5.3 Analysis with Single Determinant Methods
159
The sum of the three contributions is close to the magnetic coupling constant that
is obtained in a standard calculation when the Kohn-Sham orbitals are optimized
without imposing any restriction on the variational process.
J ≈ J DE + J KE + J SP
(5.28)
There is no ap r i o r ireason to determine the different contributions in this order.
Alternatively, the spin polarization can be calculated before relaxing the magnetic
orbitals (inverting step 2 and 3) or independently, both taking the orbitals of step 1 as
starting point. However, the examples given in Ref. [5] show that the order chosen
here gives the smallest deviation from the fully relaxed energy difference, and hence,
includes the largest part of the physics.
5.4 Analysis of Complex Interactions
The analysis of the interaction between magnetic moments is not restricted to the
isotropic bilinear exchange of two S = 1/2 centers but can also be applied to systems
with higher spins and more magnetic centers. In this section, we will first decompose
the magnetic coupling between two Ni 2+ (S = 1) ions with a sizeable biquadratic
exchange to pinpoint the origin of the deviations to the standard Heisenberg Hamiltonian. Secondly, we will focus attention on the four-spin cyclic exchange, and finally,
we will describe how these interactions can be estimated within the DFT framework.
5.4.1 Decomposition of the Biquadratic Exchange
One of the central assumptions of the Heisenberg model Hamiltonian is that the local
spin states are well separated in energy from excited spin states. We will demonstrate that non-Heisenberg behavior emerges as soon as this is no longer true. The
biquadratic exchange is in general a rather small term in the total interaction of the
spins in polynuclear TM-3d complexes. There are only a few examples where it
is important to include them for obtaining an accurate description of the lowestenergy levels. Returning to the binuclear complexes with a double azido bridge,
we will here analyze the magnetic coupling of the Ni 2+ model complex shown in
Fig. 5.12. The interaction of the spins strongly depends on the δ-angle and ranges from
approximately 100 cm −1 for δ = 0 ◦ to nearly zero for δ = 45 ◦ , which is accurately
reproduced with DDCI [6]. More interestingly, the singlet, triplet and quintet DDCI
energies do not strictly follow the expected Landé pattern, especially for small δ.
Deviations up to 3 % are observed and we will use the corresponding wave functions
to analyze the origin of the deviations to the standard Heisenberg spacing.
The magnetic orbitals are expressed again in orthogonal atomic-like orbitals,
denoted ϕ 1 , ϕ 2 for the two magnetic orbitals on site A and ϕ 3 , ϕ 4 for the orbitals on
159
The sum of the three contributions is close to the magnetic coupling constant that
is obtained in a standard calculation when the Kohn-Sham orbitals are optimized
without imposing any restriction on the variational process.
J ≈ J DE + J KE + J SP
(5.28)
There is no ap r i o r ireason to determine the different contributions in this order.
Alternatively, the spin polarization can be calculated before relaxing the magnetic
orbitals (inverting step 2 and 3) or independently, both taking the orbitals of step 1 as
starting point. However, the examples given in Ref. [5] show that the order chosen
here gives the smallest deviation from the fully relaxed energy difference, and hence,
includes the largest part of the physics.
5.4 Analysis of Complex Interactions
The analysis of the interaction between magnetic moments is not restricted to the
isotropic bilinear exchange of two S = 1/2 centers but can also be applied to systems
with higher spins and more magnetic centers. In this section, we will first decompose
the magnetic coupling between two Ni 2+ (S = 1) ions with a sizeable biquadratic
exchange to pinpoint the origin of the deviations to the standard Heisenberg Hamiltonian. Secondly, we will focus attention on the four-spin cyclic exchange, and finally,
we will describe how these interactions can be estimated within the DFT framework.
5.4.1 Decomposition of the Biquadratic Exchange
One of the central assumptions of the Heisenberg model Hamiltonian is that the local
spin states are well separated in energy from excited spin states. We will demonstrate that non-Heisenberg behavior emerges as soon as this is no longer true. The
biquadratic exchange is in general a rather small term in the total interaction of the
spins in polynuclear TM-3d complexes. There are only a few examples where it
is important to include them for obtaining an accurate description of the lowestenergy levels. Returning to the binuclear complexes with a double azido bridge,
we will here analyze the magnetic coupling of the Ni 2+ model complex shown in
Fig. 5.12. The interaction of the spins strongly depends on the δ-angle and ranges from
approximately 100 cm −1 for δ = 0 ◦ to nearly zero for δ = 45 ◦ , which is accurately
reproduced with DDCI [6]. More interestingly, the singlet, triplet and quintet DDCI
energies do not strictly follow the expected Landé pattern, especially for small δ.
Deviations up to 3 % are observed and we will use the corresponding wave functions
to analyze the origin of the deviations to the standard Heisenberg spacing.
The magnetic orbitals are expressed again in orthogonal atomic-like orbitals,
denoted ϕ 1 , ϕ 2 for the two magnetic orbitals on site A and ϕ 3 , ϕ 4 for the orbitals on