3.4 Complex Interactions
101
complicated when magnetic centers are considered with more than one unpaired
electron. Then the single-ion anisotropy discussed in Chap. 2 has to be included in
the model
ˆ
H =−J ˆ
S 1 · ˆ
S 2 + ˆ
S 1 D 1 ˆ
S 1 + ˆ
S 2 D 2 ˆ
S 2 + ˆ
S 1 D 12 ˆ
S 2 + d ˆ
S 1 × ˆ
S 2
(3.98)
and it has been shown that even biquadratic anisotropic interactions can play an
important role in the description of the low-energy physics of the complex [21]. The
corresponding operator is
ˆ
κ = ( ˆ
S 1 ˆ
S 1 )D aabb ( ˆ
S 2 ˆ
S 2 )
(3.99)
where D aabb is tensor of rank 4 with 81 (3 4 ) parameters. However by choosing the
proper magnetic axes frame this number is strongly reduced and when the system has
a certain degree of symmetry one can eventually characterize the tensor with not more
than nine parameters. Again one can resort to the numerical effective Hamiltonian
to determine these parameters.
Problems
3.1 Overlap: Demonstrate that c 1 /c 2 in Eq. 3.7 is equal to 1 − S ab /1 + S ab , where
S ab ==φ a |φ b and φ a and φ b are the orbitals of Eq. 3.17.
3.2 From delocalized to localized: Transform the following determinants and CSFs
from a delocalized to a localized orbital basis. Determine the percentage of ionic and
neutral character of the wave function. Are the wave functions eigenfunctions of ˆ
S 2 ?
a. Φ 1 =|g 1 g 1 |; Φ 2 =|g 1 g 2 |; Φ 3 =|g 1 u 1 |
b. Ψ 1 = (|g 1 g 1 |+|u 1 u 1 |)/
√
2; Ψ 2 = (|g 1 g 1 |−|u 1 u 1 |)/
√
2
c. Φ 4 =|g 1 u 1 |; Φ 5 =|g 1 u 1 v 1 |
d. Ψ 3 = (2|g 1 u 1 v 1 |−|g 1 u 1 v 1 |−|g 1 u 1 v 1 |)/
√
6
with g i =
1
√
2
(a i + b i ); u i =
1
√
2
(a i − b i ); v i = c i . a i , b i and c i are orbitals localized
on centers A, B and C, respectively.
3.3 Singlet and triplet eigenvalues: Calculate the eigenvalues of the Heisenberg
Hamiltonian given in Eq. 3.31 of Φ(T ) =|αα| and Φ(S) = (|αβ|−|βα|)/
√
2.
3.4 Extracting J -values for a three-center system: The following wave functions
Ψ k were obtained from an ab initio calculation on a system with three S = 1/2
magnetic centers. Each magnetic orbital φ i is localized on center i and has the same
spatial part in all five wave functions.
101
complicated when magnetic centers are considered with more than one unpaired
electron. Then the single-ion anisotropy discussed in Chap. 2 has to be included in
the model
ˆ
H =−J ˆ
S 1 · ˆ
S 2 + ˆ
S 1 D 1 ˆ
S 1 + ˆ
S 2 D 2 ˆ
S 2 + ˆ
S 1 D 12 ˆ
S 2 + d ˆ
S 1 × ˆ
S 2
(3.98)
and it has been shown that even biquadratic anisotropic interactions can play an
important role in the description of the low-energy physics of the complex [21]. The
corresponding operator is
ˆ
κ = ( ˆ
S 1 ˆ
S 1 )D aabb ( ˆ
S 2 ˆ
S 2 )
(3.99)
where D aabb is tensor of rank 4 with 81 (3 4 ) parameters. However by choosing the
proper magnetic axes frame this number is strongly reduced and when the system has
a certain degree of symmetry one can eventually characterize the tensor with not more
than nine parameters. Again one can resort to the numerical effective Hamiltonian
to determine these parameters.
Problems
3.1 Overlap: Demonstrate that c 1 /c 2 in Eq. 3.7 is equal to 1 − S ab /1 + S ab , where
S ab ==φ a |φ b and φ a and φ b are the orbitals of Eq. 3.17.
3.2 From delocalized to localized: Transform the following determinants and CSFs
from a delocalized to a localized orbital basis. Determine the percentage of ionic and
neutral character of the wave function. Are the wave functions eigenfunctions of ˆ
S 2 ?
a. Φ 1 =|g 1 g 1 |; Φ 2 =|g 1 g 2 |; Φ 3 =|g 1 u 1 |
b. Ψ 1 = (|g 1 g 1 |+|u 1 u 1 |)/
√
2; Ψ 2 = (|g 1 g 1 |−|u 1 u 1 |)/
√
2
c. Φ 4 =|g 1 u 1 |; Φ 5 =|g 1 u 1 v 1 |
d. Ψ 3 = (2|g 1 u 1 v 1 |−|g 1 u 1 v 1 |−|g 1 u 1 v 1 |)/
√
6
with g i =
1
√
2
(a i + b i ); u i =
1
√
2
(a i − b i ); v i = c i . a i , b i and c i are orbitals localized
on centers A, B and C, respectively.
3.3 Singlet and triplet eigenvalues: Calculate the eigenvalues of the Heisenberg
Hamiltonian given in Eq. 3.31 of Φ(T ) =|αα| and Φ(S) = (|αβ|−|βα|)/
√
2.
3.4 Extracting J -values for a three-center system: The following wave functions
Ψ k were obtained from an ab initio calculation on a system with three S = 1/2
magnetic centers. Each magnetic orbital φ i is localized on center i and has the same
spatial part in all five wave functions.