
√
6
2 (D 13 + iD 23 )
2, 0|
√
6
2 (D 11 − D 22 − 2iD 12 )
−
√
6
2 (D 13 − iD 23 )
3(D 11 + D 22 )
2, −1|
0
3
2 (D 11 − D 22 − 2iD 12 )
√
6
2 (D 13 − iD 23 )
2, −2|
00
√
6
2 (D 11 − D 22 − 2iD 12 )
© 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
217