Pseudopotentials Ashcroft empty core

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. 5.15 An ionic lattice of Ashcroft empty core pseudopotentials immersed in a free-electron gas.

Table 6.3 Contributions to the binding energy (in Ry per atom) of sodium, magnesium, and aluminium within the second order real-space representation, eqn (6.73), using Ashcroft empty-core pseudopotentials. L/gf is defined by eqn (6.75). The numbers in brackets correspond to the simple expression, eqn (6.77), for = 0) and to the experimental values of the binding energy and negative cohesive energy respectively.

The wave vector, k , and the screening length, 1/ , depend only on the density of the free-electron gas through the poles of the approximated inverse dielectric response function, whereas the amplitude, A , and the phase shift, a , depend also on the nature of the ion-core pseudopotential through eqs (6.96) and (6.97). For the particular case of the Ashcroft empty-core pseudopotential, where tfj fa) = cos qRc, the modulus and phase are given explicitly by... 

The resultant pair potentials for sodium, magnesium, and aluminium are illustrated in Fig. 6.9 using Ashcroft empty-core pseudopotentials. We see that all three metals are characterized by a repulsive hard-core contribution, Q>i(R) (short-dashed curve), an attractive nearest-neighbour contribution, 2( ) (long-dashed curve), and an oscillatory long-range contribution, 3(R) (dotted curve). The appropriate values of the inter-atomic potential parameters A , oc , k , and k are listed in Table 6.4. We observe that the total pair potentials reflect the characteristic behaviour of the more accurate ab initio pair potentials in Fig. 6.7 that were evaluated using non-local pseudopotentials. We should note, however, that the values taken for the Ashcroft empty-core radii for Na, Mg, and Al, namely Rc = 1.66, 1.39, and... 

Fig. 6.17 The structural-energy differences of a model Cu-AI alloy as a function of the band filling N, using an average Ashcroft empty-core pseudopotential with / c = 1.18 au. The dashed curves correspond to the three-term analytic pair-potential approximation. The full curves correspond to the exact result that is obtained by correcting the difference between the Lindhard function and the rational polynomial approximation in Fig. 6.3 by a rapidly convergent summation over reciprocal space. (After Ward (1985).)...

Explain the concept of a pseudopotential. Aluminium is fee with a lattice constant of a = 7.7 au. It is well described by an Ashcroft empty core pseudopotential of core radius 1.1 au. Show that the lattice must be expanded by 14% for the 2n/a(200) Fourier component of the pseudopotential to vanish. 

This repulsive term tends to cancel the true potential of the core itself, a feature noted very early in the development of the pseudopotential theory that we have described. Ashcroft (1966) took advantage of this feature in proposing the empty-core model of the pseudopotential. In this model, the repulsive term of Eq. (15-12) is combined with the Coulomb potential of a point ion and the core potential of Eq. (15-8) to give a potential due to each ion, which is approximated by... 

The vibration spectrum for potassium calculated by Ashcroft (1968), who used the empty-core pseudopotential from Eq. (15-13) and = 1.13 A (rather than the value of 1.20 A from Table 16-1). Experimental points are from Cowley, Woods, and Dolling (1966). 

There have been a very large number of applications of this theory to the vibration spectra (see Heine and Weaire, 1970, for a review) since the earliest studies (Harrison, 1964, and Sham, 1965). Perhaps the most relevant here is the calculation by Ashcroft (1968) for the alkali metals that calculation makes use of the empty-core model of the pseudopotential, described here. The results of his calculation, along with experimental points, are shown in Fig. 17-4 for potassium similar results were obtained for sodium. 

---