3.2 Model Spin Hamiltonians for Isotropic Interactions
71
Fig. 3.5 Three center
S = 1/2 system with a
quartet (Q) and two doublets
(D 1 , D 2 ). The three J -values
cannot be determined from
the two energy differences
Ψ S = 0.6776|φ 1 ¯
φ 2 |−0.6776| ¯
φ 1 φ 2 |+0.1287|φ 1 ¯
φ 1 |+0.1287|φ 2 ¯
φ 2 |+...
Ψ T = 0.7029|φ 1 ¯
φ 2 |+0.7029| ¯
φ 1 φ 2 |+...
E S =−29.441750 E h
E T =−29.438299 E h
Here, φ 1 and φ 2 are basis functions localized on site 1 and 2, respectively. The
model space of the Heisenberg Hamiltonian is spanned by the M S = 0 determinants
|φ 1 ¯
φ 2 | and | ¯
φ 1 φ 2 | and the matrix representation of the Hamiltonian can be obtained
by calculating the matrix elements φ 1 ¯
φ 2 | ˆ
H |φ 1 ¯
φ 2 and φ 1 ¯
φ 2 | ˆ
H | ¯
φ 1 φ 2 . For this
purpose, we first substitute the ˆ
S x and ˆ
S y operators by the ladder operators ˆ
S ±
ˆ
H =−J
1
2
ˆ
S
+
1
ˆ
S
−
2 + ˆ
S
−
1
ˆ
S
+
2
+ ˆ
S z,1 ˆ
S z,2
(3.31)
The action of the different operators on the model space determinants gives:
ˆ
S
+
1
ˆ
S
−
2 |φ 1 ¯
φ 2 |=0
ˆ
S
+
1
ˆ
S
−
2 | ¯
φ 1 φ 2 |=|φ 1 ¯
φ 2 |
ˆ
S
−
1
ˆ
S
+
2 |φ 1 ¯
φ 2 |=| ¯
φ 1 φ 2 |
ˆ
S
−
1
ˆ
S
+
2 | ¯
φ 1 φ 2 |=0
ˆ
S z,1 ˆ
S z,2 |φ 1 ¯
φ 2 |=−
1
4
|φ 1 ¯
φ 2 |
ˆ
S z,1 ˆ
S z,2 | ¯
φ 1 φ 2 |=−
1
4
| ¯
φ 1 φ 2 |
(3.32)
and from this the matrix elements can directly be written down:
φ 1 ¯
φ 2 | ˆ
H |φ 1 ¯
φ 2 =−J φ 1 ¯
φ 2 |
0 +
1
2
| ¯
φ 1 φ 2 −
1
4
|φ 1 ¯
φ 2
=
1
4
J
φ 1 ¯
φ 2 | ˆ
H | ¯
φ 1 φ 2 =−J φ 1 ¯
φ 2 |
0 +
1
2
|φ 1 ¯
φ 2 +0
=−
1
2
J
(3.33)
In matrix form:
|φ 1 φ 2 | φ 1 φ 2
φ 1 φ 2 |
1
4 J
−
1
2 J
φ 1 φ 2 |
−
1
2 J
1
4 J
(3.34)