Electron density contour diagrams

Electron density contour diagrams have been derived " (Figure 2) and potential energy curves and surfaces calculated In addition to this impressive quantity of accurate work on the parent compound a number of simplified schemes were proposed by Forster, Coulson and Moffit " and by Walsh these schemes have been widely employed for the interpretation of the photoelectron spectra and other properties of the derivatives of cyclopropane. 

The constant electron density contour diagrams of one homonuclear and five heteronuclear diatomic molecules presented in Fig. 8.6 have been obtained by reasonably accurate quantum chemical calculations. The electric dipole moment of F2 is zero by symmetry. The electric dipole moments of heteronuclear molecules like LiFl, LiF, HF, CIF or CO may be calculated from their electron densities using equation (5.2). These dipole moments, in turn, allow us to calculate the ionic characters gic(calc) = /Xei(calc)(calc). In Table 8.1 we compare the calculated ionic characters of 21 heteronuclear diatomic molecules with their experimental counterparts. The agreement between experiment and calculations is good the average deviation between experimental and calculated values is less than 0.02 a.u, the maximum deviation (in KLi) is 0.05 a.u. 

Fig. 8.6. Constant electron density contour diagrams of LiH, F2, LiF, HF, CIF and CO. Contour lines have been drawn at densities equal to 1.00 x 10 and 3.16 x 10 atomic units (electrons per a ) with n ranging from —3 to 3-2 in steps of unity (note that3.16 = VTO). Bonds are indicated by lines between the nuclei, bond critical points (BCPs) by diamonds ( ). Heavy lines indicate boundaries between the atoms according to AIM theory [2].

Figure Z5 Electron density contour diagrams for HjO (a) in the molecular plane and (b) in the perpendicular plane from the ah initio SCF calculation given in [14]. (Redrawn from [14] with permission from the American Chemical Society.)...

Fig. 2.7 (a) Pictorial representation or the electron density in a hydrogen-like 2p orbital compared with lb) the electron density contours Tor the hydrogen-like 2pr orbital of carbon. Contour values are relative to the electron density maximum The xy plane is a nodal surface. The signs (+ and —) refer to those of the original wave function. [The contour diagram is from Ogryzlo, E. A4 Porter G. B J. Client. Ethic. 1963.40. 258. Reproduced with permission. ... 

Consider Fig. 6.32 which is an electron density contour map of the sodium cyanide crystal Interpret this diagram in terms of everything that you know about the structure of solid sodium cyanide. 

In a later paper, the authors218 analysed the charge re-distribution in more detail, using electron density contour maps. An energy-level correlation diagram shows no correlations between bonding levels and antibonding levels in the product. 

The simplest example of the LCAO approximation is the combination of two s-AOs to form s- (bonding) and (antibonding) MOs. This is shown in Fig. 2.2 for the dihydrogen molecule. Figure 2.2a shows the simple orbital interaction diagram whereas Fig. 2.2b shows 3D-electron density contour plots for the AOs and MOs. 

Figure 3.10 Representations of the electron density ip2 of the Is orbital and the 2p orbital of the hydrogen atom. (b,e) Contour maps for the xe plane. (c,f) Surfaces of constant electron density. (a,d) Dot density diagrams the density of dots is proportional to the electron density. (Reproduced with permission from the Journal of Chemical Education 40, 256, 1963 and M. J. Winter, Chemical Bonding, 1994, Oxford University Press, Fig. 1.10 and Fig. 1.11.)...

The density is a maximum in all directions perpendicular to the bond path at the position of a bond CP, and it thus serves as the terminus for an infinite set of trajectories, as illustrated by arrows for the pair of such trajectories that lie in the symmetry plane shown in Fig. 7.2. The set of trajectories that terminate at a bond-critical point define the interatomic surface that separates the basins of the neighboring atoms. Because the surface is defined by trajectories of Vp that terminate at a point, and because trajectories never cross, an interatomic surface is endowed with the property of zero-flux - a surface that is not crossed by any trajectories of Vp, a property made clear in Fig. 7.2. The final set of diagrams in Fig. 7.1 depict contour maps of the electron density overlaid with trajectories that define the interatomic surfaces and the bond paths to obtain a display of the atomic boundaries and the molecular structure. 

FIG. 10.11 Electron density in the metal-ligand plane of dichromium tetraacetate, (a) Molecular diagram, (b) deformation density through Cr—Cr and the acetyl group averaged over equivalent regions. Contours are at 0.10 eA-3. Negative contours are broken lines. Source Benard et al. (1980). 

The qualitative study of electronic structure through the electron (number) density p(r) relies heavily on linecut diagrams, contour plots, perspective plots, and other representations of the density and density differences. There is a review article by Smith and coworkers [302] devoted entirely to classifying and explaining the different techniques available for the pictorial representation of electron densities. Beautiful examples of this type of analysis can be seen in the work of Bader, Coppens, and others [303,304]. 

All this explains why the shape of an orbital depends on the orbital angular quantum number, t. All s orbitals ( = 0) are spherical, all p orbitals ( - 1) are shaped like a figure eight, and d orbitals ( = 2) are yet another different shape. The problem is that these probability density plots take a long time to draw—organic chemists need a simple easy way to represent orbitals. The contour diagrams were easier to draw but even they were a little tedious. Even simpler still is to draw just one contour within which there is, say, a 90% chance of finding the electron. This means that all s orbitals can be represented by a circle, and all p orbitals by a pair of lobes. 

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