Gradient vector field of the charge density

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 

An atomic surface 5 comprises the union of a number of interatomic surfaces separating two neighbouring basins and some portions which may be infinitely distant from the attractor. The atomic surface of a carbon atom in cyclopropane, for example, consists of four interatomic surfaces, two with the other carbon atoms and two with hydrogen atoms. 

If the topological property which defines an atom is also one of physical significance, then it should be possible to obtain from quantum mechanics an equivalent mechanical definition. This can be accomplished through a generalization of the quantum action principle to obtain a statement of this principle which applies equally to the total system or to an atom within the system. The generalization of the action principle to a subsystem of some total system is unique, as it applies only to a region that satisfies a particular 

Because of the dominant topological property exhibited by a molecular charge distribution—that it exhibits local maxima only at the positions of the nuclei—the imposition of the quantum boundary condition of zero flux leads directly to the topological definition of an atom. Indeed the interatomic surfaces, along with the surfaces found at infinity, are the only closed surfaces in a three dimensional space which satisfy the zero flux surface condition of equation 6. 

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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... 

The atomic statement of the principle of stationary action, eqn (6.3), yields a variational derivation of the hypervirial theorem for any observable G, a derivation which applies only to a region of space H bounded by a surface satisfying the condition of zero flux in the gradient vector field of the charge density,... 

The atomic statements of the Ehrenfest force law and of the virial theorem establish the mechanics of an atom in a molecule. As was stressed in the derivations of these statements, the mode of integration used to obtain an atomic average of an observable is determined by the definition of the subsystem energy functional i2]. It is important to demonstrate that the definition of this functional is not arbitrary, but is determined by the requirement that the definition of an open system, as obtained from the principle of stationary action, be stated in terms of a physical property of the total system. This requirement imposes a single-particle basis on the definition of an atom, as expressed in the boundary condition of zero flux in the gradient vector field of the charge density, and on the definition of its average properties. 

The vanishing of the Lagrangian integral if° [T, t] is seen to be a consequence of the vanishing of the flux in the gradient vector field of the charge density at the infinite boundary of the system... 

In this work we explore using only properties derived from the gradient vector field of the charge density, using the theory of atoms in molecules (AIM) In this way we demonstrate the usefulness of the charge density to understand such properties as metallicity when ice is in such extreme conditions (> 100 GPa, since above this pressure the tunneling effect of the proton ceases). The pressure is applied via an external isotropic force of compression. 

Since calculation of the metallicity measure (rb) only depended on being able to obtain the gradient vector field of the charge density, this quantity it may be useful to experimentalists in this field. 

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