H = H$_J$+ H$_\alpha$ + H$_D$ + H$_{J'}$ ............(1)
where
H$_J$ = $\frac{1}{2}$$\sum_{<ij>}^{}S_i S_j$ (here, $\alpha$ > 0) ............(2)
and
H$_\alpha$ = $\frac{1}{2}$$\sum_{<ij>}^{} (S_i S_j)^2$ ............(3)
The D and J' terms arise from spin-orbital interactions:
D term is the single-site anisotropy energy, specifically
H$_D$ = D$\sum_{<i>}^{}(S_i^z)^2$ ............(4)
J' term is
H$_{J'}$ = $\frac{1}{2}$$\sum_{<ij>}^{} J_{ij}^{uv} S_i^{u} S_j^{v}$ ............(5)
where J$_{ij}^{uv}$ are bond-dependent exchange. $<ij>$ runs over the nearest-neighbour bonds of the FCC lattice. For FCC lattice, the last term can be recast as:
H$_{J'}$ = $\sum_{unit}{}$ $ {H} $
In each unit cell (four spin sites), we have:
H = $\sum_{t}^{}\sum_{ij \in t}^{}S_i J^t S_j $
where t referes to a tetrahedron and $J^t$ is a set of six 3 $\times$ 3 matrices. In a FCC lattice, each unit cell has eight tetrahedra. They are called as 0, X, Y, Z, XY, YZ, XY, and XYZ.
For tetrahedron 0, the six 3 $\times$ 3 $J^0$ matrices are:
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