Electron density statistics

Electron density statistics. At high resolution we know the shape of the electron density of an atom, in which case we only need to know its exact location to reconstruct the electron density in its immediate vicinity. At lower resolution we can impose an expected shape on the uni- or multivariate distributions of electron density within the protein region in a procedure that is known as histogram matching. 

In many materials, the relaxations between the layers oscillate. For example, if the first-to-second layer spacing is reduced by a few percent, the second-to-third layer spacing would be increased, but by a smaller amount, as illustrated in figure Al,7,31b). These oscillatory relaxations have been measured with FEED [4, 5] and ion scattering [6, 7] to extend to at least the fifth atomic layer into the material. The oscillatory nature of the relaxations results from oscillations in the electron density perpendicular to the surface, which are called Eriedel oscillations [8]. The Eriedel oscillations arise from Eenni-Dirac statistics and impart oscillatory forces to the ion cores. 

Deviations from this generalization may have several sources, including charge repulsion, steric effects, statistical factors, intramolecular hydrogen bonding, and other structural effects that alter electron density at the reaction site. Hague - ° P has discussed these effects. 

In many cases, however, the ortho isomer is the predominant product, and it is the meta para ratio which is close to the statistical value, in reactions both on benzenoid compounds and on pyri-dine. " There has been no satisfactory explanation of this feature of the reaction. One theory, which lacks verification, is that the radical first forms a complex with the aromatic compound at the position of greatest electron density that this is invariably cither the substituent or the position ortho to the substituent, depending on whether the substituent is electron-attracting or -releasing and that when the preliminary complex collapses to the tr-complex, the new bond is most likely to be formed at the ortho position.For heterocyclic compounds such as pyridine it is possible that the phenyl radical complexes with the nitrogen atom and that a simple electronic reorganization forms the tj-complex at the 2-position. 

At the center of the approach taken by Thomas and Fermi is a quantum statistical model of electrons which, in its original formulation, takes into account only the kinetic energy while treating the nuclear-electron and electron-electron contributions in a completely classical way. In their model Thomas and Fermi arrive at the following, very simple expression for the kinetic energy based on the uniform electron gas, a fictitious model system of constant electron density (more information on the uniform electron gas will be given in Section 6.4) ... 

According to the latter model, the crystal is described as formed of anumber of equal scatterers, all randomly, identically and independently distributed. This simplified picture and the interpretation of the electron density as a probability distribution to generate a statistical ensemble of structures lead to the selection of the map having maximum relative entropy with respect to some prior-prejudice distribution m(x) [27, 28],... 

The first step in our procedure is to compute an optimized structure for each molecule and then to use this geometry to compute the electronic density and the electrostatic potential. A large portion of our work in this area has been carried out at the SCF/STO-5G //SCF/STO-3G level, although some other basis sets have also been used. We then compute V(r) on 0.28 bohr grids over molecular surfaces defined as the 0.001 au contour of the electronic density (Bader et al. 1987). The numbers of points on these grids are converted to surface areas (A2), and the and Fs min are determined. Our statistically based interaction in-... 

Beginning way back in the 20s, Thomas and Fermi had put forward a theory using just the diagonal element of the first-order density matrix, the electron density itself. This so-called statistical theory totally failed for chemistry because it could not account for the existence of molecules. Nevertheless, in 1968, after years of doing wonders with various free-electron-like descriptions of molecular electron distributions, the physicist John Platt wrote [2] We must find an equation for, or a way of computing directly, total electron density. [This was very soon after Hohenberg and Kohn, but Platt certainly was not aware of HK by that time he had left physics.]... 

Simple dependence between PE-parameter and electron density at the distance r, can be obtained (according to the statistic model of atom) ... 

Based on the results [5] the values of PE-parameters numerically equal (within 2%) total energy of valence electrons (U) by the atom statistic model. Using the well-known ratio between electron density (J3) and inneratomic potential by the atom statistic model, we can obtain the direct dependence of PE-parameter upon the electron density at the distance rt from the nucleus ... 

The reliability of initial equations and regulations was proved with numerous calculations and comparisons. In particular, it was shown [8] that PE-parameter numerically equals the energy of valence electrons in a statistical atom model and is a direct characteristic of electron density in the atom at the given distance from the nucleus. 

These are 13 atoms of hexagons, the central atom of which has the coordination of 12 atoms. The process of rolling flat carbon systems into NT is, apparently, determined by polarizing effects of cation-anion interactions resulting in statistic polarization of bonds in a molecule and shifting of electron density of orbitals in the direction of more electronegative atoms. 

The crystallographic data contain all the information about the data that were used for determining the model. The most important information is the resolution. This refers to the minimum d spacing (O Eq. 22.1) and indicates the smallest distance between two atoms that can be resolved, i.e., completely separated based on electron density. The table also contains space group (P2i2i2i) and unit-cell information along with the statistical measurements for the reflection data. 

To elevate p-selectivity in nitration of toluene is another important task. Commercial production of p-nitrotoluene up to now leads with twofold amount to the unwanted o-isomer. This stems from the statistical percentage of o m p nitration (63 3 34). Delaude et al. (1993) enumerate such a relative distribution of the unpaired electron densities in the toluene cation-radical—ipso 1/3, ortho 1/12, meta 1/12, and para 1/3. As seen, the para position is the one favored for nitration by the attack of NO (or NO2 ) radical. A procednre was described (Delande et al. 1993) that used montmorillonite clay supported copper (cupric) nitrate (claycop) in the presence of acetic anhydride (to remove excess humidity) and with carbon tetrachloride as a medinm, at room temperature. Nitrotoluene was isolated almost quantitatively with 23 1 76 ratio of ortho/meta/para mononitrotoluene. 

It is impossible to directly measure phases of diffracted X-rays. Since phases determine how the measured diffraction intensities are to be recombined into a three-dimensional electron density, phase information is required to calculate an electron density map of a crystal structure. In this chapter we discuss how prior knowledge of the statistical distribution of the electron density within a crystal can be used to extract phase information. The information can take various forms, for example ... 

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