6.3 A Quantum Chemical Approach to Magnetic Interactions in the Solid State
195
diagonal, since ˆ
S z,m and ˆ
S z,n result now in
1
2 and −
1
2 . With this, the expression is
readily written down as
E AF (a, b, c) = 2J d −
1
2
J a −
1
2
J b
(6.41)
and from the energy difference of the two calculations, we can determine J d
E AF (a, b, c) − E F (a, b, c) = 4J d
(6.42)
The calculation of J b (and J a ) cannot be done with the simple unit cell, no other spin
flips can be made. Therefore, we double the unit cell in the b direction to obtain a
new magnetic unit cell, the super cell (a, 2b, c), represented in Fig. 6.10. The energy
of the fully ferromagnetic supercell is in principle exactly twice E F (a, b, c),b u ti t
is highly recommendable to repeat the HF or DFT calculation for this double unit
cell due to numerical precision issues. Subsequently, we flip the spins on the copper
ions in the middle of the cell to obtain a spin arrangement with antiferromagnetic
ordering along the b-axis. A careful analysis of the interactions contained in these
two magnetic unit cells gives the energy expressions of the Ising Hamiltonian
E F (a, 2b, c) = 2E F (a, b, c) =−4J d − J a − J b
(6.43)
E AF (a, 2b, c) =−J a + J b
(6.44)
Fig. 6.10 Magnetic unit cell
obtained by doubling the
simple unit cell along the b
direction. Symmetry
equivalent copper ions have
spins with the same gray
scale. The antiferromagnetic
unit cell is obtained by
flipping the dark gray spins
at the lattice positions (0,1,0)
and (
1
2 ,
3
2 ,
1
2 )