Valence charge densities

This ionic potential is periodic. A translation of r to r + R can be acconnnodated by simply reordering the sunnnation. Since the valence charge density is also periodic, the total potential is periodic as the Hartree and exchange-correlation potentials are fiinctions of the charge density. In this situation, it can be shown that the wavefiinctions for crystalline matter can be written as... 

Reciprocal lattice vectors are usefiil in defining periodic fimctions. For example, the valence charge density, p (r), can be expressed as... 

Since and depend only on die valence charge densities, they can be detennined once the valence pseudo- wavefiinctions are known. Because the pseudo-wavefiinctions are nodeless, the resulting pseudopotential is well defined despite the last temi in equation Al.3.78. Once the pseudopotential has been constructed from the atom, it can be transferred to the condensed matter system of interest. For example, the ionic pseudopotential defined by equation Al.3.78 from an atomistic calculation can be transferred to condensed matter phases without any significant loss of accuracy. 

Extra radial flexibility has been proved necessary in order to model the valence charge density of metal atoms, in minerals [6,11], and coordination complexes [5], and similar evidence of the inability of single-exponential deformation functions to account for all the information present in the observations have also been found in studies of organic [12, 13] and inorganic [14] molecular crystals. 

Chen, R., Trucano, P. and Stewart, R.F. (1977) The valence-charge density of graphite, Acta Cryst., A33, 823-828. 

Fig. 16 (a) Calculated valence charge density for a Pd monolayer (top) and clean Ta(110). (b) Calculated valence charge density for the Pd/Ta(110) system, (c) Charge density difference obtained by subtracting the superposition of the charge densities of the Pd monolayer and Ta(l 10) from that of Pd/Ta(110). Dashed lines indicate a decrease in the electron density. Reprinted from ref [33]. 

The STM simulations are based on a simplified atomic charge model [31], in which a spherical shape of the valence charge density is assumed for the atoms. These spherical shells of fixed radius (solid sphere) are superimposed in three dimensions for all atomic positions considered in the simulation. From this superposition the z(x,y) (z parallel to the plane normal) contour of the uppermost shells is... 

Figure 7, Shift in the valence charge density (top) and the one-electron potential (bottom) in displacement of O2 atoms in the axial breathing normal mode.

Figure 1 Contour plot in the xz vertical plane of valence charge density at Ep for La2Cu04.

The evidence for covalent T-Al bonding can also be visualized in real space by the valence charge density distribution as an example, this is shown for YbNiAl with hexagonal ZrNiAl structure in fig. 6. The accumulation of charge is clearly discernible between A1 and Ni with the charge density shifted to the more electronegative nickel atoms. 

Figure 14.7 Theoretical calculations of the valence charge density of semiconductors showing the formation of the covalent bond and the progression to more ionic character in the series Ge-GaAs-ZnSe. Data of Chelikovsky and Cohen (after (1)).

Fig. 4-24. Total valence charge density along the B-N bond in p-BN. The dashed line gives the contribution from the upper three valence bands, the dotted line the contribution from the first band [4].

Valence charge density contour maps from pseudopotential (LDA) calculations are presented in [4, 5, 12] see also [7]. 

Figure 5.8b. Pseudo valence charge density contour plots of the (a) RuO2(110) surface in comparison with (b) the RuO2(001) surface cut through the cus-Ru atoms. These plots are defined as the difference between the total valence electron density and a linear superposition of radially symmetric atomic charge densities. Contours of constant charge density are separated by 0.15 eV/A. Electron depletion and accumulation are marked by dashed and solid lines, respectively. In addition, regions of electron accumulation are shadowed...

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