Gradient vector field of the electron density

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. 

If such a partition scheme is to be employed, the atomic domains 2 need to be defined exphdtly. The most frequently apphed decomposition is Bader s Atoms in Molecules (AIM) approach [174,175] where the atomic domains 2 are divided by surfaces that are determined on the basis of the topology of the electron density. As the atomic volumes (or domains) are determined from the properties of the electron density, the AIM scheme can be apphed to any electron density calculated from a general wave function. In AIM theory, the boundary condition for an atom in a molecule requires the gradient vector field of the electron density Vp(r) to have zero flux [174,176],... 

Electron density plots of trans-Ammne, HN=NH, as relief plots (a) in the plane of symmetry and (b) perpendicular to it, and the same represented as contour line plots (c) and (d). (e) Gradient vector field of the electron density of trans-diimine, HN=NH. (f) Contour plot of the electron density with interatomic surface lines partitioning the molecular space into atomic basins (interatomic surfaces (IAS) and atomic interaction lines overlaid. All plots are based on calculations at the MP2/6-311G level of theory. 

At the very end of this section on the topological atom, and on the wider topological approach with its fundamental characteristics and consequences, we put the topology to rest and look at energy instead. Energy is a quanmm mechanical observable and the main question is how it can be partitioned. This is the topic of the next section, where we forget about the gradient vector field of the electron density, at least at the start. 

QTAIM locates the various critical points in the density and uses each bond critical point (BCP) as a starting point for the search of the inter-atomic surfaces of zero-flux in the gradient vector field of the electron density separated and shared by... 

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 9 Maps of the gradient vector field of the electron density along the symmetrical dissociation path for water illustrating the bifurcation mechanism of structural change. Also shown for structures a. b. and c are profiles of the electron density along the C2 symmetry axis. In a there are no cps along this axis in b there is a single degenerate cp where both first and second derivatives of p(r ) varysh in c there are two stable cps, the maximum associated with the H-H bond cp, the minimum with the ring cp...

Gradient vector field of the electron density obtained by the trajectories traced out by the gradient vectors. 

Fig. 3.3. Gradient vector field of the electronic charge density for the unstable conflict structure shown in Fig. 3.1(e). The plane shown contains the symmetry axis and is perpendicular to the plane of the nuclei. In the lower portion of the diagram, trajectories terminate at the (3, — 1) critical point between the hydrogen nuclei. In the upper portion, trajectories terminate at the pseudo (3, — 3) critical point at the oxygen nucleus. These two critical points are linked by the pair of trajectories which originate at the central (3, — 1) critical point indicated by the dot This is an unstable intersection of the one-dimensional manifold of this (3, — 1) critical point with the two-dimensional manifold of the (3, — 1) critical point between the protons.

Fig. 3.6. Displays of the gradient vector fields of the electronic charge density for configurations along the isomerization reaction coordinate for HCN. The label refers to the angle 0 formed between the C-N axis and the vector from the proton to the CN centre of mass. The bond path from the proton switches attractors, from the carbon nucleus to the nitrogen nucleus for some configuration lying between 6 - 72.1° and 72.4°.

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... 

Figure 4 displays the electron density distribution in the molecular plane of anthracene and contrasts it with that of phenanthrene along with the associated gradient vector field of the latter. In this figure, one can see the curved bond path linking the nuclei of the two hydrogen atoms, H4 and H5, in phenanthrene as well as the zero-flux interatomic surface they share, features lacking in the map of anthracene. 

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 critical point is the point at which the gradient vector field for the charge density is zero, that is, either a maximum or minimum along N. The condition Vp(r) N(r) = 0 applied to other paths between two atoms defines a unique surface that can represent the boundary of the atoms within the molecule. The electron density within these boundaries then gives the atomic charge. The combination of electron density contours, bond paths, and critical points defines the molecular graph. This analysis can be applied to electron density calculated by either MO or DFT methods. For a very simple molecule such as Hj, the bond path is a straight line between the nuclei. The... 

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