The Empty-Core Pseudopotential

The free-clectron approximation described in Chapter 15 is so successful that it is natural to expect that any effects of the pseudopotential can be treated as small perturbations, and this turns out to be true for the simple metals. This is only possible, however, if it is the pscudopotential, not the true potential, which is treated as the perturbation. If we were to start with a frcc-electron gas and slowly introduce the true potential, states of negative energy would occur, becoming finally the tightly bound core states these arc drastic modifications of the electron gas. If, however, we start with the valence-electron gas and introduce the pscudopotential, the core states arc already there, and full, and the effects of the pseudopotential arc small, as would be suggested by the small magnitude of the empty-core pseudopotential shown in Fig. 15-3. 

It should be noticed that curves like those shown in Fig. 16-1 arcdclined only for q 0. The q = 0 value is best obtained by combining all of the terms in the energy, as we did in the discussion in Chapter 15. Notice, however, that in the discussion of cohesion in Chapter 15 we used the same that we use here for the discussion of w, for q 0. This remarkable feature appears to be special to the empty-core pseudopotential. 

The lowest-lying s state for the empty-core pseudopotential is illustrated in F ig. 16-13,a. Its energy f can be computed by numerical integration and obtained as a function of Z and c. We shall make an approximate solution. 

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. 

Making first a comparison of the covalent energy, notice that in homopolar semiconductors, Wy, becomes simply w, The various geometrical factors in the empty-core pseudopotential may be directly evaluated. Then, the pseudopotential matrix element becomes... 

Polarities predicted from the empty-core pseudopotential and the relations given at Eq, 18-5. Values from LCAO theory (Table 4-1) are given in parentheses. 

A plot of the form factor for the empty-core pseudopotential, from Eq. (16-7), for aluminum. For comparison, the points give the model potential (Animalu and Heine, 1965). The choice of i is such that the two curves cross the axis at the same point. 

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). 

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