Electron, density

In the course of our explorations, we will introduce two new types of calculations  

Visualizing the electron density or the electrostatic potential can be a useful first step in exploring the reactivity of a molecular system. 

The nitration of nitrobenzene and of chlorobenzene are known to occur via the same mechanism the ring is initially attacked by NO2, yielding a cation intermediate for each isomer. When the nitration process is fully complete, the distribution of the various isomers of the final product varies greatly for the two compounds  

We will examine the meta and para cation intermediates for the two substances. 

Exploring Chemistry with Electronic Structure Methods 

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Another usefiil quantity is defining the electronic structure of a solid is the electronic density of states. In general the density of states can be defined as... 

The value of at zero temperature can be estimated from the electron density ( equation Al.3.26). Typical values of the Femii energy range from about 1.6 eV for Cs to 14.1 eV for Be. In temis of temperature (Jp = p//r), the range is approxunately 2000-16,000 K. As a consequence, the Femii energy is a very weak ftuiction of temperature under ambient conditions. The electronic contribution to the heat capacity, C, can be detemiined from... 

Kim Y S, Kim S K and Lee W D 1981 Dependence of the closed-shell repulsive interaction on the overlap of the electron densities Chem. Phys. Lett. 80 574... 

The reason that relaxation occurs can be understood in tenus of the free electron character of a metal. Because the electrons are free, they are relatively uuperturbed by the periodic ion cores. Thus, the electron density is homogeneous... 

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. 

The most popular of the scanning probe tecimiques are STM and atomic force microscopy (AFM). STM and AFM provide images of the outemiost layer of a surface with atomic resolution. STM measures the spatial distribution of the surface electronic density by monitoring the tiumelling of electrons either from the sample to the tip or from the tip to the sample. This provides a map of the density of filled or empty electronic states, respectively. The variations in surface electron density are generally correlated with the atomic positions. 

Above approximately 80 km, the prominent bulge in electron concentration is called the ionosphere. In this region ions are created from UV photoionization of the major constituents—O, NO, N2 and O2. The ionosphere has a profound effect on radio conmumications since electrons reflect radio waves with the same frequency as the plasma frequency, f = 8.98 x where 11 is the electron density in [147]. The... 

As the number of atoms in the asyimnetric unit increases, the solution of a structure by any of these phase-independent methods becomes more difficult, and by 1950 a PhD thesis could be based on a single crystal structure. At about that time, however, several groups observed that the fact that the electron density must be non-negative everywhere could be exploited to place restrictions on possible phases. The first use of this fact was by D Marker and J S Kasper [24], but their relations were special cases of more general relations introduced by J Karle and H Hauptman [25]. Denoting by A. the set of indices h., k., /., the Karle-Hauptman condition states that all matrices of the fonu... 

X-ray scattering arises from fluctuations in electron density. The general expression of the absolute scattered intensity (simplified as I(q) from now on) from the tliree-dimensional objects iimnersed in a different... 

For homogeneous particles, it represents the number of distances within the particle. For inhomogeneous particles, it has to take into account the different electron density of the volume elements. Thus it represents the number of pairs of difference in electrons separated by the distance r. A qualitative description of shape and internal structure of the... 

If we consider the scattering from a general two-phase system (figure B 1.9.10) distinguished by indices 1 and 2) containing constant electron density in each phase, we can define an average electron density and a mean square density fluctuation as ... 

Figure Bl.9.12. The schematic diagram of the relationships between the one-dimensional electron density profile, p(r), correlation fiinction y (r) and interface distribution fiinction gj(r).

In order to understand the tendency to fomi a dipole layer at the surface, imagine a solid that has been cleaved to expose a surface. If the truncated electron distribution originally present within the sample does not relax, this produces a steplike change in the electron density at the newly created surface (figme B1.26.19(A)). 

Since the electron density p(x) oc /(v)p, where /(v) is die electron wavefiinction, this implies that the electron wavefiinction varies in a similarly step-wise fashion at the interface. This indicates that d i //dx, where s indicates that the derivative is evaluated at the surface, becomes infinite. Since the electron kinetic... 

Figure Bl.26.19. The variation of the electron density (A) from an iimelaxed surface and (B) showing the smoothing of the electron density to lower the kinetic density.

Here pyy r ) represents the probability density for finding the 1 electrons at r, and e / mutual Coulomb repulsion between electron density at r and r. ... 

The polarization fiinctions are essential in strained ring compounds because they provide the angular flexibility needed to direct the electron density mto the regions between the bonded atoms. 

The so-called Flohenberg-Kolm [ ] theorem states that the ground-state electron density p(r) describing an A-electron system uniquely detemiines tlie potential V(r) in the Flamiltonian... 

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