Atomic sphere

Figure Bl.21.1. Atomic hard-ball models of low-Miller-index bulk-temiinated surfaces of simple metals with face-centred close-packed (fee), hexagonal close-packed (licp) and body-centred cubic (bcc) lattices (a) fee (lll)-(l X 1) (b)fcc(lO -(l X l) (c)fcc(110)-(l X 1) (d)hcp(0001)-(l x 1) (e) hcp(l0-10)-(l X 1), usually written as hcp(l010)-(l x 1) (f) bcc(l 10)-(1 x ]) (g) bcc(100)-(l x 1) and (li) bcc(l 11)-(1 x 1). The atomic spheres are drawn with radii that are smaller than touching-sphere radii, in order to give better depth views. The arrows are unit cell vectors. These figures were produced by the software program BALSAC [35]-...

Figure B3.2.4. A schematic illustration of an energy-independent augmented plane wave basis fimction used in the LAPW method. The black sine fimction represents the plane wave, the localized oscillations represent the augmentation of the fimction inside the atomic spheres used for the solution of the Sclirodinger equation. The nuclei are represented by filled black circles. In the lower part of the picture, the crystal potential is sketched.

In the connnonly used atomic sphere approximation (ASA) [79], the density and the potential of the crystal are approximated as spherically synnnetric within overlapping imifiBn-tin spheres. Additionally, all integrals, such as for the Coulomb potential, are perfonned only over the spheres. The limits on the accuracy of the method imposed by the ASA can be overcome with the fiill-potential version of the LMTO (FP-LMTO)... 

The space filling model developed by Corey, Pauling, and Koltun is also known as the CPK model, or scale model [197], It shows the relative volume (size) of different elements or of different parts of a molecule (Figure 2-123d). The model is based on spheres that represent the "electron cloud . These atomic spheres can be determined from the van der Waals radii (see Section 2.10.1), which indicate the most stable distance between two atoms (non-bonded nuclei). Since the spheres are all drawn to the same scale, the relative size of the overlapping electron clouds of the atoms becomes evident. The connectivities between atoms, the bonds, are not visualized because they are located beneath the atom spheres and are not visible in a non-transparent display (see Section 2.10). In contrast to other models, the CPK model makes it possible to visualize a first impression of the extent of a molecule. 

Metal atoms tend to behave like miniature ball-bearings and tend to pack together as tightly as possible. F.c.c. and c.p.h. give the highest possible packing density, with 74% of the volume of the metal taken up by the atomic spheres. However, in some metals, like iron or chromium, the metallic bond has some directionality and this makes the atoms pack into the more open b.c.c. structure with a packing density of 68%. 

Tomasi s Polarized Continuum Model (PCM) defines the cavity as the union of a series of interlocking atomic spheres. The effect of polarization of the solvent continuum is represented numerically it is computed by numerical integration rather... 

The first step is to choose a molecular geometry, the atomic sphere radii and the exchange parameters for each atom and the outer sphere region. I have summarized these in Table 12.1. 

We have used the multisublattice generalization of the coherent potential approximation (CPA) in conjunction with the Linear-MufRn-Tin-Orbital (LMTO) method in the atomic sphere approximation (ASA). The LMTO-ASA is based on the work of Andersen and co-workers and the combined technique allows us to treat all phases on equal footing. To treat itinerant magnetism we have employed for the local spin density approximation (LSDA) the Vosko-Wilk-Nusair parameterization". 

Second, using the fully relativistic version of the TB-LMTO-CPA method within the atomic sphere approximation (ASA) we have calculated the total energies for random alloys AiBi i at five concentrations, x — 0,0.25,0.5,0.75 and 1, and using the CW method modified for disordered alloys we have determined five interaction parameters Eq, D,V,T, and Q as before (superscript RA). Finally, the electronic structure of random alloys calculated by the TB-LMTO-CPA method served as an input of the GPM from which the pair interactions v(c) (superscript GPM) were determined. In order to eliminate the charge transfer effects in these calculations, the atomic radii were adjusted in such a way that atoms were charge neutral while preserving the total volume of the alloy. The quantity (c) used for comparisons is a sum of properly... 

The consequences of this approximation are well known. While E s is good enough for calculating bulk moduli it will fail for deformations of the crystal that do not preserve symmetry. So it cannot be used to calculate, for example, shear elastic constants or phonons. The reason is simple. changes little if you rotate one atomic sphere... 

The muffin-tin potential around each atom in the unit cell has been calculated in the framework of the Local-Spin-Density-Approximation using the ASW method. The ASW method uses the atomic sphere approximation (ASA), i.e. for each atom a sphere radius is chosen such that the sum of the volumes of all the overlapping spheres equals the unit cell volume. The calculation yields the expected ferromagnetic coupling between Cr and Ni. From the self-consistent spin polarized DOS, partial and total magnetic moment per formula unit can be computed. The calculated total magnetic moment is 5.2 pg in agreement with the experimental value (5.3 0.1 e calculations presented here have been performed... 

Antiphase boundary (APB) conservative vacancy segregation at Arrhenius plot Asymmetrical mixtures Atomic-sphere approximation (ASA) ASA-LSDA... 

For all PCM-UAHF calculations, the number of initial tesserae/atomic sphere has been set to 60 by default. For comparative purposes, C-PCM calculations of the... 

Perhaps the most widely discussed source of uncertainty in electrostatic calculations is the location of the solute/solvent boundary. The most common treatment is to place the boundary at the surface of a set of overlapping spheres centered at the nuclei. But what radius should one use for those spheres One common answer is van der Waals radii times I.2.46 In our own quantum mechanical solvation models,12 27 and those of several others59, 69, these radii are empirical parameters. Recently Barone et al.70 have modified the PCM to use charge-dependent united-atom spheres instead of all-atom spheres, and they optimized the electrostatic radii for a... 

Philosophers and historians interested in science mostly have focused on physics, biology (especially evolutionary biology), and the social sciences, with only occasional mention of atomic spheres, benzene hexagons, and the periodic table. 1 Attention to chemical epistemology largely has centered on a... 

In the cP2-W type (CN 8) structure Vsph is 0.68 Vat (only a portion of the available space is occupied by the atomic sphere ). In the cF4-Cu type and in the ideal hP2-Mg type (CN 12) structures, Vsph is 0.74 Vat. Considering now the previously reported relationship between RCs n and i CN8, we may compute for a given element very little volume (Vat) change in the allotropic transformation from a form with CN 12 to the form with CN 8, because the radius variation is nearly... 

Figure 10.6. Structural features of D. vulgaris Rbr (deMare et al. 1996). A, Rbr subunit with protein backbone and iron atoms (spheres). B, Diiron-oxo site with amino acid side chain ligands. Models generated via RASMOL (Sayle and Milner-White 1995) and coordinates from IRYT in the Protein Databank.

Figure 2.8 Structures of the catalytic core domain ofJMJD2A. (a) The catalytic core without substrate (PDBcode2gp3) butinthe presenceof iron and zinc atoms (spheres in red and blue respectively), (b) The catalytic core in the presence of a-ketoglutarate (pdb code 2gp5).

The important point of (12) is, however, that it retains atomic information in the one electron wave functions of (11). In LMTO - ASA, e.g. the bandwidth W will depend essentially on the charge density (()ap evaluated at the surface of the atomic sphere, being large when ( ) p is high (the electron wave is hardly contained within the sphere), small when is low (the electron wave is mostly contained within the sphere). 

The results are conveniently and clearly expressed in a thermodynamic formalism this is why they find their place in this chapter. They depend however on parameters which are drawn from band-theory, especially from the LMTO-ASA (Linear Muffin-Tin Orbitals-Atomic Sphere Approximation) method. 

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