# Topological Analysis

**Mathematical Framework**

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(

**r**

_{c})), which is a real, symmetric 3 x 3 matrix of the pure and mixed second derivatives of η(

**r**

_{c}):

Diagonalization of ** H**(η(

**r**

_{c})) gives eigenvalues

*h*,

_{xx}*h*,

_{yy}*h*, 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 "(

_{zz}*r*,

*s*)". The

**rank**

*r*is defined as the number of non-zero eigenvalues of the hessian and the

**signature**

*s*= ∑

*h*⁄ |

_{ii}*h*|. A critical point with one or more hessian eigenvalues equal to zero is called a

_{ii}**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(**r**_{s}) = 0, where **n** is the normal vector of the surface
and **r**_{s} 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 **η_{ic}.
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 η_{ic}.

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 η

_{ic}=

*f*. For a characteristic value η

_{c}(

**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 η

_{ic}=

*f*.

**Conceptual Framework**

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.

**synaptic order**of a valence basin is defined by the number of core basin sets with which it has a common separatrix

*A,*is written V

_{1}(

*A*), a disynaptic basin touching cores of A and B is indicated as V

_{2}(

*A*,

*B*), and so on .... In simple main group molecules the monosynaptic basins can be attributed to "lone pairs" (be careful, the electronic basin population need not even be approximately equal to 2, e.g. a Ne atom has one basin for the valence shell containing 8 electrons), while the polysynaptic basins are attributed to chemical bonds. The question whether the polysynaptic order is associated (necessary or sufficient condition) with polycentric bonding is still a matter of debate (see section

*How to interpret*). With the classification of basins as

**core basins**and

**valence basins**, which is necessary for the definition of a

**synaptic order**of the latter, we leave the field of pure mathematics and enter the field of chemistry. A unique definition cannot be given without some assumptions based on intuition or experience. This is discussed in more detail in section

*Shell Structure*.

last update: 15.07.2002