90
3 Two (or More) Magnetic Centers
Table 3.3 Effect of ( ˆ
S 1 · ˆ
S 2 ) 2 on the determinants that form the quintet, triplet and singlet spin
functions of a binuclear system with S = 1
Operator
αααα
ααββ
ββαα
αβ αβ
αββ α
βαβα
βααβ
ˆ
S
+
1
ˆ
S
−
2
ˆ
S
+
1
ˆ
S
−
2
0
0
4ααββ
0
0
0
0
ˆ
S
+
1
ˆ
S
−
2
ˆ
S
−
1
ˆ
S
+
2
0
4ααββ
0
κ
κ
κ
κ
ˆ
S
−
1
ˆ
S
+
2
ˆ
S
+
1
ˆ
S
−
2
0
0
4ββαα
κ
κ
κ
κ
ˆ
S
−
1
ˆ
S
+
2
ˆ
S
−
1
ˆ
S
+
2
0
4ββαα
0
0
0
0
0
ˆ
S
+
1
ˆ
S
−
2
ˆ
S z,1 ˆ
S z,2
0
0
−κ
0
0
0
0
ˆ
S
−
1
ˆ
S
+
2
ˆ
S z,1 ˆ
S z,2
0
−κ
0
0
0
0
0
ˆ
S z,1 ˆ
S z,2 ˆ
S
+
1
ˆ
S
−
2
0
0
0
−ααββ −ααββ −ααββ −ααββ
ˆ
S z,1 ˆ
S z,2 ˆ
S
−
1
ˆ
S
+
2
0
0
0
−ββαα −ββαα −ββαα −ββαα
ˆ
S z,1 ˆ
S z,2 ˆ
S z,1 ˆ
S z,2 αααα
ααββ
ββαα
0
0
0
0
κ = αβ αβ + αββ α + βαβα + βααβ
λ( ˆ
S 1 ˆ
S 2 )
2 1
2
√
3
2(ααββ + ββαα) − αβαβ − αββα − βααβ − βαβα
=
λ
2
√
3
1
4
(8ααββ + 8ββαα) −
1
2
2κ + 2ααββ +
1
4
(8ααββ + 8ββαα)
−
1
2
2κ + 2ββαα − 4(
1
4
2κ −
1
2
ααββ −
1
2
ββαα)
=
λ
2
√
3
8(ααββ + ββαα) + 4(−αβαβ − αββα − βααβ − βαβα)
= 4λS
(3.74c)
and the eigenvalues of the Heisenberg Hamiltonian extended with a term for the
biquadratic exchange are
− J ˆ
S 1 · ˆ
S 2 + λ( ˆ
S 1 · ˆ
S 2 )
2
Q = (−J + λ)Q
(3.75a)
− J ˆ
S 1 · ˆ
S 2 + λ( ˆ
S 1 · ˆ
S 2 )
2
T = (J + λ)T
(3.75b)
− J ˆ
S 1 · ˆ
S 2 + λ( ˆ
S 1 · ˆ
S 2 )
2
S = (2J + 4λ)S
(3.75c)
3.4.2 Four-Center Interactions
Magnetic interactions are not restricted to the exchange of the spin moments on two
magnetic centers, but can be extended to the simultaneous interaction of three or
four magnetic centers. These interactions are in general smaller than the two-body
interactions but can not always be neglected. A clear example is given by the magnetic