Local electronic density

Implementation of the Kohn-Sham-LCAO procedure is quite simple we replace the standard exchange term in the HF-LCAO expression by an appropriate Vxc that will depend on the local electron density and perhaps also its gradient. The new integrals involved contain fractional powers of the electron density and cannot be evaluated analytically. There are various ways forward, all of which... 

Figure. 3 (a) Partial pair correlation function.s gij(B.) in liquid K-Sb alloys, (b) Total, partial, and local electronic densities of states in liquid Ko.soSbo.so- Cf. text. 

Mezey PG (1999) Local Electron Densities and Functional Groups in Quantum Chemistry. 203 167-186... 

Two working modes are used for the STM first, the constant height-mode, in which the recorded signal is the tunneling current versus the position of the tip over the sample, and the initial height of the STM tip with respect to the sample surface is kept constant (Fig. 22(a)). In the constant currentmode, a controller keeps the measured tunneling current constant. In order to do that, the distance between tip and sample must be adjusted to the surface structure and to the local electron density of the probed sample via a feedback loop (Fig. 22(b)). 

From the early advances in the quantum-chemical description of molecular electron densities [1-9] to modem approaches to the fundamental connections between experimental electron density analysis, such as crystallography [10-13] and density functional theories of electron densities [14-43], patterns of electron densities based on the theory of catastrophes and related methods [44-52], and to advances in combining theoretical and experimental conditions on electron densities [53-68], local approximations have played an important role. Considering either the formal charges in atomic regions or the representation of local electron densities in the structure refinement process, some degree of approximate transferability of at least some of the local structural features has been assumed. 

It is possible, however, to avoid any violation of these fundamental properties, and derive a result on the local electron densities of non-zero volume subsystems of boundaryless electron densities of complete molecules [159-161]. A four-dimensional representation of molecular electron densities is constructed by taking the first three dimensions as those corresponding to the ordinary three-space E3 and the fourth dimension as that representing the electron density values p(r). Using a compactifi-cation method, all points of the ordinary three- dimensional space E3 can be mapped to a manifold S3 embedded in a four- dimensional Euclidean space E4, where the addition of a single point leads to a compact manifold representation of the entire, boundaryless molecular electron density. 

As scattering intensity is computed from p (r) in this book, the symbol p (r) has two different meanings. Only in the field of WAXS it is identical to the plain electron density. However, in the area of SAXS it indicates the electron density difference1, i.e., the deviation of the local electron density from the average electron density (p (r))v in the irradiated volume V. 

The mobilities of dislocations are determined by interactions between the atoms (molecules) within the cores of the dislocations. In pure simple metals, the interactions between groups of adjacent atoms depend very weakly on the configuration of the group, since the cohesive forces depend almost entirely on the local electron density, and are of long range. 

The hardness of WC is associated with the fact that the array of W-atoms in the cores of glide dislocations changes from hexagonal prismatic to quasi-octahedral so the coordination number of the C-atoms changes from approximately six to approximately eight. This increases the local electron density so dislocation motion is resisted. 

In their model, N is the local electron density measured over an appropriate volume, which they argue is given by the Mott criterion discussed in Sect. 3.4.1. Thus, with the Mott radius of 20 A for uniform spheres, these spheres will constitute the appropriate volume over which to measure the local electron density N. 

Further inspection of Fig. 4.5 demonstrates that the O atoms are surrounded by dark zones. The STM technique probes not only the atomic topography but also the electronic structure, and the dark zones reflect the modification of the local electronic density in the vicinity of the adsorbates, this modification being responsible for the operation of indirect interactions (which may be either repulsive or attractive) between adsorbed particles mediated through the substrate. 

An alternate approach, which has proven to be extremely useful for metals, has been developed by Daw, Baskes and Foiles - (and to a lesser extent, by Ercolessi, Tosatti and Parrinello ). Called the embedded atom method (EAM) (or the glue model by the second group), the interactions in this approach are developed by considering the contribution of each individual atom to the local electron density, and then empirically determining an energy functional for each atom which depends on the electron density. This circumvents the problem of defining a global volume-dependent electron density. 

While the embedded atom method has been formally derived by Daw and Baskes the functions used in computer simulations are t3pically empirically determined. The description presented here will therefore treat this approach as an empirical method. The first step in determining the potential is to define a local electron density at each atomic site in the solid. A simple sum of atomic electron densities has proven to be adequate, and so in most cases a sum of free atom densities is used . The second step is to determine an embedding... 

The surface basicity of a solid catalyst can be defined in a way analogous to that applied to conventional bases. Thus, a surface Lewis base site is one that is able to donate an electron pair to an adsorbed molecule. If we take the definition of surface basicity in a more general way, it could be said that the active surface corresponds to sites with relatively high local electron densities. This general definition will include not only Lewis basicity but also single electron donor sites. We emphasize that the literature of heterogeneous catalysis often reports that both single-electron and electron-pair donor sites exist on basic catalysts. 

In the remainder of this section, we give a brief overview of some of the functionals that are most widely used in plane-wave DFT calculations by examining each of the different approaches identified in Fig. 10.2 in turn. The simplest approximation to the true Kohn-Sham functional is the local density approximation (LDA). In the LDA, the local exchange-correlation potential in the Kohn-Sham equations [Eq. (1.5)] is defined as the exchange potential for the spatially uniform electron gas with the same density as the local electron density ... 

Based on the first-principles study of helium adsorption on metals (Zaremba and Kohn, 1977), Esbjerg and Nprskov (1980) made an important observation. Because the He atom is very tight (with a radius about 1 A), the surface electron density of the sample does not vary much within the volume of the He atom. Therefore, the interaction energy should be determined by the electron density of the sample at the location of the He nucleus. A calculation of the interaction of a He atom with a homogeneous electron distribution results in an explicit relation between the He scattering potential V r) and the local electron density p(r). For He atoms with kinetic energy smaller than 0.1 eV, Esbjerg and Nprskov (1980) obtained... 

First-principles calculations of an STM, including a real tip and a real sample, clearly show that within the normal tip-sample distances (3.-6 A from nucleus to nucleus), in the gap region, the local electronic density resembles neither that of the tip nor that of the sample. Substantial local modifications are induced by the strong interaction. An example is the system of an A1 sample with an A1 tip, calculated by Ciraci, Baratoff, and Batra (1990a), as shown in Fig. 8.1. As the tip-sample distance is reduced to 8 bohr, the electron density begins to show a substantial concentration in the middle of the gap. This phenomenon becomes much more pronounced when the tip-sample distance is reduced to about 7 bohr. These distances are exactly the normal distances where atom-resolved images are obtained. 

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