Pseudopotential screened

Fig. 7.17 Plot of the calculated dielectric constant in silicon crystallites of different size. The broken curve corresponds to calculations based on the Penn model [Tsl], the dotted line corresponds to pseudopotential calculations [Wa5], while the full line is based on self-consistent linear screening calculation of hydrogenic impurities [AI4]. Redrawn from [AI4].

Returning to the block diagram, the screening potential VSCR is added to the electron-core potential which is now called the pseudopotential, and this new total potential is used together with the new structure to start the loop over again. When input and output agree, the calculation is self-consistent. ... 

Thus, the screened pseudopotential form factor of aluminium normalized by the Fermi energy will approach q = 0 at —2/3 as observed in Fig. 5.12. 

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 generalization of the pseudopotential method to molecules was done by Boni-facic and Huzinaga[3] and by Goddard, Melius and Kahn[4] some ten years after Phillips and Kleinman s original proposal. In the molecular pseudopotential or Effective Core Potential (ECP) method all core-valence interactions are approximated with l dependent projection operators, and a totally symmetric screening type potential. The new operators, which are parametrized such that the ECP operator should reproduce atomic all electron results, are added to the Hamiltonian and the one electron ECP equations axe obtained variationally in the same way as the usual Hartree Fock equations. Since the total energy is calculated with respect to this approximative Hamiltonian the separability problem becomes obsolete. 

How to proceed with these matrix elements will depend upon which property one wishes to estimate. Let us begin by discussing the effect of the pseudopotential as a cause of diffraction by the electrons this leads to the nearly-free-electron approximation. The relation of this description to the description of the electronic structure used for other systems will be seen. We shall then compute the screening of the pseudopotential, which is necessary to obtain correct magnitudes for the form factors, and then use quantum-mechanical perturbation theory to calculate electron scattering by defects and the changes in energy that accompany distortion of the lattice. 

Figure 12 shows the idea of the pseudopotential. The coulombic attraction of a valence electron by the nucleus is screened by the core. However, this stiU strong attraction is... 

Figure 12 The various parts of the pseudopotential (1) is the Pauli repulsion between the electrons in the valence orbital (Sb, 1 = 0 in this case) with the core (2) is the coulomb attraction of the electrons for the nucleus screened by the core to give (3) (4), the pseudopotential, is the sum of (l)and (3) Ro is the crossing point of the pseudopotential. (Reprinted from Ref 12. 1981, with permission from Elsevier)...

Recently, Zunger and coworkers [18, 58-63] employed the semi-empirical pseudopotential method to calculate the electronic structure of Si, CdSe [60] and InP [59] quantum dots. Unlike EMA approaches, this method, based on screened pseudopotentials, allows the treatment of the atomistic character of the nanostructure as well as the surface effects, while permitting multiband and intervalley coupling. The atomic pseudopotentials are extracted from first principles LDA calculations on bulk solids. The single particle LDA equation,... 

The outcome of a calculation such as that given above is not entirely unambiguous. In particular, the choice of atomic pseudopotential and the treatment of screening effects can lead to potentials with different qualitative features. For the sake of discussion, in fig. 4.4 we show a series of pair potentials for A1 developed using different schemes for handling these effects. For our purposes, the key insight to emerge from this analysis is that we have seen how, on the basis of a free electron picture of metals, one may systematically construct effective pair potentials. 

In the late 1980s, Feibelman developed his Green s function scattering method using LDA with pseudopotentials to describe adsorption on two-dimensionally infinite metal slabs [175]. based on earlier work by Williams et al [176]. The physical basis for the technique is that the adsorbate may be considered a defect off which the Bloch waves of the perfect substrate scatter. The interaction region is short-range because of screening by the electron gas of the metal. Feibelman has used this technique to study, for example, the chemisorption of an H2 molecule on Rh(OOl) [177]. S adatoms on Al(331) [178] and Ag adatoms on Pt(l 11)... 

Values for are given in the Solid State Table. Their origin will be discussed in Chapter 16. Here we have u ed w rather than v, this is customary for pseudopotentials, and the superscript zero indicates that we have not added the screening corrections. 

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... 

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