70
3 Two (or More) Magnetic Centers
E(S) =−
1
2
J
S(S + 1) − S 1 (S 1 + 1) − S 2 (S 2 + 1)
(3.27)
3.6 Calculate the eigenvalues of the Heisenberg Hamiltonian of the spin eigenfunctions with maximum and minimum spin moment of a dimeric system with
S 1 = S 2 .
Since the reference point of energy can be chosen arbitrarily, the expression for the
eigenvalues is usually simplified by adding a constant factor equal to −
1
2 J
S 1 (S 1 +
1) + S 2 (S 2 + 1)
, leading to
E(S) =−
1
2
JS(S + 1)
(3.28)
From this it is easily derived that the difference between two subsequent eigenvalues
is given by
E(S − 1) − E(S) = JS
(3.29)
where S runs from S 1 + S 2 to |S 1 − S 2 |. This regular Landé pattern gives the exact
energy differences as long as the interaction with excited electronic configurations
is negligible and is the basis for extracting magnetic coupling parameters from electronic structure calculations. The generalization of the Heisenberg Hamiltonian to
multiple sites is straightforward
ˆ
H =
i> j
−J ij ˆ
S i ˆ
S j
(3.30)
In systems with multiple magnetic sites, the number of energy differences between
the different spin functions is not always enough to determine all the J -values. An
obvious example is the three-center/three-electron case as depicted in Fig. 3.5.The
Hamiltonian has three different J -values while the quartet and the two doublet states
only define two energy differences. The effective Hamiltonian theory described in
Chap. 1 is a more general approach to extract J -values, because it not only uses
the energies but also information contained in the wave function. To illustrate the
procedure we first treat a simple biradical model with two S =
1
2 spins and after that
focus on the three center problem.
The following results for the singlet and triplet states were obtained from an ab
initio calculation on a dimer with two S =
1
2 centers:
3 Two (or More) Magnetic Centers
E(S) =−
1
2
J
S(S + 1) − S 1 (S 1 + 1) − S 2 (S 2 + 1)
(3.27)
3.6 Calculate the eigenvalues of the Heisenberg Hamiltonian of the spin eigenfunctions with maximum and minimum spin moment of a dimeric system with
S 1 = S 2 .
Since the reference point of energy can be chosen arbitrarily, the expression for the
eigenvalues is usually simplified by adding a constant factor equal to −
1
2 J
S 1 (S 1 +
1) + S 2 (S 2 + 1)
, leading to
E(S) =−
1
2
JS(S + 1)
(3.28)
From this it is easily derived that the difference between two subsequent eigenvalues
is given by
E(S − 1) − E(S) = JS
(3.29)
where S runs from S 1 + S 2 to |S 1 − S 2 |. This regular Landé pattern gives the exact
energy differences as long as the interaction with excited electronic configurations
is negligible and is the basis for extracting magnetic coupling parameters from electronic structure calculations. The generalization of the Heisenberg Hamiltonian to
multiple sites is straightforward
ˆ
H =
i> j
−J ij ˆ
S i ˆ
S j
(3.30)
In systems with multiple magnetic sites, the number of energy differences between
the different spin functions is not always enough to determine all the J -values. An
obvious example is the three-center/three-electron case as depicted in Fig. 3.5.The
Hamiltonian has three different J -values while the quartet and the two doublet states
only define two energy differences. The effective Hamiltonian theory described in
Chap. 1 is a more general approach to extract J -values, because it not only uses
the energies but also information contained in the wave function. To illustrate the
procedure we first treat a simple biradical model with two S =
1
2 spins and after that
focus on the three center problem.
The following results for the singlet and triplet states were obtained from an ab
initio calculation on a dimer with two S =
1
2 centers: