The Valence Shell Charge Concentration

We can understand how —d2fix)/dx2 shows us where fix) is locally concentrated or depleted in another way by approximating it by the finite difference formula  

This approximation becomes more accurate the smaller the interval Ax. The term t[/(xo + Ax) + /(x0 - Ax)] is effectively the average of the function s values in the local neighborhood of x0. As a result, —f (x) will be positive if f(x) is greater than the values in its immediate neighborhood (since (Ax)2 0). In other words, if the function is locally concentrated, then —f (x) 0. On the other hand, if f (x) 0, the function is locally depleted. 

The four maxima and the saddle point are critical points in the function L(r) analogous to the maxima and saddle points in p(r) discussed in Chapter 6. Every point on the sphere of maximum charge concentration of a spherical atom is a maximum in only one direction, namely, the radial direction. In any direction in a plane tangent to the sphere, the function L does not change therefore the corresponding curvatures are zero. When an atom is part of 

The two bonding maxima and the two nonbonding maxima in the valence shell charge concentration of the sulfur atom in SC12 have an approximately tetrahedral arrangement just like the two bonding domains and the two nonbonding domains of the VSEPR model. As we shall see, this correspondence is found for many other molecules, and so it seems reasonable 

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Figure 7.6 Contour maps of L for H2O, (left in the molecular plane right, perpendicular to the molecular plane), NH3 (in a symmetry plane through N and one H), and CH4 (in a symmetry plane through C and two H s). The maxima in the valence shell charge concentration are indicated by the dots.

The localization of the valence shell electrons of the central atom into pairs that is the basis of the VSEPR model can be clearly seen in the contour map of L for a plane through the carbon atom and two Cl ligands for the CC14 molecule given in Figure 8.8. There is a maximum in the valence shell charge concentration of both carbon and chlorine along each... 

The postulates of VSEPR theory are consistent with the partitioning of electron density according to Bader s atoms-in-molecules method [173], in which the electron pairs return as the valence shell charge concentration. 

The outer quantum shell of an atom is divided into an inner region over which < 0 and an outer one over which > 0. The portion of the shell over which V p < 0, is called the valence-shell charge concentration or VSCC. Within this shell is the sphere over whose surface the valence electronic charge is maximally and uniformly concentrated. 

The Laplacian thus displays where the electronic charge is locally concentrated or depleted [25, 26]. The topology of the valence shell charge concentration (VSCC), the region of the outer shell of an atom over which V p < 0, is in accordance with the Lewis and valence shell electron pair repulsion model. To each local maximum in the VSCC, a pair of bonded or non-bonded electrons can be assigned (Fig. 2). 

Figure 10.53 shows plots of L for our example diimine, HN=NH, in relief and contour modes. The nitrogen atoms show a shell structure but the hydrogen atoms do not. This is the usual appearance of atoms from different rows of the periodic table. More interesting are the valence-shell charge concentrations (CCs). 

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