Laplace concentration

If the second derivative, and hence the curvature of/, is negative at x, then / at x will be larger than the average of / at all neighbouring points, i.e. / concentrates at point x73. Therefore - V2p(r), which is the second derivative of a function depending on three coordinates x, y and z, has been called the Laplace concentration of the electron density distribution. Furthermore, the Laplacian of pir) provides the link between electron density p(r) and energy density Hir) via a local virial theorem (equation 8)67,... 

Cremer and Kraka97 based their model on a principal of avoidance of geminal and vicinal charge concentrations , which they derived from the analysis of the Laplace concentration, -V2p(r), in the valence shell of bonded atoms. As noted in Section IV. C, the Laplacian of p(r) reflects the shell structure of an atom. Upon bond formation, the valence... 

Figure 15 presents a schematic view of how the atomic subspaces Cl, C6 and Cl 1 of 1,6-methanojl Ojannulene (35) change upon an approach of Cl to C6. Bond paths (solid lines between atoms), bond critical points (dots) and the traces of the zero-flux surfaces S (A, B) (perpendicular to bond paths) that separate the atomic subspaces are shown in Figure 15a. Clearly, the subspace C11 extends less and less into the region between C1 and C6 until the surfaces of C1 and C6 coincide and a bond path between C1 and C6 is formed. At the same time, the Laplace concentration between Cl and C6 gradually increases and coverges to the one found for a three-membered ring. As shown in Figure 15b, this change corresponds to the valence tautomerism of the l,6-methano[10]annulene to bisnorcaradiene27,54.

Through-space interactions are confirmed by the Laplace concentration -V2p(r) that reveals polarization of the electron density at the interacting atoms. 

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 analysis of the Laplace concentration, (r), yields information about the extent of through-space interactions and the concomitant changes in the molecular properties. Hence, a clear distinction between the various modes of intermolecular interactions should be possible. However, a quantification of these changes again needs an appropriate reference, something which in most cases is not present. Therefore, a description of homoconjugative interactions in terms of the Laplace concentrations has only been applied in selected cases" but has not been worked out to a more general description of no-bond homoaromaticity. 

The analysis of p(r) leads to useful information about chemical bonding. Additional information on the electroiiic structure of Ng molecules can be gained by analyzing not only p(r) but also V p(r), the Laplacian of the electron density distribution. The Laplacian of a scalar hinction f indicates where this function concentrates (V f < 0) and where it is depleted (V f > 0) [33]. For f = p(r), the Laplace concentration — V p(r) reveals where the electrons lump together in the molecule [34]. 

It is important to note that the Laplace concentration reflects the effects of all occupied MOs and, therefore, provides a more reliable description than the frontier MOs that often must be supplemented by the next lower (higher) orbital to the HOMO (LUMO) to reproduce experimental findings. However, it must be born in mind that the analysis of the Laplace concentration as well as the frontier orbital model are both not sufficient to distinguish between bonding and nonbonding situations and to draw a border line between electrostatic and covalent Ng,X bonding. 

In Fig. 3, perspective drawings of the Laplace concentration of He, Ne, and Ar are shown. The He atom possesses a single concentration peak surrounded by a sphere of depletion of negative charge (Fig. 3a). For Ne (Fig. 3b), two concentration and two depletion spheres can be distinguished. The inner concentration sphere is at the nucleus while the outer concentration sphere is about in the valence region. Again, one can associate the inner shell with the Is electrons and the outer concentration shell with the valence electrons of Ne. Similarly one can speak of an inner shell depletion sphere and a valence shell depletion sphere. 

For Ar (Fig. 3c), three pairs of spheres with concentration or depletion of negative charge exist in the Laplace concentration — V p(r) The most inner can be associated with the Is electrons of Ar the next with the 2s2p electrons, and the most outer with the valence electrons of Ar. 

Fig.7aandb. Perspective drawings of the Laplace concentration, —. V p(r), of (a) He and(b)H/B2 ions at their equilibrium distance shown with regard to a plane containing the nudd...

Investigation of the Laplace concentration — V p(r) of Hex " provides additional information. In Fig. 8, perspective drawings and contour line diagrams of — V p(r) are given for both the X E ground state and the excited state of HeN" ". The valence shell concentration of N ( P) is highly anisotropic. There are concentration lumps in the direction of the singly occupied 2p(7t) orbitals and deep concentration holes in the direction of the unoccupied 2p(a)... 

Increase of the positive charge at atom X enhances the acceptor ability for X. All HeX dications are covalently bonded according to the calculated pg and Hg values listed in Table 7. This holds also for X ions with isotropical charge distribution, i.e. species without holes in the valence sphere. The Laplace concentration of C ( S) is shown in Fig. 9a. Although isotropical, the charge concentration in the valence shell is so small that the C nucleus is insufficiently shielded and an oncoming He atom is attracted by C ( S). A weakly covalent bond is formed (D = 17.9 kcal/mol) which shows up in the Laplace concentration only by distortion in the valence shell concentration of (Fig. 9c). 

Fig. lOa-f. Contour line diagrams and perspective drawings of the Laplace concentration, — V p(r), of ArC (a, b) state, (c, d) state, direction of filled a-orbital and (e, f) 11 state, direction of unfilled Jt-orbital. In the contour line diagrams inner shell concentrations are no longer shown. Ref. [13]... 

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