Chemistry Reference
In-Depth Information
crystallochemical nets. For example, the RCSR symbols dia (the corresponding
EPINET code
sqc
6) and dia-b are assigned to both the diamond net and the isotypic
binary compound (sphalerite, ZnS), with the same topology, but with two chemi-
cally and crystallographically nonequivalent atoms. The s-
d
-
G
-
n
symbol nia-5,5-
Pna
2
1
designates the subnet derived from the RCSR nia (NiAs) net with two
topologically different 5-coordinated atoms, and the maximum-symmetry embed-
ding of the net has the space group
Pna
2
1
. If a net does not fall in any topological
classification, we designate it by the Cambridge Structural Database Reference
Code or by the chemical formula of the corresponding compound.
When performing the topological taxonomy, the problem that emerges is how to
compare two infinite objects that are periodic nets. The representation of a periodic
net as a finite object is achieved by means of the concept of
labeled quotient graph
[
24
]. The labeled quotient graph is a graph whose vertices and edges correspond to
infinite sets of both translation-equivalent nodes and edges of the net; the
corresponding atoms and bonds occupy the same primitive cell of the crystal
structure. Hence, the labeled quotient graph is finite, but it stores all the information
about the net topology. It is extremely important to develop the theory of labeled
quotient graphs; this theory could play the same role for crystal chemistry as space-
group theory did for crystallography. Such attempts have been undertaken in recent
Topological index
of a net is a set of numbers that is related to the net topology.
The net topology is completely described by the
adjacency matrix
of the labeled
quotient graph; the adjacency matrix contains the information about all edges
(chemical bonds) of the net. Thus, the adjacency matrix can be used as the strictest
topological index to check the isomorphism of the nets [
28
] and also to relate the
crystal structures to the same topological type. For a quick comparison of the net
topologies or to find non-strict topological similarities, one can use other kinds of
N
2
,
representing the atoms in the first, second, etc., coordination shells around a
ered for the topological classification. The cumulative number of atoms in the first
ten coordination shells averaged over all nonequivalent atoms (TD
10
) is defined as
the
topological density
of the net; the larger TD
10
, the denser the net, i.e., the more
neighbors are in a local area of the net node.
Point symbol
lists the shortest circuits
(closed chains of connected atoms) meeting at all bond angles of each nonequiva-
lent atom in the net.
Extended
point symbol gives the information on circuits in
more detail, while
net
point symbol summarizes the point symbols for all nonequiv-
alent atoms with the corresponding stoichiometric coefficients.
Vertex symbol
gives
the information similar to extended point symbol but for rings (circuits without
shortcuts). If the nets have equal sets of all these indices, they are assumed to be
topologically isotypic [
23
]. If the sets of indices are different, but close to each
other, the nets can be considered topologically similar. The same can be said of
different atoms in the same structure: if their indices are identical, the atoms are
topologically equivalent even if they are not related by a space-group operation.
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