Empty-core model

Core radii for the empty-core model, in A. These are listed also in the Solid State Table. 

There have been a very large number of applications of this theory to the vibration spectra (see Heine and Weairc, 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 pseudopolential, 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. 

There the relation was made in terms of the splitting at F rather than X, since the corresponding formulae are simpler.) The first comparison we make is between the LCAO values and the empty-core pseudopotential. We shall find only qualitative correspondence between the values because of errors in the empty-core model, which become large here. We shall then go on to consider other properties, using pseudopotential matrix elements obtained without resort to the empty-core model. 

The trends are the same, but the magnitudes differ by a factor of about two. This discrepancy arises from the inaccuracy of the empty-core model in fitting the pseudopotential in this range, as can be seen by considering the third column in Table 18-1, where Empirical Pseudopotential Method matrix elements are listed directly. These are in fact taken from the same calculated pseudopotential which was fitted to the empty-core model to obtain the values in the second column. A look at Fig. 16-1, where an accurate pseudopotential is plotted along with the empty-core fit indicates that the region near qfk = 1.108 (corresponding to the... 

It will be useful later to notice that the core states arc eigenstates of the Hamiltonian, so that // 11> can be replaced by c>, but we can most clearly see the appropriateness of the empty-core model of the pseudopotential by leaving /f as a kinetic and potential energy. Wc have seen that the pscudopotential always enters our calculations in a matrix element between plane waves, so wc write such a matrix clement as... 

The set of transformations made in Eqs. (D-5)-(D-7) rationalize the empty-core model. If one wishes to do better than that model, it is best to return to Eq. (D-3) and take advantage of the fact that the cores arc eigenstates, to write... 

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... 

We may also compare the polar energies. This is particularly simple in the context of the empty-core model. Then, in gallium arsenide, for example, we may expect the unscreened pseudopotential (both Z and r ) for both gallium and arsenic to be the same as in each pure material. The screening depends only upon the Fermi wave number, and since the bond length does not change appreciably in an isoelectronic series, the entire denominator should be the same for... 

Figure 5.7 The Ashkroft pseudopotential. This is an empty core model potential, in which the potential is zero inside a radius / c(0 which is different for each /. / is the azimuthal quantum number.

Figure S. Vibration spectrum of K with the empty-core model for rc = 1.13A [Ref. 25],...

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