The two-band approach can be applied to magnetic materials, where the bands spin up and spin down bands have the same capacity (N = nJ = N# = 5). If in addition we assume that the bands have the same width and shape (see, e. g., Figure 11), there is a remarkable collapse of the model onto the single­band EAM form, with a modified embedding function.

The formalism here extends to the two-band model, but the physics is analogous to other magnetic potentials.27 For simplicity, consider a rectangular d-band of full width W centered on E0. The bond energy for a single spin-up band relative to the free atom is given by •E = Z/N —1 w

2 NE/ W dE

— W/2

[16]

where Z" is the occupation of the band and uparrows denote ‘spin up.’

To describe the ferromagnetic case, it is assumed that there are two independent d-bands corresponding to opposite spins, and that these can be projected onto an atom to form a local density of states. For a free atom atomic case, Hund’s rules determine a high-spin case (e. g., S = 2 for iron), and there is an energy Ux associated with transferring an electron to a lower spin state. In the solid, the simplest method is to set Ux to be proportional to the spin with the coefficient of proportionality being an adjustable parameter, E0.

Ux = — EoZ"- Z#I [17]

Defining the spin, S = Z" — Z# and assuming charge neutrality (T = (Z" + Z#)), the two-d-band binding energy on a site i is then as follows:

Ui = u" + U# + ux = + WN (T2 + S2) — TWi/2 — EoSi

[18]

Differentiating this equation about Si gives us the optimal value for the magnetization of a given atom of Si = 2NE0/Wj, and the many-body energy of an atom with T = 6, N = 5 (suppressing the i label) as

U =—6W/5 — 5E2/W E0/Wi < 0.4 = —2W/5 — 4E0 E0/ Wi > 0.4 [19]

where neither band is allowed to have occupancy more than 5 or less than 0. For a material with T d-electrons (where T > 5), transfer of electrons between the spin bands becomes advantageous for W > 10E0/(10 — T). For smaller W, the spin " band is full and the energy is simply proportional to the bandwidth of the # band as in the FS model. Similar cases apply to the T< 5 case when the minority band may be empty.

There has been some controversy about the expression for Ux. In the two-band model, this is a promotion energy from the minority spin band to the majority. In the atomic case, it is the energy to violate Hund’s rule, and the implicit reference state is the high-spin atom. Electron transfer is bound by the number ofelectron, so the function has discontinuous slope at Si = 0.4. By contrast, the approach of Dudarev and coworkers27 uses a Stoner model for the spin energy, which introduces quadratic and quartic terms in Ux(S,). In that case, the implicit reference state is the nonmagnetic solid, and any value of Si is acceptable.

Within the second-moment model, the bandwidth W is given by the square root of p, the sum of the squares of the hopping integral. Applying this, and the usual pairwise repulsion V(r), gives an expres­sion for the two-band energy

и = EV (rj) — ^pj

/

— Б/ yfpjH(2W — 5Eo)

— 4E0H(5E0 — 2 W) [20]

where H is the Heaviside step function, B is a con­stant, and the zero of energy corresponds to the nonmagnetic atom.

Note that this form does not explicitly include S, and that it has the EAM form with an embedding function F(x) = v/x(1 — B/x).

Although this model incorporates magnetism and provides a way to calculate the magnetic moment at each site, it is possible to use it without actually calculating S. The additional many-body repulsive term is similar to, for example, the many-body potential method of Mendelev et al. It is also inter­esting that in the original FS paper, it was not possi­ble to fit the properties of the magnetic elements Fe and Cr; an extra term was added ad hoc. Later para — meterizations of FS potentials for iron with a pure square root for F have not exactly reproduced the elastic constants.28

The implication of this work is that for second — moment type models, there should be a one-to — one relation between the local density p and the magnetic moment. Figure 12 shows this relation for two parameterizations, p from Dudarev-Derlet and Mendelev et al. and the magnetic moment cal­culated with spin-dependent DFT projected onto atoms. It shows that there are two cases. For atoms associated with local defects, the density varies quite sharply with p, while for the crystal under pressure the variation is slower. It is noteworthy that the same broad features are present in both potentials, even though the Mendelev et al. potential was fitted with­out consideration of magnetic properties, albeit with a FS-type embedding function. This suggests that the magnetic effects were unwittingly captured in the fitting process.