Electronic charge density gradient vector field

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°.

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... 

Laplacian of the electron density, W p(r) scalar derivative of the gradient vector field of p(r). It determines where electronic charge is locally concentrated, < 0, and depleted, > 0. The Laplacian can be partitioned into energy densities. It can be demonstrated that there is a local virial theorem... 

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