Local pseudopotential calculation

Table 3 A comparison of semi-local and non-local pseudopotential calculations of ionization energies o/Fe (a.u.)...

One current limitation of orbital-free DFT is that since only the total density is calculated, there is no way to identify contributions from electronic states of a certain angular momentum character /. This identification is exploited in non-local pseudopotentials so that electrons of different / character see different potentials, considerably improving the quality of these pseudopotentials. The orbital-free metliods thus are limited to local pseudopotentials, connecting the quality of their results to the quality of tlie available local potentials. Good local pseudopotentials are available for the alkali metals, the alkaline earth metals and aluminium [100. 101] and methods exist for obtaining them for other atoms (see section VI.2 of [97]). 

One of the most conspicuous differences between computational results is in the degree to which a normal H—Si chemical bond is formed. In the local-density pseudopotential calculations, the Si—H separation is about 1.6 A. This is much larger than the predictions of MNDO, Hartree-Fock, or PRDDO calculations, which are much closer to the molecular Si—H distance. It is not clear at this point whether the H—Si bond is, in fact, weaker than a conventional bond when in this configuration and therefore is overestimated by the Hartree-Fock-like calculations, or whether the strength is being underestimated in the local-density calculations. 

Fig. 5. Contour plot of the adiabatic potential-energy surface of an H atom in the (110) plane for the neutral H—B pair from a local-density pseudopotential calculation. The boron atom is at the center. For every hydrogen position, the B and Si atoms are allowed to relax, but only unrelaxed positions are indicated in the figure (Reprinted with permission from the American Physical Society, Denteneer, P.J.H., Van de Walle, C.G., and Pantelides, S.T. (1989). Phys. Rev. B 39, 10809.)...

The hydrogen frequency appropriate to the H—B pair was supported by the local-density pseudopotential calculations of Chang and Chadi (1988)... 

The semi-local pseudopotential used for the calculations of Table 3 was based on a parameterization for the neutral atom. Melius, Olafson, and Goddard also included in their paper some calculations based on the single valence electron ion Fe +. As expected, this parameterization leads to far worse results, which differ on average from the all-electron results by 0.05 a.u., thus emphasizing the importance of the contribution of valence-valence interaction to the effective potential [equation (51)]. 

Van Camp et al [13] also calculated the first and second pressure derivatives by the ab-initio norm-conserving non-local pseudopotential method of the energy differences between the T X-, and L-states of the valence and lowest conduction band and the top of the valence band in ri5 for 3C-SiC. The first-order pressure coefficients for the direct (ri5v-ric) and indirect (T15v - Xu) bandgaps are 61.7 meV GPa 1 and -1.1 meV GPa 1, respectively. The signs of these values are the same for all semiconductors by Paul s rule. However, the magnitudes are quite different from those of other semiconductors. 

The gap of p-BN increases upon volume compression as in the case of diamond, while in most semiconductors the reverse is true [4]. This is due to the fact that boron and nitrogen atoms have deep and localized pseudopotentials as compared with the atoms of other rows of the periodic table. It can be calculated [5] by the same technique described in [4] that at very high pressures boron nitride should favor the rock salt phase, while the p-BN structure is unstable even at highly compressed volumes [5]. Calculations using a nonlocalized pseudopotential model lead to a band gap of 4.41 eV for p-BN for further details, see [6]. Electronic properties of p-BN have been derived by ab initio, norm-conserving, nonlocal pseudopotential... 

The alert reader will have realized that almost all examples given in this chapter are from Na. Of course this was on purpose two of the most important conditions to be fulfilled for the excellent validity of the jellium model, namely (a) that the pseudopotential is local and (b) that the geometrical parts of the ionic arrangement are weak, are best met in NaA. In trying other elements we found only one other material that works comparably well — potassium. But the important point to note is that this simple model serves as a guideline for more complex cases. After electronic shells and plasmons have been found in Na, they have been found in almost all metal clusters. In order to get the same quantitative agreement as in the case of Na one has either to do all-electron calculations or to use non-local pseudopotentials, as has been done in the case of Li [39, 40]. But in these... 

The temperature dependence of the moments has been calculated from the data given in Figure 5.10. The oscillator strength turns out to be nearly temperature-independent, a surprising result if one looks how the spectra change, but in agreement with the spirit of the local pseudopotential extension of the jellium model [38, 39]. 

If the ions are treated explicitly, the ion-electron interaction is described by pseudopotentials. This allows us to eliminate all core electrons from the actual calculation and to deal only with the valence electrons. There exists a wide variety of pseudopotentials. The most elaborate of them are nonlocal operators because they project out of the occupied core electron states, see e.g. [18]. Separable approximations to projection can simplify the handling [19]. The simplest to use are, of course, local pseudopotentials, as e.g. the old empty-core potential of [20] or the more recent and extensive adjustment of [21]. It is a welcome feature that most simple metals, except for Li, can be treated fairly well with local pseudopotentials [22]. It is to be remarked that all these choices have been optimized with respect to structural properties. The performance concerning dynamical features has not yet been explored systematically. In fact, nSost of the available pseudopotentials tend to produce a blue-shifted plasmon position. An optimization of pseudopotentials for simple metals with simultaneous adjustment of static and dynamic properties is presently under way [23]. For noble metals, it is known that pseudopotentials alone (even when nonlocal) cannot reproduce the proper plasmon position. One needs to explicitly take into account the considerable polarizability of the ionic core, in particular of the rather soft last d shell [24]. 

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