Electron charge concentration depletion

Fig. 5.9. Molecular graphs showing the Laplacian —V p(r) and bond paths. Bond paths are shown as solid lines and bond critical points as dots. Dashed contours show areas of electronic charge concentration and solid contours show regions of charge depletion, (a) cyclopropane, (b) oxirane, (c) protonated oxirane, and (d) fluorine-bridged cation. Reproduced from J. Am. Chem. Soc., 107, 3800 (1985), by permission of the American Chemical Society.

Figure 7.3 Truncated representation of p versus the distance from the nucleus for a spherically symmetric electron density of a free sulfur atom (3P). (b) Truncated representation of L(r) at the same scale as (a). This function reveals the three shells K, L, and M constituting the sulfur atom. Each shell consists of a region of local charge concentration (dark areas) and a region of local charge depletion (light...

Electrons in the core of an atom are fully localized into spherical shells but not into opposite-spin pairs. In an isolated atom the valence shell electrons are similarly localized into a spherical shell. The Laplacian shows that in each of these spherical shells there is a spherical region of charge concentration and a spherical region of charge depletion. But in these regions there is no localization of electrons of opposite spin into pairs. There are no Lewis pairs or electron pair domains in an inner shell. The domain of each electron is spherical and fully delocalized through the shell. 

Fig. 10. Molecular and electronic structure of cation 74. (a) Perspective view on the cation 74. (b) Coordination sphere of the Pb atom, selected interatomic distances (pm) and angles (deg) Pb-(center CC, C7), 281.7 Pb-(center C6 CT), 280.7 Pbl-C2, 227(2) Pbl-C2 221.1(15) Pbl-C8, 231.0(19) C6-C7, 133(2) CC-CT 132(2) (center C6, C7)-Pb-(center C6, C7 163.6. (c) Contour plots of the Laplacian distribution [V p(r)] in the plane containing the atoms Pb, C6 and C7. Solid and dotted lines designate regions of local charge concentration and depletion, respectively. The bond paths are indicated by the solid back lines, bond critical points are marked with a black square. (Reprinted with permission from Ref. 53. Copyright 2003, Wiely-VCH.)...

Alternatively, one could investigate the Laplace concentration of the electron density, -V p (r), rather than p (r) itself. The Laplace concentration indicates regions in the molecule in which negative charge concentrates and is depleted . Therefore, it is the correct quantity to reveal changes in the electronic structure due to through-space interactions leading to homoaromaticity. 

An n-type semiconductor at room temperature has free electrons in the conduction band compensated by positively charged immobile ionized donors (Section 4.3.6). The concentration of free electrons (donors), buik, is usually between lO and lO cm", hence 10 — 10 smaller than in a metal. A common situation is that the electrons are transferred from the semiconductor to a redox system, so that equilibrium is achieved. The surface region of the solid is then depleted of free electrons (Figure 19) a depletion layer of width rfsc is formed. The concentration of free electrons in the depletion layer, w(x), is much smaller than the concentration of the immobile donors buik- Thus, the charge density in the depletion layer per unit volume is e buik- Therefore, the surface charge density at the solid side is ... 

Where then to look for the Lewis model, a model which in the light of its ubiquitous and constant use throughout chemistry must most certainly be rooted in the physics governing a molecular system If one reads the introductory chapter on fields in Morse and Feshbach s book Methods of theoretical physics (1953), one finds a statement to the effect that the Laplacian of a scalar field is a very important property, for it determines where the field is locally concentrated and depleted. The Laplacian of the charge density at a point r in space, the quantity V p(r), is defined in eqn (2.3). This property of the Laplacian of determining where electronic charge is locally concentrated and depleted follows from its definition as the limiting difference between the two first derivatives which bracket the point in question as defined in eqn (2.2) and illustrated in Fig. 2.2. 

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