Surface of zero flux

Also indicated by arrows are the two trajectories that terminate at the BCP in this symmetry plane. They are members of the infinite set of such trajectories that define the interatomic surface of zero-flux in Vp between the boron and fluorine atoms. 

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. 

Chemists have long been intrigued by the question, Does an atom in a molecule somehow preserve its identity An answer to this question comes from studies on the topological properties of p(r) and grad p(r). It has been shown that the entire space of a molecule can be partitioned into atomic subspaces by following the trajectories of grad p(r) in 3D space. These subspaces themselves extend to infinity and obey a subspace virial theorem (2 (7) + (V) = 0). The subspaces are bounded by surfaces of zero flux in the gradient vectors of p(r), i.e., for all points on such a surface,... 

Since the surface is not crossed by any gradient lines, it is referred to as the surface of zero flux. As further discussed below, the virial theorem is satisfied for each of the regions of space satisfying the zero-flux boundary condition. 

For a basin Q defined by the surface of zero flux, and therefore implicitly also for the whole system, integration over G(r) gives the kinetic energy K, defined by... 

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 constraint on the variation of its action integral. The constraint requires that the subsystem be bounded by a surface of zero flux in the gradient vectors of the charge density, i.e. 

Since the atom Q is bounded by a surface of zero flux, Z,(Q) = 0 and one obtains the atomic statement of the virial theorem,... 

It has been shown that the principle of stationary action for a stationary state applies to a system bounded at infinity and to one bounded by a surface of zero flux in Vp(r). It is demonstrated in Chapter 8, through a variation of the action integral, that the same boundary conditions are obtained in the general time-dependent case. One may seek the most general solution to the problem of defining an open system by asking for the set of all possible subsystems to which the principle of stationary action is applicable. Thus, one must consider the variation of the energy functional f2 3 defined as... 

The second equality given in eqn (6.70) follows from the definition of X(r) in eqn (5.49). The integration of this energy density over a region of space bounded by a surface of zero flux in Vp yields an energy e( ) which will satisfy the various statements of the atomic virial theorem,... 

Complete localization is, of course, possible only for an isolated system. What is remarkable, however, is the extent to which the electrons of atoms in an ionic molecule approach this limit of perfect localization, with /(fi) values in excess of 95 per cent not being uncommon. In systems, such as the fluorides and chlorides of lithium and sodium displayed in Fig. E7.2, the atomic surface of zero flux is found to minimize the fluctuation in the atomic populations and, thus, the magnitude of the correlation hole per particle is an extremum for such atoms. The properties of the number and pair densities for these... 

We return to eqn (8.118) for the case where D refers to a region of space bounded by a surface of zero flux in the gradient vector field of the charge density. In addition to retaining the terms at the time end-points which arise from the generalization of the variation and which are present for a total system as well, we must consider the terms arising from the non-vanishing of... 

I) Isolated singularities that must be of the (3, 1)-type (see Figure 9) and are associated with separa-trices that are toplological spheres. A separatrix is a surface of zero flux when this is a topological sphere (i.e., a closed surface that may be continually deformed into a sphere), it defines a region of space where charge circulation may occur but outside of... 

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 bond path is always found to be accompanied by a shadow graph, the virial path, first discovered by Keith, Bader, and Aray [17]. The virial path is a line of maximally-negative potential energy density in three-dimensional space that links the same pair of atoms that share a bond path and an interatomic surface of zero-flux. No theoretical basis has ever been provided that requires the presence of a virial path as a doppelganger of every bond path that links two chemically bonded atoms, however, there is no known computational violation of this observation to date known to the authors. The presence of the virial path links the concept of chemical bonding directly with the concept of energetic stability as amply discussed in literature on QTAIM. 

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