Laplacian distribution

Under the conditions of maximum localization of the Fermi hole, one finds that the conditional pair density reduces to the electron density p. Under these conditions the Laplacian distribution of the conditional pair density reduces to the Laplacian of the electron density [48]. Thus the CCs of L(r) denote the number and preferred positions of the electron pairs for a fixed position of a reference pair, and the resulting patterns of localization recover the bonded and nonbonded pairs of the Lewis model. The topology of L(r) provides a mapping of the essential pairing information from six- to three-dimensional space and the mapping of the topology of L(r) on to the Lewis and VSEPR models is grounded in the physics of the pair density. 

It is Lewis complementarity, on the other hand, that is operative when the mating of the molecules is determined by acid-base interactions, one that is described and predicted by the complementary mating of the lumps with the holes in the two associated Laplacian distributions. A molecule s reactive surface is defined by the zero envelope of the Laplacian distribution, the envelope that separates the shells of charge concentration from those of charge depletion. The reactive surfaces make immediately clear the locations of the lumps, the nucleophilic sites, and the holes, the electrophilic sites, that are brought into juxtaposi-... 

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.)...

The contour maps of the Laplacian distribution of the electronic charge density... 

Figure 6.4 Relief plot of the negative Laplacian distribution for acetamide in the molecular plane, from [179].

Figure 16 Contour line diagrams of the Laplacian distribution V p(r) of (a) ArF (b) ArH (c) HArF. Solid lines give areas of charge concentration (V p(r) < 0), while dashed lines give areas of charge depletion (V p(r) > 0). The solid line connecting the atomic nuclei is the bond path. The solid lines that cross the bond path at the bond critical point show the zero-flux surface crossing the shown plane. (Ref 37. Reproduced by permission of Springer Verlag)...

Figure 7.2 displays the value of the charge density and of its associated Laplacian distribution as a function of position in the y, z-plane for an argon atom. Since the charge distribution is spherically symmetric, the same display is obtained for any plane containing the nucleus. The charge density exhibits... 

The presence of a shell of charge concentration and one of charge depletion in the Laplacian distribution for each quantum shell is demonstrated in Fig. 7.3 for the four quantum shells of the Kr atom. While the diagram displays the behaviour of — V V along a given radial line, it should be borne in mind that this is a representation of the structure present in three-dimensional... 

FiO. 7,3. Schematic representation of the profile of — for the krypton atom along a radial line. Unlike a radial distribution plot, which is a one-dimensional function, the Laplacian distribution displays shell structure in three-dimensional space. 

Fig. 7.4. Representations of the Laplacian distributions of methane and methylfluoride. The figures in (a) are displays of the zero envelope of V p(r), those in (b) of the atomic graphs. The envelope encompassing the inner shell charge concentration on carbon appears as a small sphere. The envelopes of the bonded maxima in the VSCC of carbon also encompass the protons in CH and CH3F. There is a transfer of charge from C to F in CHjF and the bonded maximum along the C F axis is reduced to the small region lying between the C nucleus and the envelope on F. An atomic graph displays the connectivity of the critical points in a VSCC. The carbon nucleus is denoted by a solid cross, the positions of the remaining nuclei by open crosses. There is a bonded maximum, a (3, — 3) critical point in — V p, at each of the four vertices.

Fig. 7.14. Relief plots of the negative of the Laplacian distributions for triplet and singlet states of CFj. The lower diagrams are for the plane containing the nuclei, the upper ones for the perpendicular symmetry plane containing the C nucleus, the plane containing the non-bonded charge maxima. There are two non-bonded maxima in the triplet, one in the singlet. The point labelled a is a (3, — 1) critical point in the VSCC of triplet carbon. There is no radial maximum or lip at the point labelled h and its mirror image and the VSCC of singlet carbon exhibits holes at these two points. The maxima present in the VSCCs of the F atoms are not shown as they are larger by a factor of 10 than those on the carbons.

Properties of the charge density at the (3, — 1) critical point for a number of diatomic molecules are given in Table 7.4 and, for some polyatomic molecules, in Table 7.5. Contour and relief maps of the Laplacian distributions for some of these molecules are shown in Figs 7.15-7.17. 

The Laplacian distributions for and S2N2 in Fig. 7.13 provide further examples of shared interactions, the unequal sharing of the charge concentration in the latter molecule being an example of a polar system. The final molecule in this figure, illustrates both types of interactions. This... 

Extraordinarily small values of G(r )/p(rJ, of the order of 0.03 au, are exhibited by the non-nuclear maxima, the pscudoatoms, in the metallic clusters of Li and Na atoms illustrated in Fig. 2.11. The same small values of kinetic energy per electron are reflected in the ratio of the average values of r(f2) to Al(f2) for the pseudoatoms and, in accordance with the Heisenberg uncertainty principle, they indicate that the charge density of the pseudoatoms is loosely bound and unconfined. The Laplacian distributions for these... 

Fig. 7,16. Relief maps of the negative of the Laplacian distribution of p to contrast the distinguishing features of the shared and closed-shell limits of atomic interactions as represented by Nj and Arj, respectively. The map for Fj is intermediate between the two limits. While V pfr.,) is positive for F2 as found for Ar2, its value of p(r J Is three times larger than that forArj. Electronic charge is accumulated in the binding region of F2, as is typical of a shared interaction, but is concentrated in the atomic basins, as is typical of a closed-shell interaction. While V pfrJ > 0 for both Ar2 and Fj, the Laplacian distribution is a minimurn at r, for Arj, but a maximum at the same point in Fj. The charge densities are calculated using a 6-31IG (2d, 2p)...

The Laplacian distributions for the shared interactions illustrated in Figs 7.15 to 7.17 clearly indicate the predominance of the concentration of electronic charge over the binding region for such interactions. In all of these molecules < 0 over the binding region for each interacting pair of nuclei. 

The differing properties of the charge distributions for bound and unbound states are made very clear by comparing the Laplacian distributions of p for and Hc2 (Fig. 7.15). In the former, the atomic interaction is dominated by the contractions in p perpendicular to the bond path while in the latter it is dominated by the contractions in p towards each of the nuclei. 

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