102
3 Two (or More) Magnetic Centers
Ψ 1
Ψ 2
Ψ 3
Ψ 4
Ψ 5
|φ 1 φ 2 φ 3 |− 0.4426
−0.6583
0.5774
−0.1465
0.1135
|φ 1 φ 2 φ 3 |
0.7706
−0.0661
0.5774
0.0367
−0.2476
|φ 1 φ 2 φ 3 |− 0.3280
0.7243
0.5774
0.1098
0.1341
|φ 1 φ 1 φ 2 |
0.0102
0.0234
0.0000
−0.0440
0.0017
|φ 1 φ 1 φ 3 |− 0.0725
−0.0495
0.0000
0.1244
0.0341
|φ 1 φ 2 φ 2 |
0.2243
−0.1120
0.0000
0.7653
−0.5685
|φ 2 φ 2 φ 3 |
0.2017
0.1336
0.0000
−0.5805
−0.7636
|φ 1 φ 3 φ 3 |− 0.0789
0.0407
0.0000
−0.1472
0.0147
|φ 2 φ 3 φ 3 |
0.0076
0.0508
0.0000
0.0579
0.0127
The energies (in E h )a r eE 1 =− 27.9611962, E 2 =− 27.9601927, E 3 =
−27.9596947, E 4 =−27.8326257, E 5 =−27.83169141.
a. Determine the M S quantum numbers of the determinants and identify Ψ 3 as a
spin eigenfunction with S = 3/2.
b. Extract the J -values from the energies of the lowest three states under the assumption that J 12 = J 23 = J 13 (see Eq. 3.44).
c. Write down the determinants that span the model space of the Heisenberg Hamiltonian and determine the norm of the projections of Ψ k on this model space.
d. Select the three roots with the largest norm and orthogonalize the projections
Ψ k
e. Construct the 3 × 3 effective Hamiltonian and extract the different J -values by
comparing with the matrix elements of the Heisenberg Hamiltonian given in
Eq. 3.39.
3.5 Heisenberg twice. (a) Use the eigenvalues of Q, T and S for ˆ
H =−J ˆ
S 1 · ˆ
S 2
to compute the eigenvalues of Q, T and S for the operator ˆ
S 1 · ˆ
S 2 . (b) From this,
compute the eigenvalues of Q, T and S for the biquadratic operator ( ˆ
S 1 · ˆ
S 2 ) 2 and
check the validity of Eq. 3.75.
3.6 Biquadratic interactions: Do the following total energies follow the regular
spacing predicted by the Heisenberg Hamiltonian? E Q =−139.48992180 E h , E T =
−139.49305142 E h and E S =−139.49443101 E h . Calculate J and λ (in meV) from
the energy differences.
References
1. W. Heitler, F. London, Z. Phys. 44, 455 (1927)
2. B. Bleaney, K.D. Bowers, Proc. R. Soc. Lond. Ser. A 214, 451 (1952)
3. R. Boˇ ca, Theoretical Foundations of Molecular Magnetism (Elsevier, Amsterdam, 1999)
4. H. Bethe, Z. Phys. 71, 205 (1931)
5. J.C. Bonner, M.E. Fisher, Phys. Rev. 135(3A), A640 (1964)
6. J.W. Hall, W.E. Marsh, R.R. Weller, W.E. Hatfield, Inorg. Chem. 20, 1033 (1981)
7. S.Eggert,I.Affleck,M.T akahashi,Phys.Rev .Lett.73(2), 332 (1994)
8. R. Georges, J.J. Borrás-Almenar, E. Coronado, J. Curély, M. Drillon, in Magnetism: Molecules
to Materials, ed. by J.S. Miller, M. Drillon (Wiley-VCH, Weinheim, 2001), pp. 1–47, chap. 1
3 Two (or More) Magnetic Centers
Ψ 1
Ψ 2
Ψ 3
Ψ 4
Ψ 5
|φ 1 φ 2 φ 3 |− 0.4426
−0.6583
0.5774
−0.1465
0.1135
|φ 1 φ 2 φ 3 |
0.7706
−0.0661
0.5774
0.0367
−0.2476
|φ 1 φ 2 φ 3 |− 0.3280
0.7243
0.5774
0.1098
0.1341
|φ 1 φ 1 φ 2 |
0.0102
0.0234
0.0000
−0.0440
0.0017
|φ 1 φ 1 φ 3 |− 0.0725
−0.0495
0.0000
0.1244
0.0341
|φ 1 φ 2 φ 2 |
0.2243
−0.1120
0.0000
0.7653
−0.5685
|φ 2 φ 2 φ 3 |
0.2017
0.1336
0.0000
−0.5805
−0.7636
|φ 1 φ 3 φ 3 |− 0.0789
0.0407
0.0000
−0.1472
0.0147
|φ 2 φ 3 φ 3 |
0.0076
0.0508
0.0000
0.0579
0.0127
The energies (in E h )a r eE 1 =− 27.9611962, E 2 =− 27.9601927, E 3 =
−27.9596947, E 4 =−27.8326257, E 5 =−27.83169141.
a. Determine the M S quantum numbers of the determinants and identify Ψ 3 as a
spin eigenfunction with S = 3/2.
b. Extract the J -values from the energies of the lowest three states under the assumption that J 12 = J 23 = J 13 (see Eq. 3.44).
c. Write down the determinants that span the model space of the Heisenberg Hamiltonian and determine the norm of the projections of Ψ k on this model space.
d. Select the three roots with the largest norm and orthogonalize the projections
Ψ k
e. Construct the 3 × 3 effective Hamiltonian and extract the different J -values by
comparing with the matrix elements of the Heisenberg Hamiltonian given in
Eq. 3.39.
3.5 Heisenberg twice. (a) Use the eigenvalues of Q, T and S for ˆ
H =−J ˆ
S 1 · ˆ
S 2
to compute the eigenvalues of Q, T and S for the operator ˆ
S 1 · ˆ
S 2 . (b) From this,
compute the eigenvalues of Q, T and S for the biquadratic operator ( ˆ
S 1 · ˆ
S 2 ) 2 and
check the validity of Eq. 3.75.
3.6 Biquadratic interactions: Do the following total energies follow the regular
spacing predicted by the Heisenberg Hamiltonian? E Q =−139.48992180 E h , E T =
−139.49305142 E h and E S =−139.49443101 E h . Calculate J and λ (in meV) from
the energy differences.
References
1. W. Heitler, F. London, Z. Phys. 44, 455 (1927)
2. B. Bleaney, K.D. Bowers, Proc. R. Soc. Lond. Ser. A 214, 451 (1952)
3. R. Boˇ ca, Theoretical Foundations of Molecular Magnetism (Elsevier, Amsterdam, 1999)
4. H. Bethe, Z. Phys. 71, 205 (1931)
5. J.C. Bonner, M.E. Fisher, Phys. Rev. 135(3A), A640 (1964)
6. J.W. Hall, W.E. Marsh, R.R. Weller, W.E. Hatfield, Inorg. Chem. 20, 1033 (1981)
7. S.Eggert,I.Affleck,M.T akahashi,Phys.Rev .Lett.73(2), 332 (1994)
8. R. Georges, J.J. Borrás-Almenar, E. Coronado, J. Curély, M. Drillon, in Magnetism: Molecules
to Materials, ed. by J.S. Miller, M. Drillon (Wiley-VCH, Weinheim, 2001), pp. 1–47, chap. 1