Chemistry Reference
In-Depth Information
topologically equal (similar, different) to
” is usually drawn after a visual analy-
sis, comparing only a few local topological properties of the structures, while the
overall structure topology is the natural crystallochemical subject. In short, crystal
chemistry begins when the crystal structure topology is taken into consideration.
Below, we briefly consider the simplest formalization of the crystal space
topology perceived as a periodic graph. A more detailed terminology on this
Graph
is a set of vertices (points); each ordered pair of the vertices determines an
edge of the graph. The set of all edges is called the
topology
of the graph. Since in
crystal chemistry the graph vertices and edges correspond to atoms and interatomic
bonds, respectively, the chemical term “topology” means merely the set of all
interatomic bonds in the crystal structure.
Net
is a kind of
infinite
graph that is
simple
(it has no loops, multiple or directed
edges) and
connected
(any pair of vertices in the graph is connected by a chain of
edges); vertices of the graph are often called
nodes
of the net. The
degree n
of a
node is the number of edges incident to the node; it is equivalent to coordination
number of the corresponding atom, so that the node (atom) is called
n
-coordinated.
A
subnet
(
supernet
) of the net
A
is a net whose sets of nodes and edges are subsets
(supersets) of corresponding sets of
A
. Two nets are called
isomorphic
if there is
one-to-one mapping between the sets of their nodes and edges. The symmetry of the
net is described by an
automorphism group
that enumerates all possible permuta-
tions of nodes giving rise to isomorphic nets; this symmetry does not involve any
geometrical realization of the net and, hence, can be considered as “topological”
symmetry of the corresponding chemical structure. Net is
n-periodic
if its automor-
phism group contains a subgroup being isomorphic to a group of
n
-independent
translations. When we consider some arrangement of the net nodes in the space, we
speak about an
embedding
of the net. The symmetry of the net embedding can
be lower (but not higher) than the symmetry of the net.
Coordination figure
of the
node is the solid formed by nodes incident to this node; if the coordination figure
has a polyhedral form, it corresponds to the coordination polyhedron of the
corresponding atom. The net is
uninodal
(
bi
-,
tri
-,
...
,
polynodal
) if all its nodes
are equivalent by symmetry (or there are two, three,
...
...
, many nonequivalent
nodes).
If the crystal structures have isomorphic nets, they are called (topologically)
isomorphism is strictly determined, the concept of topological type is a way to an
objective taxonomy (independent of the visual analysis) of crystal structures. Note
that the oldest primary classification taxon in crystal chemistry,
structure type
,
arranges crystal structures according to their geometrical properties (unit cell
dimensions, space group, atomic positions). In contrast, topological type includes
all structures with the same system of chemical bonds irrespective of their embed-
dings. To designate the topological type, we can use the name of one of its
representatives as well as the symbol of the net corresponding to the type. In this
chapter, we will use three kinds of nomenclature for net symbols: (1) three-letter
