Heine-Abarenkov pseudopotential

As is seen from the behaviour of the more sophisticated Heine-Abarenkov pseudopotential in Fig. 5.12, the first node q0 in aluminium lies just to the left of (2 / ) / and g = (2n/a)2, the magnitude of the reciprocal lattice vectors that determine the band gaps at L and X respectively. This explains both the positive value and the smallness of the Fourier component of the potential, which we deduced from the observed band gap in eqn (5.45). Taking the equilibrium lattice constant of aluminium to be a = 7.7 au and reading off from Fig. 5.12 that q0 at 0.8(4 / ), we find from eqn (5.57) that the Ashcroft empty core radius for aluminium is Re = 1.2 au. Thus, the ion core occupies only 6% of the bulk atomic volume. Nevertheless, we will find that its strong repulsive influence has a marked effect not only on the equilibrium bond length but also on the crystal structure adopted. 

Fig. 5.12 The Heine-Abarenkov (1964) pseudopotential for aluminium which has been normalized by the Fermi energy. The term q0 gives the position of the first node. The two large dots mark the zone boundary Fourier components, vDS(111) and (200).

The occurrence of a fairly well-defined molecular group TI2 Te or of the ionic assembly Tf, Tf, Te" cannot be approached in terms of perturbation theory because it involves the formation of bound states. Stem (1966) and others have shown how these can arise if the atomic potentials of the two species are very different. A rough measure of the differences in potentials can be obtained from published tables of electronegativities or stability ratios (Sanderson (I960)), or the well depths which characterise the Heine-Abarenkov-Animalu model potential (see, for instance, Animalu and Heine (1965)). The connection between pseudopotentials and electronegativity implied by this remark has recently been justified by Heine and Weaire (1970). 

As mentioned at the end of the previous section, the factors containing the energy can be replaced by an arbitrary expression of the Austin-Heine-Sham type. This fact can be further exploited, as is e.g. done in the Heine-Animalu pseudopotential, based on the Heine-Abarenkov derivation (see Ref. 26-27 of the previous chapter). Heine and Animalu determined the ionic contribution to the pseudopotential by imposing that it correctly describes the electron scattering from the atom. This potential is parametrized as... 

Figure 5.8 The pseudopotential proposed by Heine and Abarenkov that is a square-well model potential with value A inside a cutoff radius r .

The model pseudopotential of Heine and Abarenkov is presented in Figure 5.8. The distance rj is interpreted as the radius of the ion core. One may at first select Yi from a range of reasonable values. After a value for rj is chosen, it remains fixed throughout the rest of the calculations. The parameter A (Figure 5.8) is then varied to give the proper energy state. For each eigenvalue the parameter A/ has a different value, ri was foimd to lie between 1.5 and 3.0 atomic units. 

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