ELF represents a continuous and differentiable scalar field η(r) in three-dimensional space, as does the electron density ρ(r). Therefore, topological analysis of η(r) is technically similar to the one put forward by R.W.F. Bader and coworkers for ρ(r):
The topological analysis of η(r) is performed via its associated gradient vector field ∇η(r). This field is characterized by so-called critical points, where ∇η(r) = (0, 0, 0). They represent local maxima, minima and saddle points of η(r). A differentiation between the various kinds of critical points is achieved via the associated Hessian matrix H(η(rc)), which is a real, symmetric 3 x 3 matrix of the pure and mixed second derivatives of η(rc):
Diagonalization of H(η(rc)) gives eigenvalues hxx, hyy, hzz, which represent orthogonal tensor components in its local coordinate system. With this knowledge the different types of critical points can be characterized by their "rank"and "signature", symbolically written as "(r,s)". The rank r is defined as the number of non-zero eigenvalues of the hessian and the signature s = ∑ hii ⁄ | hii |. A critical point with one or more hessian eigenvalues equal to zero is called a degenerate critical point. Degenerate critical points play an important role during topology changes (see section Catastrophe Theory). In 3D space there exist four different types of nondegenerate critical points: attractor (3, -3), repellor (3, 3), and saddle points (3, 1) and (3, -1).
The gradient path ∇η(r) through each point ends (ω-limit) at either an attractor or a saddle point. The stable manifold of a critical point is the set of all points for which this critical point is an ω-limit. The stable manifold of an attractor is called a basin. In other words: A basin is the spacial region given by all points whose gradient paths end at the same attractor. The stable manifold of a saddle point is a surface (in a general meaning) which obeys the zero-flux condition: n · ∇η(rs) = 0, where n is the normal vector of the surface and rs is each point of the surface. The separatrices between two or more basins are the boundary points, lines or surfaces between two or more basins. They consist of the stable manifold of one or more saddle points interconnecting the basins and are therefore zero-flux surfaces.
The saddle points between basins are those points on the common separatrix which have the highest η-value. In this sense they are characteristic basin interconnection points (bip) with characteristic basin interconnection values ηic. One can define a basin set as the join of sets of interconnected basins including a specified basin, where the basin interconnection points are above or equal to a given value ηic.
Another term used in topological analysis is that of an f-localization domain. It is a region in space bounded by the isosurface η(r) = f. An f-localization domain is called irreducible if it contains just one attractor, otherwise it is reducible. An irreducible f-localization domain is always contained within the basin of its attractor and a reducible f-localization domain is always contained within the corresponding basin set with interconnection value ηic = f. For a characteristic value ηc(r) a reducible localization domain splits into two or more localization domains, which may still be reducible or irreducible. The number of irreducible localization domains contained in the reducible f-localization domain equals the number of members of the basin set with interconnection value ηic = f.
An important tool to characterize the topology of η(r) is the bifurcation diagram or the basin interconnection diagram. The type of information is similar in both diagrams, they differ in the amount of information contained: the bifurcation diagram is totally contained in the corresponding basin interconnection diagram. The bifurcation diagram shows in a tree-like graphics the values of the attractors and the values when a reducible localization domain becomes irreducible, i.e. the values of the highest saddle point of each basin. These values can be determined graphically via inspection of the bifurcation points of reducible localization domains. The basin interconnection diagram additionally shows the values of the lower lying saddle points of the separatrix of each basin, which cannot be uniquely determined and attributed to a basin graphically.The synaptic order of a valence basin is defined by the number of core basin sets with which it has a common separatrix
last update: 15.07.2002