6.3 A Quantum Chemical Approach to Magnetic Interactions in the Solid State
193
6.3.2 Periodic Calculations
Magnetic interactions in extended systems can also be studied without creating the
more or less approximate representation of the material with an embedded cluster.
The approach based on the translational symmetry in the crystal naturally leads to the
well-known band structures of the Bloch functions, periodic one-electron functions.
ψ k (r) =
r ′
e
ik·r ′
φ(r
′ )
(6.38)
The difficulty of constructing spin eigenfunctions with S < S max for extended systems with unpaired electrons makes that most of the periodic calculations are performed within a single determinant method and no restrictions on the spatial part of
the spin orbitals. The results are then necessarily interpreted with the Ising model
Hamiltonian described in Sect. 3.2.2. In practice, the total energy of the magnetic
unit cell (not necessarily of the same size as the structural unit cell) is calculated for
different spin orientations (that is, different M S values) and the relative energies are
compared to the matrix elements of the Ising Hamiltonian to determine the magnetic
coupling strength between the ions in the crystal. This is not necessarily limited to
isotropic bilinear coupling but can also be used to extract estimates for biquadratic
and four-center interactions.
To illustrate the procedure of extracting magnetic coupling parameters by periodic calculations, we will focus on the perovskite structure Sr 2 CuO 3 , related to the
previously used spin ladder compound SrCu 2 O 3 , although the structural motif here
is formed by CuO 4 units arranged in linear chains along the b-axis of the unit cell.
Figure 6.8 illustrates the structure of this oxide and indicates the unit cell with a
dashed box. The unit cell has two symmetry inequivalent Cu 2+ ions with an S =
1
2
spin moment each.
In the first place, we calculate the energy per unit cell with all spins aligned ferromagnetically as schematically depicted in the left panel of Fig. 6.9. This calculation
can be done within any spin unrestricted periodic computationally scheme, either
HF or DFT and it gives us E F (a, b, c). Subsequently, this energy has to be expressed
as an Ising energy. The Ising Hamiltonian for this compound is defined as
ˆ
H =−J a
i,j
ˆ
S z,i ˆ
S z,j − J b
k,l
ˆ
S z,k ˆ
S z,l − J d
m,n
ˆ
S z,m ˆ
S z,n
(6.39)
where the interaction along the c-direction is neglected and J d is the interaction along
the body diagonal of the unit cell. As can be seen in the left panel of Fig. 6.9, the unit
cell contains 8 times the interaction along the body diagonal. The four vertices along
a and b represent the J a and J b interactions, but each of these have to be counted
only for 1/4 since the vertices are shared by four unit cells. Furthermore, the copper
ion in the center of the unit cell interact with the copper ions in the adjacent unit
cells and this contributes two times 1/2J a and two times 1/2J b to the Ising energy.
In total the energy expression becomes
193
6.3.2 Periodic Calculations
Magnetic interactions in extended systems can also be studied without creating the
more or less approximate representation of the material with an embedded cluster.
The approach based on the translational symmetry in the crystal naturally leads to the
well-known band structures of the Bloch functions, periodic one-electron functions.
ψ k (r) =
r ′
e
ik·r ′
φ(r
′ )
(6.38)
The difficulty of constructing spin eigenfunctions with S < S max for extended systems with unpaired electrons makes that most of the periodic calculations are performed within a single determinant method and no restrictions on the spatial part of
the spin orbitals. The results are then necessarily interpreted with the Ising model
Hamiltonian described in Sect. 3.2.2. In practice, the total energy of the magnetic
unit cell (not necessarily of the same size as the structural unit cell) is calculated for
different spin orientations (that is, different M S values) and the relative energies are
compared to the matrix elements of the Ising Hamiltonian to determine the magnetic
coupling strength between the ions in the crystal. This is not necessarily limited to
isotropic bilinear coupling but can also be used to extract estimates for biquadratic
and four-center interactions.
To illustrate the procedure of extracting magnetic coupling parameters by periodic calculations, we will focus on the perovskite structure Sr 2 CuO 3 , related to the
previously used spin ladder compound SrCu 2 O 3 , although the structural motif here
is formed by CuO 4 units arranged in linear chains along the b-axis of the unit cell.
Figure 6.8 illustrates the structure of this oxide and indicates the unit cell with a
dashed box. The unit cell has two symmetry inequivalent Cu 2+ ions with an S =
1
2
spin moment each.
In the first place, we calculate the energy per unit cell with all spins aligned ferromagnetically as schematically depicted in the left panel of Fig. 6.9. This calculation
can be done within any spin unrestricted periodic computationally scheme, either
HF or DFT and it gives us E F (a, b, c). Subsequently, this energy has to be expressed
as an Ising energy. The Ising Hamiltonian for this compound is defined as
ˆ
H =−J a
i,j
ˆ
S z,i ˆ
S z,j − J b
k,l
ˆ
S z,k ˆ
S z,l − J d
m,n
ˆ
S z,m ˆ
S z,n
(6.39)
where the interaction along the c-direction is neglected and J d is the interaction along
the body diagonal of the unit cell. As can be seen in the left panel of Fig. 6.9, the unit
cell contains 8 times the interaction along the body diagonal. The four vertices along
a and b represent the J a and J b interactions, but each of these have to be counted
only for 1/4 since the vertices are shared by four unit cells. Furthermore, the copper
ion in the center of the unit cell interact with the copper ions in the adjacent unit
cells and this contributes two times 1/2J a and two times 1/2J b to the Ising energy.
In total the energy expression becomes