Trajectories of the gradient vector field

Since the gradient vector of a scalar points in the direction of greatest 

Every trajectory must originate or terminate at a point where Vp(r) vanishes, i.e. at a critical point in p. 

Trajectories cannot cross since Vp(r) defines but one direction at each point r. 

All but two of the trajectories shown in the Fig. 2.5 originate at infinity where Vp vanishes. The remaining two originate at the (3, — 1) critical point located between the nuclei and these, along with all but another two of the trajectories, terminate at one of the two nuclei. The remaining two trajectories in this plane terminate at the (3, — 1) critical point. 

An understanding of these properties of trajectories is sufficient for a reader to proceed with the discussion of their use in the definition of structure given in the following section. The present section is concluded with the derivation of the differential equation for Vp(r) and the corresponding integral curve which defines the points on its trajectory. 

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Fig. 5.42 Contour lines for p, the electron density distribution, in a homonuclear diatomic molecule X2. The lines originating at infinity and terminating at the nuclei and at the bond critical point C are trajectories of the gradient vector field (the lines of steepest increase in p two trajectories also originate at C). The line S represents the dividing surface between the two atoms (the line is where the plane of the paper cuts this surface). S passes through the bond critical point and is not crossed by any trajectories...

Fig. 5.43 Heteronuclear (as well as homonuclear cf. Fig. 5.42) molecules can be partitioned into atoms. S represents a slice through the zero-flux surface that defines the atoms A and B in a molecule AB. The lines with arrows are the trajectories of the gradient vector field. S passes through the bond critical point C and is not crossed by any trajectory lines...

The presence of a (3, —1) critical point in the electron density between neighbouring atoms in an equilibrium geometry signifies that the atoms are linked by a line of maximum density, a bond path, and that the atoms are bonded to one another. The bond path is defined by the unique pair of trajectories of the gradient vector field of the density Vp(r) that terminate, one each at the nuclei. The set of trajectories of Vp(r) that terminate at a (3, —1) critical point defines the interatomic surface that separates the... 

The interaction of two atoms results in the formation of a (3, — 1) or bond critical point in the charge density. The trajectories of the gradient vector field of p which originate and terminate at this critical point define the bond path and interatomic surface, respectively. Thus the properties of the charge density at such a critical point serve to... 

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... 

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...

Nucleus attractor of the gradient vector field of p r) where trajectories starting at infinity terminate. 

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. 

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. 

In the gradient vector field of a diatomic molecule AB (or any general molecule), one can distinguish three types of trajectories First, there are just two trajectories that connect the... 

Fio. 6.13. (a) The trajectories of p representing the gradient vector field of a free atom, (b), (c) The distorted gradient vector fields of bound atoms. The plane in (b) contains H-C-C-H nuclei of staggered ethane. The plane shown in (c) contains the nuclei representing the... 

The gradient vector field of the charge density is represented through a display of the trajectories traced out by the vector Vp. A trajectory of Vp, also called a gradient path, starting at some arbitrary point Tq is obtained by calculating Vp(ro), moving a... 

The gradient vector field of the electron density is illustrated for diborane in Figures 1 and 2. The trajectory diagrams on... 

Since an interatomic surface is defined by a set of trajectories of Vp that terminate at a cp and since trajectories never cross, an interatomic surface S(r) is one of local zero flux in the gradient vector field of the electron density that is, it is not traversed by any trajectories of Vp. The zero-flux property is expressed in equation (1) in terms of //(r), the unit vector... 

FIGURE 11. Gradient vector field of the HF/6-31 G(d,p) electron density distribution p (r) calculated for the plane of the cyclopropane ring. Bond critical points p are denoted by dots. There are three different types of trajectories type 1 trajectories start at infinity or the centre of the ring and end at a carbon nucleus type II trajectories (heavy lines) define the bond path linking two neighbouring carbon atoms type III trajectories form the three zero-flux surfaces between the C atoms (in the two-dimensional display only their traces can be seen). They terminate at the bond critical points... 

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