3.2 Model Spin Hamiltonians for Isotropic Interactions
73
Finally, the numerical effective Hamiltonian becomes:
|φ 1 φ 2 | φ 1 φ 2
φ 1 φ 2 | −29.440025 0.001726
φ 1 φ 2 | 0.001726 −29.440025
(3.38)
The two Hamiltonians (Eqs. 3.34 and 3.38) can only be compared when they have
the same zero of energy. Therefore the diagonal matrix elements of the Heisenberg Hamiltonian are shifted by −
1
4 J and those of the effective Hamiltonian
by −29.440025 E h . The comparison shows that there is a one-to-one correspondence between both matrices and that the magnetic coupling parameter is equal to
−2 × 0.001726 E h =− 757.6cm −1 . In this simple case, the eigenfunctions of the
Heisenberg Hamiltonian are the same as those of ˆ
S 2 ; Ψ 1 and Ψ 2 are directly the
singlet and triplet functions, respectively. Hence J is also given by the difference
of the singlet and triplet energies: J = E S − E T =− 29.441751 −−29.43830 =
−0.003452 E h =−757.6cm −1 .
However, the extraction strategy based on effective Hamiltonians is generally
applicable and in cases where the energies of the different spin states do not provide
enough information to determine all the magnetic coupling parameters one necessarily has to rely on the effective Hamiltonian procedure. The three different magnetic
coupling strengths J 12 , J 13 and J 23 of the three center/three electron case of Fig. 3.5
cannot be extracted from the two energy differences defined by the quartet and the
two doublets states. Instead an effective Hamiltonian has to be constructed from the
ab initio wave functions and compared to the matrix representation of the Heisenberg
Hamiltonian
ˆ
H =−J 12 ˆ
S 1 · ˆ
S 2 − J 13 ˆ
S 1 · ˆ
S 3 − J 23 ˆ
S 2 · ˆ
S 3
(3.39)
|φ 1 φ 2 φ 3 | φ 1 φ 2 φ 3 | φ 1 φ 2 φ 3
φ 1 φ 2 φ 3 |
1
4 (J 12 + J 13 − J 23 )
−
1
2 J 12
−
1
2 J 13
φ 1 φ 2 φ 3 |
−
1
2 J 12
1
4 (J 12 − J 13 + J 23 )
−
1
2 J 23
φ 1 φ 2 φ 3 |
−
1
2 J 13
−
1
2 J 23
1
4 (−J 12 + J 13 + J 23 )
(3.40)
The quartet spin function Q =| ααα is an eigenfunction of the Heisenberg
Hamiltonian of Eq. 3.39 with eigenvalue −
1
4 (J 12 + J 13 + J 23 ). However, the doublet
functions D 1 and D 2 defined in Eq. 1.41 are not:
ˆ
H
|ααβ−|βαα
=−
1
4
(J 12 − J 23 )
|ααβ+|βαα
+
3
4
J 13
|ααβ−|βαα
+
1
2
(J 12 − J 23 )|αβα (3.41)
73
Finally, the numerical effective Hamiltonian becomes:
|φ 1 φ 2 | φ 1 φ 2
φ 1 φ 2 | −29.440025 0.001726
φ 1 φ 2 | 0.001726 −29.440025
(3.38)
The two Hamiltonians (Eqs. 3.34 and 3.38) can only be compared when they have
the same zero of energy. Therefore the diagonal matrix elements of the Heisenberg Hamiltonian are shifted by −
1
4 J and those of the effective Hamiltonian
by −29.440025 E h . The comparison shows that there is a one-to-one correspondence between both matrices and that the magnetic coupling parameter is equal to
−2 × 0.001726 E h =− 757.6cm −1 . In this simple case, the eigenfunctions of the
Heisenberg Hamiltonian are the same as those of ˆ
S 2 ; Ψ 1 and Ψ 2 are directly the
singlet and triplet functions, respectively. Hence J is also given by the difference
of the singlet and triplet energies: J = E S − E T =− 29.441751 −−29.43830 =
−0.003452 E h =−757.6cm −1 .
However, the extraction strategy based on effective Hamiltonians is generally
applicable and in cases where the energies of the different spin states do not provide
enough information to determine all the magnetic coupling parameters one necessarily has to rely on the effective Hamiltonian procedure. The three different magnetic
coupling strengths J 12 , J 13 and J 23 of the three center/three electron case of Fig. 3.5
cannot be extracted from the two energy differences defined by the quartet and the
two doublets states. Instead an effective Hamiltonian has to be constructed from the
ab initio wave functions and compared to the matrix representation of the Heisenberg
Hamiltonian
ˆ
H =−J 12 ˆ
S 1 · ˆ
S 2 − J 13 ˆ
S 1 · ˆ
S 3 − J 23 ˆ
S 2 · ˆ
S 3
(3.39)
|φ 1 φ 2 φ 3 | φ 1 φ 2 φ 3 | φ 1 φ 2 φ 3
φ 1 φ 2 φ 3 |
1
4 (J 12 + J 13 − J 23 )
−
1
2 J 12
−
1
2 J 13
φ 1 φ 2 φ 3 |
−
1
2 J 12
1
4 (J 12 − J 13 + J 23 )
−
1
2 J 23
φ 1 φ 2 φ 3 |
−
1
2 J 13
−
1
2 J 23
1
4 (−J 12 + J 13 + J 23 )
(3.40)
The quartet spin function Q =| ααα is an eigenfunction of the Heisenberg
Hamiltonian of Eq. 3.39 with eigenvalue −
1
4 (J 12 + J 13 + J 23 ). However, the doublet
functions D 1 and D 2 defined in Eq. 1.41 are not:
ˆ
H
|ααβ−|βαα
=−
1
4
(J 12 − J 23 )
|ααβ+|βαα
+
3
4
J 13
|ααβ−|βαα
+
1
2
(J 12 − J 23 )|αβα (3.41)