Electron density ellipticity

We can measure the extent electronic charge is preferentially accumulated by a quantity called the ellipticity e. At the bond critical point it is defined in terms of the negative eigenvalues (or curvatures), Aj and A2 as e = (A1/A2) — I. As A1 < A2 < 0, we have that A ]/A2 > 1, and therefore the ellipticity is always positive. Tf e = 0 then we have a circularly symmetric electron density, which is typically found at bond critical points in linear molecules. 

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. 

Figure 6.17 Contour map of p in the interatomic surface associated with the CC bond critical point in ethene. The plane of the plot is perpendicular to the molecular plane. The C and two H nuclei are projected onto the plane of the plot to indicate the orientation of the molecule. We see that electronic charge is preferentially accumulated in the direction perpendicular to the molecular plane, giving an elliptical shape to the electron density in this plane.

Electron densities of ortho- and para-positions of biphenyl are higher than that of meta-posltions. Sterically non-controlled isopropylatlon of biphenyl at low temperature occurred predominantly at ortho and para-posltlons to give 2-and 4-IPBP because of the electrophilic nature of the alkylation. However, selective formation of 4-IPBP over HM is controlled by the sterlc restrictions depending on the elliptical pore of the zeolite and on the conformation of the transition state for the formation of products. The molecule 2-IPBP has approximately 0.75 nm of diameter In a twisted bulky conformation with an angle of 64°[8]. The formation of 2-IPBP is prevented over HM because the corresponding transition state with bulky conformation requires more space than is available at the acidic sites of HM. On the other hand, the formation of 4-IPBP proceeds unhindered because of its smaller transition state. The formation of 3-IPBP Is also less hindered because of flexible conformations at transition states In HM pore. 

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 ... 

Then, analyzing the electron density topology requires the calculation of Vp and of the hessian matrix. After diagonalization one can find the critical points in a covalent bond characterized by a (3, -1) critical point, the positive curvature X3 is associated with the direction joining the two atoms covalently bonded, and the X2, curvatures characterize the ellipticity of the bond by ... 

The same type of calculations have been performed using experimental X-ray structure factors on crystalline phosphoric acid, 7V-acetyl-a,P-dehydrophenyl-alamine methylamide, and N-acetyl-1 -tryptophan methylamide by Souhassou [60] on urea, 9-methyladenosine, and imidazole by Stewart [32] and on 1-alanine [61] and annulene derivatives [62] by Destro and co-workers. The latter authors collected their X-ray data at 16 K [63]. Stewart [32] showed that the positions of the (3, -1) critical points from the promolecule are very close to those of the multipole electron density, but that large differences appear in comparing the density, the Laplacian maps, and the ellipticities at the critical points. Destro et al. [67] showed that the results obtained may be slightly dependent on the refinement model. 

Deformation density maps give some indications of the deformation from sphericity (or ellipticity) of the electrons of the atoms in the model as a result of chemical bonding or the existence of lone-pair electrons. They should, however, be interpreted with caution, especially with respect to the resolution of the electron-density map obtained from them. The maximum value of sin 0/A should be much higher than normally used, generally requiring short-wavelength X rays and low temperatures of measurement. 

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... 

Further investigation of 2 Pb via analysis of the Laplacian of the electron density and the electron localization function (ELF) [34] reveals a strong banana P-P-bond with large p-character (the bond ellipticity is 0.45) in the plane of the ring, and partial double bond character (Wiberg bond indices = 1.2) for the PC-bonds (Fig. 5). 

---