The two-band model projects the magnetism onto each atom. It does not properly describe magnetic interactions, so it cannot distinguish between ferro, para, and antiferromagnetism. In order to do so, we need to include an interionic exchange term.

Pauli repulsion arises from electron eigenstates being orthogonal. While its nature on a single atom is complex, its interatomic effects can be modeled as a pairwise effect of repulsion between electrons of similar spins. The secondary effect of magnetization is that there are more electrons in one band than in another, and more same-spin electron pairs to repel one another, and so the repulsion between those bands is enhanced.

Conceptually, this can be captured in two pairwise effects, the standard nonmagnetic screened-Coulomb repulsion of the ions plus the core-core repulsion, and an additional Si dependent term arising from Pauli repulsion between like-spin electrons.

V (r, j) = Vo(rt,) + (S" Sj" + S# S#) Vm(rj) [21]

Note that an antiferromagnetic state with Si + Sj = 0 would have a lower repulsive energy. In the tight — binding picture, this would be compensated by a much reduced hopping integral, and hence, lower W. If we insist on Si > 0, then we suppress these solutions and can model ferromagnetic or diamagnetic iron. Also, as with DFT-GGA/LDA, the spin is Ising-like.

At the time of writing, no good parameterization of this type of potential exists. The difficulty is that determining the spins Sj is a nonlocal process: the optimal value ofthe spin on site i depends on the spin at site j. The only practical way to proceed appears to be to treat the spins as dynamical variables, in which case it is probably better to treat them as noncollinear Heisenberg moments.