Gradient vector path

R F W Bader s theory of atoms in molecules [Bader 1985] provides an alternative way to partition the electrons between the atoms in a molecule. Bader s theory has been applied to many different problems, but for the purposes of our present discussion we will concentrate on its use in partitioning electron density. The Bader approach is based upon the concept of a gradient vector path, which is a cuiwe around the molecule such that it is always perpendicular to the electron density contours. A set of gradient paths is drawn in Figure 2.14 for formamide. As can be seen, some of the gradient paths terminate at the atomic nuclei. Other gradient paths are attracted to points (called critical points) that are... 

Gradient vector paths around formamide. The paths terminate at atoms or at bond critical points (indicated by squares). 

Fig. 4 Map of the gradient vector field of the radial density for the plane containing the maximum number of nuclei. Each fine represents a trajectory traced out by the vector V Gradient paths of radial density in the xy-plane of the second period hydrides. The central atom is located at (0, 0). The hydrogen nuclei are clearly identified by the distinct boundary path that encloses the nucleus...

Vector quantities, such as a magnetic field or the gradient of electron density, can be plotted as a series of arrows. Another technique is to create an animation showing how the path is followed by a hypothetical test particle. A third technique is to show flow lines, which are the path of steepest descent starting from one point. The flow lines from the bond critical points are used to partition regions of the molecule in the AIM population analysis scheme. 

Points on the zero-flux surfaces that are saddle points in the density are passes or pales. Should the critical point be located on a path between bonded atoms along which the density is a maximum with respect to lateral displacement, it is known as a pass. Nuclei behave topologically as peaks and all of the gradient paths of the density in the neighborhood of a particular peak terminate at that peak. Thus, the peaks act as attractors in the gradient vector field of the density. Passes are located between neighboring attractors which are linked by a unique pair of trajectories associated with the passes. Cao et al. [11] pointed out that it is through the attractor behavior of nuclei that distinct atomic forms are created in the density. In the theory of molecular structure, therefore, peaks and passes play a crucial role. 

Figure 6.11 (a) Contour plot of p for the molecular plane of the ethene molecule, (b) The gradient vector field of the electron density for the same plane. All the gradient paths shown originate at infinity and terminate at one of the six nuclei. 

The key to investigating the topology of the electron density p is the gradient vector V p, which is perpendicular to a constant electron density snrface and points in the direction of steepest ascent. Then, a sequence of infinitesimal gradient vectors corresponds to a gradient path. Since gradient vectors are directed, gradient paths also have a direction They can go uphill or downhill. 

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. 

The saddle point between two atoms is a (3, — 1) critical point. The saddle point is the origin of the gradient vectors along the direction in which the density is a minimum. The gradient vectors in this direction link the (3, — 1) critical point with the atoms, and constitute the bond path, connecting the atoms. In the plane perpendicular to the bond path at the (3, — 1) critical point, the gradient vectors terminate as illustrated for the two-dimensional case in Fig. 6.7. 

If one analyses the gradient of p (r) not only at the point p but also at other points in molecular space, then the gradient vector field of p (r) will be obtained81. The gradient vector p (r) always points in the direction of a maximum increase in p (r). Thus, each such vector is directed toward some neighbouring point. By calculating Vp (r) at a continuous succession of points, a trajectory of Vp (r), the path traced out by the gradient vector of p (r), is obtained. 

The analysis of the gradient vector field of the charge density displays the trajectories traced out by Vp (gradient path). Because p is a local maximum at nuclear position ((3, -3) critical point), all the gradient paths at a proximity of a... 

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