Contour diagrams, difference density

Distributions Ap(r) have been determined for various derivatives of 1 by both ab initio and X-ray diffraction studies66. In Figure 10, a contour line diagram of Ap(r) in the ring plane of cis, cw-1,2,3-tricyanocyclopropane is shown66a. Positive difference densities are found between the three C atoms, but the Ap(r) maxima are displayed up to 0.3 A from the intemuclear axis66. The displacement of the maxima is considered to indicate the bent bond character of the CC bonds of 1. 

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. 

Figure 5. Contour diagram of difference density functions for p - Pt (5a) and p - pjy (5b) and of transition density matrix Pl.IV (5c) ( ).

Fig. 2. A comparison between contour diagrams of the density difference Ap(r) (first column) and entropy difference A (r) (second column) functions for representative diatomics (H2, N2, HF, LiF) and linear triatomics (HCN and HNC). In view of the qualitative nature of this comparison, to show more dramatically the overall similarity of the two surfaces not equidistant contour values have been selected. The corresponding profiles of A (r), for the cuts along the bond axis, are shown in the third column of the figure ([29]).

Fig. 15. Difference density (DD) maps of nitrobenzene (65) with respect to a procrystal of spherically averaged atoms, on cuts vertical to the molecular plane through the midpoints of bonds, as shown in the left top diagram. Under each DD map, the difference between this map and the same map rotated by 90° is shown (AAp), indicating the n-bond ellipticity. Contour line values are 0.05n e/A , n=l,2,3,..

FIGURE 8.1 Contour diagrams of the molecular density difference function Ap(r) = p(r) -p (r) (first column), the information-distance density As(r) = pfrl/fiefr)] (seeond eolumn), and its approximate, first-order expansion As(r) Ap r)w r) (third eolumn), for seleeted diatomie and linear triatomic molecules H2, HF, LiF, HCN, and HNC. The solid, pointed, and broken lines denote the positive, zero, and negative values, respectively, of (equally spaced) contours. (From Nalewajski, R. F. et al., Int. J. Quantum Chem., 2002, 87, 198.)... 

FIGURE 8.2 A comparison between diagrams of nonequidistant contours of the density difference function Ap(r) (first column) and the entropy difference function hp(r) (second column) for the linear molecules in Figure 8.1. The corresponding profiles of h (r) for cuts along the bond axis are shown in the third column. (Reprinted from Nalewajski, R. F. and Broniatowska, E. J. Phys. Chem. A, 2003,107, 6270.)... 

Figure 3 f " contour diagram for H2CO in a plane perpendicular to the molecular plane containing the CO bond. Drawn are the differential f"(r), the finite difference f (r) (corresponding to AN = 1 and AN = 0.01) and the LUMO density. 

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]. 

Fig. 7. Polymorphic forms of o-ethoxy cinnamic acid molecular diagrams and deformation density maps close to the mean plane of the molecules in the a- and the y terms (contours at 0.12 eA 3). Subtle differences in the cinnamoyl bond and the hydrogen bond region are noticeable. The Laplacians of the intermolecular hydrogen bonds in the acid dimer are shown in the relief maps along side (range -250 to 250 eA 5).

Figure 6 shows a contour line diagram of the electron density distribution of the van der Waals complex Hcj (He,He distance 2.74 A). The electron density at the midpoint between the two He atoms is just 0.008 e/A, quite different from the values found for a typical covalent bond between first-row elements (1-5 e/A ). Despite the smallness of p(r) in the internuclear region, the He nuclei are linked by a MED path and the midpoint is the position of a (3, — 1) critical point (Fig. 6). As pointed out above this does not imply the existence of a chemical bond. The energy density H(r) is positive at the (3, — 1) critical point, which means that the kinetic energy rather than the potential energy dominates in the internuclear region. There is no chemical bond between the He atoms.

Fig. 11. Profile plot along the internuclear axis (above) and contour line diagram (below) of the difference electron density Ap(r) = —...

Fig. 11. Features of a difference electron density projection for TMA (3) H O PA. The difference electron density which represents the enclathrated picric acid molecules is contoured at levels of 1, 2, 3 e A In TMA H2O PA the molecules lie in the (III) planes and the hydrogen bonds do not lie in the plane of the projection but are directed out of this plane. For TMA H O the present diagram serves as a slightly distorted representation of the arrangement in the layers of the framework TMA and HjO molecules, as viewed along a. The hydrogen bonds have the following lengths, averaged over corresponding bonds in TMA H O % PA and the five independent layers in TMA HjO (the values in parentheses are the RMS deviations from the means) a 2.641 (15) A b 2.632 (52) c 2.874 (44) d 2.850 (52) e 2.592 (29). (Taken from Ref. )...

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