Lines iso-density

Figure II 2 9a-s. The valence electron iso-density lines in the plane of B atoms (a-b plane) for equilibrium (a) and distorted structures (b-e). The electron density is localized at B atom positions for equilibrium structure (a). The B atoms displacements ( Af = 0.005) induce the alternating interatomic charge density delocalization, different for the particular types of the distortion (b-d). Nuclear microcirculation enables then effective charge transfer over the lattice in an external electric potential. The Fig (e) corresponds to the case of the distortion (d) over the larger lattice segment...

The ODF, f g) = /(a, /, 7), is a function of three variables. To give a picture of such a function one usually shows sections taken for a set of fixed values of one of the angles for example, sections for a = 0°, a = 10°, a = 20°, etc. In each section one uses iso-density lines, which shows the areas where f g) deviates from the random texture (/rand( ) = 1 whatever g). 

Some examples of ODF s are presented in the following figures. Figure 9 shows one section (a = 0°) of the ODF of a TiN thin layer. In this particular case (which is the case of a fiber texture) every a section will show exactly the same distribution of iso-density lines. The strong maximum (about 25 times the random distribution) is located in each section dX (3 — 55° and 7 = 45°, which corresponds to crystallites with a (111) axis parallel to C the normal to the specimen. From this figure one can appreciate the spread around this maximum. 

Our first task is to display the three-dimensional distribution of electron density, p(r), where r represents a vector describing any point in space. Figure 10.48 shows relief plots of p(r) of frani-diimine, HN=NH, drawn (a) in the plane of the molecule and (b) perpendicular to it. The molecular dimensions are drawn in the base plane and p(r) is represented as the third dimension. Plots (c) and (d) are contour plots, with contours (iso-density lines) drawn at 0.002, 0.004, 0.008, 0.02, 0.04, 0.08 and 0.2 atomic units of electron density (1 au = 1 e oo, where oq is the Bohr radius) and so on. Thus the nuclear positions are the local maxima of p(r), and the density rapidly falls away from these points. 

In Fig. 27.7, iso-density lines of highest electron density calculated for Nb3Al on crude-adiabatic level by computer code SOLID2000 [55] are shown. 

Fig. 27,7 Calculated iso-density lines of highest electron density (cut in b-c plane) at equilibrium nuclear geometry R q of NbaAl (a) and at the distorted geometry R cr of Nb atoms in Fn phonon mode when nonadiabatic e-p interactions trigger transition into antiadiabatic state and inter-sites polarization is induced. Figures (b-f) represents different geometrical fxjsitions of Nb atoms on the circumferences of the flux circles...

ELF can be visualized with different kinds of images. Colored sections through a molecule are popular, using white for high values of ELF, followed by yellow-red-violet-blue-dark blue for decreasing values simultaneously, the electron density can be depicted by the density of colored points. Contour lines can be used instead of the colors for black and white printing. Another possibility is to draw perspective images with iso surfaces, i.e. surfaces with a constant value of ELF. Fig. 10.2 shows iso surfaces with ELF = 0.8 for some molecules from experience a value of ELF = 0.8 is well suited to reveal the distribution of electron pairs in space. 

In Figure 4.6 a number of iso-Mahalanobis distance contours of a three component mixture have been depicted. The square of the variation coefficient (v) is constant. The ellipses drawn at Figure 4.5 are each contour lines with the same probability density value. This means that if a mixture is set to the centre point of one of the ellipses in the figure, than the probability that the composition of the mixture is present inside the drawn ellipse is in all the cases the same. 

Figure 2 Spatial distribution functions displayed as three-dimensional maps showing the local oxygen density in liquid water. Above TIP4P water at ambient conditions Below PPC water along the co-existence line at 2U0 C. The iso-surfaces shown are for — 1.3 where the surfaces have been cf)k>ied according to their separation from the central molecule, as discu.sse

Fig. 11. The Iso-, state of the H2+ system, (a) Binding charge. The dashed line shows the binding charge obtained from the superposed atomic density. The SA value is 0.5 e. (b) CEDs. Positive value means the inside of the nucleus. The SA values are 0.0,0.75, and -0.75 for the total, binding, and antibinding CEDs, respectively. (Reproduced from Koga et al., 1980.)...

Fig. 2 Electronic charge density map for Mg porphyrin along the molecular plane. The atomic positions are marked by crossed circles. The metal is in the center of the map, while the four N ligands are the remaining circles. Neighbor iso-lines correspond to values that differ by 0.01 atomic units (electrons per cubic bohr). The lower value iso-lines (0.01 atomic units) lie in the diagonals of the map, out of the metal-ligand directions. The highest reported value iso-lines (0.20 atomic units) lie in the direction of the nuclei, both metal and N. Close to the middle of the metal-ligand segment, a saddle point lies...

Figure 4.18 EXAFS Fourier transforms of the sulfite oxidase iodide complex. Transforms are Mo-S phase-corrected, with the solid line showing experimental data and the broken line the best fit. The weak EXAFS from the distant iodide can be seen as the small transform peak at 5 A. The DFT-computed active site structure, calculated using constraints based on the crystal structure, is shown in the inset with a computed total electron density iso-surface with a value of 0.02 electrons per cubic a.u.

The line integral method advsmced by van Leeuwen and Baerends [Phys. Rev. A 51, 170, (1995)] is applied to the calculation of correlation energies of the iso-electronic series terms with nuclear charge Z -f 1 from exact densities Peioct.Zi Pexact,z+i and the total energy of the term with nuclear charge Z. Numerical calculations are performed for He(Z = 1,2,3)-, Li(Z = 3,4,5,6,7,8,9)- and Be(Z = 4,5,6) isoelectronic series. 

Figure 1. Observed N(CH) vs. N(Hi) for various lines of sight. The results of a typical low density model (tiH=ISO cm ) and a high density model (np=2500 cm ) are included (from Banks et al. 1984).

Fig. 2.16. Temperature dependences of the dielectric relaxation times for PVAc at atmospheric pressure ( ) and at a constant volume equal to 0.847 mlg (A), 0.849 ml ( ), and 0.852 ml g (V). The slopes at the intersection of the iso-baric and isochoric lines yield values for the respective activation energies at constant pressure and constant volume a = 238 and 448kJmol (r = 2.5 s) and = 166 and 293 kJ mol (r = 0.003 s). The ratio of the isochoric and isobaric activation energies is a measure of the relative contribution of thermal energy and volume that is, this ratio would be unity if the molecular motion were thermally activated, and zero if it were strictly dominated by density. For PVAc, the ratio is 0.6, indicating that both contributions are significant. From Roland and Casalini by permission [132].

Fig. 7.5 Crystals structure, band structure, and band-decomposed charge density plots for AgBiSc2 with Ag vacancy in AF-11 (a, d, g), intermediate (b, e, h) and AF-llb (c, f, i) structure with 64 atoms, respectively. The Fermi level (Ey, red dashed line) is set at 0 eV. The empty defects states summed between 0 and -tO.2 eV with Ag vacancies. The iso-surfaces for band-decomposed charge density plots correspond to a value of 0.007 e x...

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