Topological features of the electron density

Now that we have seen how eleetron densities can be determined by high-angle X-ray diffraction, it is time to discuss methods for extracting chemically relevant information from the electron density distribution in three dimensions. The topological analysis of electron density was first mentioned in Section 2.4 in a more general context, and it should be stressed here that the mathematical tools to be discussed in this section can be applied equally to electron density distributions determined experimentally or derived quantum mechanically (Section 3.6). This direct link between theory and experiment therefore offers a unique way to view details of the electronic stmcture of chemical systems and to inspect structural details that lie beyond the standard viewpoint of three-dimensional molecular structure defined purely by the positions of atomic nuclei. In this 

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We can finally comment on the validity of the AIM analysis of bond critical point in solids. Its validity has been recently questioned [18,83] on the basis of a recent finding that the topological features of the electron density of a complex... 

This multipole refinement leads to a significant decrease in the residual indices and a reduction of features in the residual maps. Also, a topological study of the electron density described in this form allows us to obtain more quantitative information on electron distributions than can be derived from electron density maps, the bond order and its force, dipolar moments, covalent character, etc. 

Kobayashi, K., 8c Nagase, S. (1999). Bonding features in endohedral metallofullerenes. Topological analysis of the electron density distribution. Chemical Physics Letters, 302(3-4), 312-316. 

Most of the relevant features of the charge density distribution can be elegantly elucidated by means of the topological analysis of the total electron density [43] nevertheless, electron density deformation maps are still a very effective tool in charge density studies. This is especially true for all densities that are not specified via a multipole model and whose topological analysis has to be performed from numerical values on a grid. 

Hybrids constructed from hydrogenic eigenfunctions are examined in their momentum-space representation. It is shown that the absence of certain cross-terms that cause the breaking of symmetry in position space, cause inversion symmetry in the complementary momentum representation. Analytical expressions for some simple hybrids in the momentum representation are given, and their nodal and extremal structure is examined. Some rather unusual features are demonstrated by graphical representations. Finally, special attention is paid to the topology at the momentum-space origin and to the explicit form of the moments of the electron density in both spaces. 

A full description is beyond the scope of this review, but it is noted that the topological method identifies other chemical features in the electron density. The union of all bond paths gives a bond path network that is normally in a 1 1 correspondence with the chemical bond network drawn by chemists. The bond paths for bonds in strained rings are curved, reflecting their bent nature. In Figure 6, we show the gradient paths in the molecular plane of cyclopropane. The C—C bond paths are distinctly bent outward. The value of the Laplacian at the bond critical point discriminates between ionic and covalent bonding." Maps of the Laplacian field reveal atomic shell structure, lone pairs, and sites of electrophilic and nucleophilic attack. The ellipticity of a bond measures the buildup of density in one direction perpendicular to the... 

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