Gradient vector field

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. 

Figure 11. (a) The density and (b) its corresponding gradient vector field of BCI3. (See legend to Fig. 2 for contour values.)... 

Quantum mechanics applies to a segment of a system, that is, to an open system, if the segment is bounded by a surface of zero flux in the gradient vector field of the density. Thus the quantum mechanical and topological definitions of an atom coincide [1]. The quantum mechanical rules for determining the average value of a property for a molecule, as the expectation value of an associated operator, apply equally to each of its constituent atoms. 

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... 

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...

Recall that the operation of V on a scalar quantity is the gradient, which is a vector. For example, if V is operated on a scalar pressure field P, then V P is the pressure gradient vector field, which can have different values in the three spatial directions. The operation of V on a vector field can either be the divergence or the curl of the vector field. The former is obtained by the dot product (also called the scalar product) as V v or div 1 , where the result is a scalar whereas, the latter is obtained by the cross product (also called the vector product) V / v. or curl v, and the result is a vector field. 

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... 

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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