The surface of zero flux

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. 

Method/Basis Set Mulliken Lowdin Hirshfeld Bader 

Though rare, there are cases in which the total density shows minor maxima at non-nuclear positions. As all (3, — 3) critical points are attractors of the gradient field, basins occur which do not contain an atomic nucleus. These non-nuclear basins (which have been found in Si—Si bonds1 in Li metal, and some other cases, distinguish the zero-flux partitioning from other space partitioning methods. 

3 Chemical Bonding and the Topology of the Total Electron Density Distribution 

An important function of the electron density is its Laplacian, defined as 

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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... 

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. 

The density is a maximum in all directions perpendicular to the bond path at the position of a bond CP, and it thus serves as the terminus for an infinite set of trajectories, as illustrated by arrows for the pair of such trajectories that lie in the symmetry plane shown in Fig. 7.2. The set of trajectories that terminate at a bond-critical point define the interatomic surface that separates the basins of the neighboring atoms. Because the surface is defined by trajectories of Vp that terminate at a point, and because trajectories never cross, an interatomic surface is endowed with the property of zero-flux - a surface that is not crossed by any trajectories of Vp, a property made clear in Fig. 7.2. The final set of diagrams in Fig. 7.1 depict contour maps of the electron density overlaid with trajectories that define the interatomic surfaces and the bond paths to obtain a display of the atomic boundaries and the molecular structure. 

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,... 

The properties of the topologically defined atoms and their temporal changes are identified within a general formulation of subspace quantum mechanics. It is shown that the quantum mechanical partitioning of a system into subsystems coincides with the topological partitioning both are defined by the set of zero flux surfaces in Vp(r). Consequently the total energy and any other property of a molecular system are partitioned into additive atomic contributions. 

Fig. Z5. A contour map for the NaQ molecule overlaid with trajectories of Vp. With the exception of the four trajectories associated with the (3, — 1) critical point (denoted by a dotX the tiaj ories originate at infinity and terminate at one of the two nuclei. Two trajectories originate at infinity and terminate at the (3, — 1) critical point, while two others originate at this point and terminate, one each, at the nuclei. The property of zero flux in the gradient vectors of p is illustrated for the interatomic surface whose intersection with this plane is given by the two trajectories which terminate at the critical point. An arbitrarily drawn surface is shown not to have this property of zero flux. The definition of the derivative dr/dl as the limit of Ar/A/ is also shown.

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. 

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,... 

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... 

It is the use of zero-flux surfaces for the topological definition of atoms or functional grouping of atoms in molecules that maximizes the extent of transferability of their properties between systems. 

Bond points are local maxima in two directions. Thus, they act as attractors of gradient paths in these two directions. The union of these paths defines a surface (the valley of our analogy) between the two atoms connected by the bond path through the bond point. No gradient paths cross this surface, leading to the name of zero-flux surface.Mathematically, this surface is defined as the union of all points such that... 

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. 

In Sect. 3 of their paper [ 132] the authors scrutinize the justification of the use of zero-flux surfaces to define atomic basins from a subsystem variational principle. This contention is based on Hilbert and Courant s generalisation of variation calculus to the case of variable domains. A number of conditions have to be obeyed for this generalisation to be applicable. The authors claimed that one such condition is in general violated, by means of a coim-terexample. Of course, one counterexample suffices but I show here that the calculation in their counterexample is flawed. 

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