1.2 Chemistry and Topology
Many authors, among whom the authors of the following chapters of this volume,
have addressed the relevance of various aspects of topology in chemistry (see for
example: Ayers et al. [25]). From the abstract mathematical standpoint, however, a
topology is defined within the framework of set theory: given a set X, a topology T
on X is a family of parts of X, called m open sets, i.e. a subset of PðXÞ ¼ 2
X , such
that:
(i) the empty set and X are open sets;
(ii) any union of open sets is an open set;
(iii) the intersection of any finite number of open sets is an open set.
The ðX; TÞ couple is called a topological space on X. If X is a metric space
(endowed with a distance), the canonical topology is defined from the corresponding open balls of X (each of them being defined by a center and a radius).
A topological space ðX; TÞ is a topological manifold if it is separated and locally
euclidean, i.e. every point in X admits an open neighborhood homeomorphic to R
n :
n is unique and defines the dimension of X, then referred to as a nD-manifold.
Within the chemical context, the “molecular space” is the 3D space filled with
electrons and punctual nuclei restricting or straining the euclidean topology
depending of the electron model. The molecular space is thus either:
• the discrete 0D-manifold of the Lewis’ graph embedded in the 3D euclidean
space with an approximate or optimized nuclear geometry. The topology is the
one induced by the canonical graph distance.
• or the continuous 1D-manifold generated by a sufficiently regular potential
function V(r) (electron density, ELF, MESP, see below) through its gradient
field ∇V(r) straining the 3D space for some particle moving through geodesics
according to the least action principle and Euler-Lagrange equations (see
Sect. 1.4.2.1). The corresponding topology can be regarded as a generalization
of the graph topology for a continuous set of vertices: two points in R
3 define a
generalized edge if they are on the same gradient path of V. The distance
between two points is either infinite if the points do not ly on the same path, or
equal to the length of the gradient arc between the two points if the points are on
the same path (the gradient paths are actually oriented and thus formally define
generalized edges of directed graphs). The open balls of the metric topology are
therefore arcs of gradient paths, where a neighborhood of any wandering point is
homeomorphic to a neighborhood of the tangent space defined by the gradient
direction in the dual space.
Application of the gradient dynamical system theory allows delineation of
non-overlapping basins forming a partition of the molecular space in R
3 from which
the basin adherences are removed. This partition defines a topology (unions of basin
interiors completed by the empty set), which is metric in the Lewis’ sense (the
connectivity between basins being related to their synapticity), and which is also a
1 Topological Approaches of the Bonding in Conceptual Chemistry
5
Many authors, among whom the authors of the following chapters of this volume,
have addressed the relevance of various aspects of topology in chemistry (see for
example: Ayers et al. [25]). From the abstract mathematical standpoint, however, a
topology is defined within the framework of set theory: given a set X, a topology T
on X is a family of parts of X, called m open sets, i.e. a subset of PðXÞ ¼ 2
X , such
that:
(i) the empty set and X are open sets;
(ii) any union of open sets is an open set;
(iii) the intersection of any finite number of open sets is an open set.
The ðX; TÞ couple is called a topological space on X. If X is a metric space
(endowed with a distance), the canonical topology is defined from the corresponding open balls of X (each of them being defined by a center and a radius).
A topological space ðX; TÞ is a topological manifold if it is separated and locally
euclidean, i.e. every point in X admits an open neighborhood homeomorphic to R
n :
n is unique and defines the dimension of X, then referred to as a nD-manifold.
Within the chemical context, the “molecular space” is the 3D space filled with
electrons and punctual nuclei restricting or straining the euclidean topology
depending of the electron model. The molecular space is thus either:
• the discrete 0D-manifold of the Lewis’ graph embedded in the 3D euclidean
space with an approximate or optimized nuclear geometry. The topology is the
one induced by the canonical graph distance.
• or the continuous 1D-manifold generated by a sufficiently regular potential
function V(r) (electron density, ELF, MESP, see below) through its gradient
field ∇V(r) straining the 3D space for some particle moving through geodesics
according to the least action principle and Euler-Lagrange equations (see
Sect. 1.4.2.1). The corresponding topology can be regarded as a generalization
of the graph topology for a continuous set of vertices: two points in R
3 define a
generalized edge if they are on the same gradient path of V. The distance
between two points is either infinite if the points do not ly on the same path, or
equal to the length of the gradient arc between the two points if the points are on
the same path (the gradient paths are actually oriented and thus formally define
generalized edges of directed graphs). The open balls of the metric topology are
therefore arcs of gradient paths, where a neighborhood of any wandering point is
homeomorphic to a neighborhood of the tangent space defined by the gradient
direction in the dual space.
Application of the gradient dynamical system theory allows delineation of
non-overlapping basins forming a partition of the molecular space in R
3 from which
the basin adherences are removed. This partition defines a topology (unions of basin
interiors completed by the empty set), which is metric in the Lewis’ sense (the
connectivity between basins being related to their synapticity), and which is also a
1 Topological Approaches of the Bonding in Conceptual Chemistry
5