Bond ellipticity

It was discussed in Section III that the structure of a system, as characterized by its molecular graph, is determined by the number of bond, ring and cage critical points. These are stable critical points in the sense that they retain their properties and hence the charge density retains its corresponding form as the nuclei are displaced. A structure which contains only stable critical points will thus persist over some range of all possible 

The sign and magnitude of the Laplacian of the charge density at the bond critical point, the quantity V Pb provide a classification of the mechanics of the atomic interactions involved in chemical bond formation. This aspect of the classification of the bonds in hydrocarbon molecules is now presented following an introduction to the properties of the Laplacian of a molecular charge distribution. 

---

The value of the C—P bond ellipticity e calculated for phosphabenzene with the 6-31G basis set to be 0.2558 (89MI4) exceeds that of the P—C bond in phosphine CH3PH2 (0.1457), which points to a somewhat doublebond character of the CP bond in (95). 

Another insight into the nature of a covalent bond is provided by analysing the anisotropy of the electron density distribution p (r) at the bond critical point p. For the CC double bond, the electron density extends more into space in the direction of the n orbitals than perpendicular to them. This is reflected by the eigenvalues 2, and k2 of the Hessian matrix, which give the curvatures of p (r) perpendicular to the bond axis. The ratio 2, to /.2 has been used to define the bond ellipticity e according to equation 8S0 ... 

Although a distinction between a and it electrons is no longer appropriate when analysing p (r), it is nevertheless appealing to relate the bond ellipticity e to the n character of a double bond80,82. 

SCHEME 8. Schematic presentation of bond ellipticities in the cases of benzene, 1,3-butadiene and cyclobutadiene. Major axes of bond ellipticities are indicated by double-headed arrows... 

FIGURE 14. CC bond orders n and bond ellipticities i of cyclopropyl homoconjugated molecules (a) norcaradiene, (b) bicyclo[2.1.0]pentene, (c) bicyclo[3.1. OJhexenyl cation. On the right, the preferred mode of electron delocalization is indicated by dashed lines. Also given is the number of delocalized electrons as calculated from topological bond orders. See text... 

The 7r-character of the closing bond as measured by the bond ellipticity e is larger than that of cyclopropane. 

SCHEME 15. Surface delocalization in homoaromatic and homoantiaromatic molecules. Major axes of bond ellipticities are indicated by arrows the direction of surface delocalization in the three-membered ring is given by a bold arrow... 

Bond ellipticity Anisotropy of p (p) (local property) Molecular graph Molecular structure... 

Fig. 6.8. A comparison of the C-C bond and atomic properties of the carbon atoms in the pentadienyl cation with the corresponding fragment (as indicated by the numbering of the atoms) in the benzenium ion, protonated benzene. The bond properties compared are bond order n, bond ellipticity e, and the Laplacian at the bond critical point, The atomic properties compared are the net charges on the carbons g(C) and their quadrupole moments Q,.(C). Also given are the differences in energy of the carbon atoms, A (C) = FfClCCeH. ] - ElQCCjH ...

---