Zero flux surface

Bader et al. have developed a theory of molecular structure [8], based on the topological properties of the electron density p(r). In this theory, a molecule may be partitioned into atoms or fragments by using zero-flux surfaces that satisfy the condition... 

Points on the zero-flux surfaces that are saddle points in the density are passes or pales. Should the critical point be located on a path between bonded atoms along which the density is a maximum with respect to lateral displacement, it is known as a pass. Nuclei behave topologically as peaks and all of the gradient paths of the density in the neighborhood of a particular peak terminate at that peak. Thus, the peaks act as attractors in the gradient vector field of the density. Passes are located between neighboring attractors which are linked by a unique pair of trajectories associated with the passes. Cao et al. [11] pointed out that it is through the attractor behavior of nuclei that distinct atomic forms are created in the density. In the theory of molecular structure, therefore, peaks and passes play a crucial role. 

The interatomic (zero-flux) surfaces partition the molecule into separate nonoverlapping atoms (atomic basins), which... 

Atomic volumes play an important role in relating physicochemical properties to biological effects. Most atoms in molecules are not entirely bounded by interatomic surfaces and an atomic volume is defined as a measure of the space enclosed by the intersection of the atom s zero-flux surfaces with some outer envelope of the density. The envelope with a value of 0.001 au is generally chosen as this has been shown to yield molecular sizes in good agreement with experimentally assigned van der Waals radii [16, 17]. A related property is the van der Waals surface area, which QTAIM determines by integrating an atom s exposed contribution to a molecule s isovalued surface. 

Fig. 7.5 The serinyl group NHCH(CH2OH)C(=0) cut from the glycine mold represented by the intersection of its van derWaals 0.001 au isodensity surface with the -C(C=0) or C-surface at the top left and the NH- or N-surface at the bottom center. These are the complementary sides of the amidic zero-flux surface characteristic of a polypeptide. All properties of the residue are defined and make additive contributions to the molecule constructed from it. The residue has a net charge of -0.006 e.

The nuclei of neighboring atoms and molecules in crystals are separated by the E(r) by zero-flux surfaces S ( r) ... 

Figure 7.. Distributions of ESP (left) and ED for (100) plane of LiF. CPs (3,-1) are denoted by dotes, (3,+l) - by triangles. The lines of the intersection of the zero-flux surfaces with the plane of the figure are shown.

Comparison of forms of atomic fragments limited by the zero flux surfaces in ESP and electron density (Fig. 7) displays the role of different factors in the formation of the crystal structure. So in crystals with NaCl-type structure the exchange and correlation of electrons decrease the size of the cation and enlarge the size of the anion which leads to the structureforming interactions anion-anion in the (001) plane of the electron density maps. In ESP-maps the big cations and small anions are seen. 

It is important that L(r) vanishes when the integration is performed over the zero-flux surface atomic basin. This is because the integral over (r) can be replaced by the surface integral over the flux at the surface (Bader 1990) ... 

Alternatively if all four kinds of critical point are chosen as vertices, one gets a partitioning into fragments which each contain the flux lines of a single bond. The surfaces of the bond fragment are zero flux surfaces, i.e. no field lines cross into or out of the bond fragment. In this interpretation, each bond occupies a finite space and every point in space belongs to one and only one bond. 

Figure 9.4 Electron density gradient paths in a plane containing the atoms of the HCN molecule. The solid lines are the intersections of the zero-flux surfaces with the plane. The large black dots are the bond critical points...

Interatomic surface Zero-flux surface S Internuclear surface through which the flux of Vp(r) is zero (see equation 6)... 

FIGURE 11. Gradient vectorfield 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... 

Atoms A and B have to be connected by a MED path. The existence of a MED path implies a saddle point p of the electron density distribution p (r) as well as a zero-flux surface S (AB) between atoms A and B (necessary condition). 

Figure 15 presents a schematic view of how the atomic subspaces Cl, C6 and Cl 1 of 1,6-methanojl Ojannulene (35) change upon an approach of Cl to C6. Bond paths (solid lines between atoms), bond critical points (dots) and the traces of the zero-flux surfaces S (A, B) (perpendicular to bond paths) that separate the atomic subspaces are shown in Figure 15a. Clearly, the subspace C11 extends less and less into the region between C1 and C6 until the surfaces of C1 and C6 coincide and a bond path between C1 and C6 is formed. At the same time, the Laplace concentration between Cl and C6 gradually increases and coverges to the one found for a three-membered ring. As shown in Figure 15b, this change corresponds to the valence tautomerism of the l,6-methano[10]annulene to bisnorcaradiene27,54.

Fig. 5.43 Heteronuclear (as well as homonuclear cf. Fig. 5.42) molecules can be partitioned into atoms. S represents a slice through the zero-flux surface that defines the atoms A and B in a molecule AB. The lines with arrows are the trajectories of the gradient vector field. S passes through the bond critical point C and is not crossed by any trajectory lines...

For example, Figure 2 illustrates the total charge distribution p(r) of the HsO system at various points along the minimum energy Czv insertion pathways, (ro = distance of oxygen from midpoint of H-H and m is H-H separation). Shown on these diagrams are the zero-flux surfaces S(r), i.e. those surfaces satisfying... 

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. 

Nucleus + basin Q Volume K of basin Atomic charge Atomic dipole moment Atomic energy Zero-flux surface S Bond path... 

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