Electron density distribution ellipticity

The shape of the electron density distribution in a plane through the bond critical point and perpendicular to the bond as measured by its ellipticity e. 

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 ... 

Figure 2. A summary of the properties of the eleetron density distribution for the skeletal SiOSi dimers in a set of H6Si207 moleeules with geometries fixed at those observed for the Si207 dimers in eoesite. In (a), (b) and (e), respectively, the individual SiO bond lengths, R(SiO), observed for eoesite are plotted against Ai, A2 and A.i, the curvatures of the electron density distribution calculated for the molecules at their saddle points Fc. In (d), R(SiO) is plotted against the magnitude of the electron density, p(rc). In (e) R(SiO) is plotted against G r /p tc) where G(rc)/p(rc) is the kinetic energy density and in (f) R(SiO) is plotted vs. ellipticity, f, of the bonds.

It is found that multiple bonds do not appear as such in the topology of the electron density. However, the value of the charge density at a bond critical point reflects the bond multiplicity and can indeed be empirically correlated with bond orders (refs. 93 and 94). As expected, it is also found that the charge distribution in the CC interatomic surface in ethylene has an elliptical nature associated with the presence of a tt bond. 

As mentioned in Chapter 8 (page 172), the double bond is associated with an elliptical distribution of electronic charge in the plane perpendicular to the CC nuclear axis and containing its mid-point where the electronic density has a local maximum (critical point in the theory of Bader). The relief diagram and the contour plots of Fig. 9.7 taken from the work of Bader et al. (ref. 92) show the distribution of the electronic charge density in the nuclear plane of the molecule. 

FIGURE 5. Summary of electronic distribution in aniline, (a) Bond distances (A), NBO charges [bracket, in au] and Wiberg indices (parentheses, in au). (b) Topology of the electron density determined from atom-in-molecule calculations p(r) = electron density, L = Laplacian of the density defined as L(r) = —V2p(r) and e = ellipticity of the bond critical point, (c) Laplacian map of the density, (d) Iso-surfaces of the electron localization function, ELF = 0.87 the values are the populations of the valence basins... 

There is one further descriptor that can be used to characterize double bonds. This is bond ellipticity. Figure 10.48(a) and (b) show an uneven distribution of the electron density along the N=N bond in HN=NH. This is more apparent in Figure 10.51, which shows a plot of the electron density in a plane perpendicular to the AIL at the BCP. The plot is elhptical. In the standard valence-bond picture a double bond consists of a rotational symmetry of electron density about the line linking the two bonded nuclei) and a it bond of two overlapping p orbitals perpendicular to this line. The preferential accumulation of electronic charge, or the amount of deviation from a circular symmetric distribution of p(r), is called... 

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