Maximum electron density path

Bond critical points represent extremes of electronic density. For this reason, these points are located in space where the gradient vector V p vanishes. Then the two gradient paths, each of which starts at the bond critical point and ends at a nucleus, will be the atomic interaction line. When all the forces on all the nuclei vanish, the atomic interaction line represents a bond path. In practice, this line connects two nuclei which can consequently be called bonded [5]. In terms of topological analysis of the electron density, these critical points and paths of maximum electron density (atomic interaction lines) yield a molecular graph, which is a good representation of the bonding interactions. 

Here, pb is the bond critical point (saddle point in three dimensions, a minimum on the path of the maximum electron density). In Eq. (44), and A.2 are the principal curvatures perpendicular to the bond path. The parameters A and B in Eq. (45) determined using various basis sets are given in Bader et al. [83JA(105)5061]. Convenient parameters in the quantitative analysis of a conjugation effect are the relative 7r-character tj (in %) of the CC formal double or single bonds determined with reference to the bond of ethylene (90MI2) ... 

Type II trajectories start at a point p in the internuclear region between two bonded atoms and end at one of the two nuclei in question. There are just two trajectories per bond, which together define a path of maximum electron density (MED path) that is visible in the perspective drawing of p r) shown in Figure 9. Each lateral displacement from the MED path leads to a decrease of p(r). The point p corresponds to the minimum of p(r) along the path and to a saddle point of p(r) in three dimensions. 

There is no path of maximum electron density between the interacting atoms which, according to Cremer-Kraka27 82,83, is a necessary condition for covalent bonding. However, interaction indices derived from the electron density distribution are as large as 30% of the bond order of a normal single bond. 

The bond critical point is the origin of two special gradient paths each one traces the ridge of maximum electron density from the bond critical point to one of the two neighboring nnclei. The nnion of these two gradient paths is the bond path, which nsnally connects atoms that are joined by a chemical bond. 

R. F. W. Bader and associates at Canada s McMaster University have derived a means of describing the electron distribution associated with specific atoms in a molecule, called the atoms in molecules (AIM) method. The foundation of this approach is derived from quantum mechanics and principles of physics. It uses the methods of topology to identify atoms within molecules. The electron density of a molecule is depicted by a series of contours. Bond paths are the paths of maximum electron density between any two atoms. The critical point is a point on the bond path where the electron density is a maximum or a minimum with respect to dislocation in any direction. The bond critical point is defined by the equation... 

In addition to the electron density contour lines in Figures 4.65 and 4.66, there are also essentially straight lines cormecting the nuclei. Each of these lines indicates a bond path, which follows the path of maximum electron density from one nucleus to another. The concept of a bond path is thus quite... 

The path of maximum electron density between two nuclei is called the bond path by Bader and the minimum along... 

Bond paths are observed between bonded atoms in a molecule and only between these atoms. They are usually consistent with the bonds as defined by the Lewis structure and by experiment. There are, however, differences. There is only a single bond path between atoms that are multiply bonded in a Lewis structure because the electron density is always a maximum along the internuclear axis even in a Lewis multiple bond. The value of pb does, however, increase with increasing Lewis bond order, as is shown by the values for ethane (0.249 au), ethene (0.356 au), and ethyne (0.427 au), which indicate, as expected, an increasing amount of electron density in the bonding region. 

Fig. 7.1 The electron density p(t) is displayed in the and

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. 

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