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Gradient trajectory, path

Figure 1. Gradient trajectory map of the crystalline urea in the molecular plane. Bond-paths are indicated by heavy lines, those corresponding to the interaction surface crossing the plane are drawn by weaker, while those originating from a (3,-l-3)CP and terminating at a (3,-3)CP by the weakest lines. Reprinted with permission from Ref. [78]. Figure 1. Gradient trajectory map of the crystalline urea in the molecular plane. Bond-paths are indicated by heavy lines, those corresponding to the interaction surface crossing the plane are drawn by weaker, while those originating from a (3,-l-3)CP and terminating at a (3,-3)CP by the weakest lines. Reprinted with permission from Ref. [78].
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. [Pg.127]

At any given instant the equation S(x, t) = const, defines a surface in Euclidean space. As t varies the surface traces out a volume. At each point of the moving surface the gradient, VS is orthogonal to the surface. In the case of an external scalar potential the particle trajectories associated with S are given by the solutions mx = VS. It follows that the mechanical paths of a moving point are perpendicular to the surface S = c for all x and t. A family of trajectories is therefore obtained by constructing the normals to a set of... [Pg.106]

FIGURE 11. Gradient vectorfield 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... [Pg.64]

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. [Pg.375]

Jet trajectory. A free liquid jet in air will describe a trajectory, or path under the action of gravity, with a vertical velocity component which is continually changing. The trajectory is a streamline consequently, if air friction is neglected, Bernoulli s theorem may be applied to it, with all the pressure terms 0. Thus, the sum of the elevation and velocity head must be the same for all points of the curve. The energy gradient is a horizontal line at distance V2/2g above the nozzle, where V is the velocity leaving the nozzle. [Pg.434]

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... [Pg.296]

Atoms and bonds are defined, respectively, by surfaces and lines embedded in three-dimensional space. A surface and a line are submanifolds of dimensions two and one, respectively, of R. It is necessary that the surface or line be embedded smoothly in R, i.e. that it possess a unique tangent hyperplane at each point. This condition of smoothness is certainly satisfied in the present case since an interatomic surface and a bond path are defined in terms of the trajectories of the gradient vectors of the charge density associated with a (3, — 1) critical point, as the surface and axis of a ring structure are defined by the trajectories associated with a (3, + 1) critical point. One can picture the tangent plane to a point of the curved interatomic surface illustrated in Fig. 2.6, a plane defined by the gradient vector of p at that point on the surface. [Pg.91]

A gradient path of V has a simple physical interpretation. It is a line of force—the path traversed by a test charge moving under the influence of the potential F(r X). At a critical point other than a (3, — 3) critical point, the force vanishes. Thus a critical point in the field V(r X) denotes a point of electrostatic balance between the attractors of the system. Since trajectories defining the surface which separates neighbouring basins satisfy the zero-flux condition... [Pg.99]

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... [Pg.310]


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See also in sourсe #XX -- [ Pg.444 , Pg.445 ]




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