density or electron localization function (ELF). In this chapter, we reexamine these
models in order to escape from the quantum mechanical dilemma and we show how
topological analyzes enable to recover these models.
1.1 Introduction
Chemistry thinks the matter as being made of atoms linked one to another by bonds.
This description has been initiated in the first half of the XIXth century by John
Dalton [1] who drew caloric forces between bonded atoms. It was consistently
improved all along the century with the introduction of many fundamental concepts. The concept of isomerism, proposed by Berzélius in order to account for
structural differences between species having the same stoichiometry but different
properties, has been addressed by Alexander Crum Brown [2, 3]. The concept of
valence due to Frankland and Kolbe which gives a rationale to the bonding connectivity between atoms, has been an important step ahead in the development of
structural chemistry where the important contributions of Kekulé, Kolbe, Couper,
Butlerov, Lodschmidt, Crum Brown, Hofmann, Le Bel and Van’t Hoff yield the
contemporary representations. The emerging picture of a molecule is, therefore, that
of a discrete network where the nodes are occupied by the elemental atoms. The
possible valences of the elements are given by their position in the Periodic Table.
In this respect, the determination of the structural formulas of the possible isomers
corresponding to a given stoichiometry appears to be first a discrete topology
problem (before being a geometry problem for the particular case of stereoisomers),
which can be mathematically formalized within the framework of finite graph
theory [4].
1 In this context, a bridge spanning the traditional gap between chemistry
and mathematics was early recognized by the mathematician James J. Sylvester. In
his benchmark article “Chemistry and Algebra” [5], 21 years after Kekulé on
carbon tetravalence [6] and 41 years before Lewis on “The Atom and the
Molecule” [7], he gave a tribute to Frankland while stating: “It may not be wholly
without interest to some of the readers of NATURE to be made acquainted with an
analogy that has recently forcibly impressed me between branches of human
knowledge apparently so dissimilar as modern chemistry and modern algebra (…).
I hardly ever take up Dr. Frankland’s exceedingly valuable “Notes for Chemical
Students”, which are drawn up exclusively on the basis of Kekulé’s exquisite
conception of valence, without deriving suggestions for new researches in the
theory of algebraical forms”. And farther: “Every (quantic) invariant and co-variant
thus becomes expressible by a graph precisely identical with a Kekulean diagram or
chemicograph”. The invariant and co-variant are here assignable to a set of atoms of
given valences and a set of bonds, respectively.
1
The concept of infinite graph applicable to non-covalent molecular materials being less directly
fruitful because of the ambiguity in the definition of the eigenvalue spectrum.
2
B. Silvi et al.
models in order to escape from the quantum mechanical dilemma and we show how
topological analyzes enable to recover these models.
1.1 Introduction
Chemistry thinks the matter as being made of atoms linked one to another by bonds.
This description has been initiated in the first half of the XIXth century by John
Dalton [1] who drew caloric forces between bonded atoms. It was consistently
improved all along the century with the introduction of many fundamental concepts. The concept of isomerism, proposed by Berzélius in order to account for
structural differences between species having the same stoichiometry but different
properties, has been addressed by Alexander Crum Brown [2, 3]. The concept of
valence due to Frankland and Kolbe which gives a rationale to the bonding connectivity between atoms, has been an important step ahead in the development of
structural chemistry where the important contributions of Kekulé, Kolbe, Couper,
Butlerov, Lodschmidt, Crum Brown, Hofmann, Le Bel and Van’t Hoff yield the
contemporary representations. The emerging picture of a molecule is, therefore, that
of a discrete network where the nodes are occupied by the elemental atoms. The
possible valences of the elements are given by their position in the Periodic Table.
In this respect, the determination of the structural formulas of the possible isomers
corresponding to a given stoichiometry appears to be first a discrete topology
problem (before being a geometry problem for the particular case of stereoisomers),
which can be mathematically formalized within the framework of finite graph
theory [4].
1 In this context, a bridge spanning the traditional gap between chemistry
and mathematics was early recognized by the mathematician James J. Sylvester. In
his benchmark article “Chemistry and Algebra” [5], 21 years after Kekulé on
carbon tetravalence [6] and 41 years before Lewis on “The Atom and the
Molecule” [7], he gave a tribute to Frankland while stating: “It may not be wholly
without interest to some of the readers of NATURE to be made acquainted with an
analogy that has recently forcibly impressed me between branches of human
knowledge apparently so dissimilar as modern chemistry and modern algebra (…).
I hardly ever take up Dr. Frankland’s exceedingly valuable “Notes for Chemical
Students”, which are drawn up exclusively on the basis of Kekulé’s exquisite
conception of valence, without deriving suggestions for new researches in the
theory of algebraical forms”. And farther: “Every (quantic) invariant and co-variant
thus becomes expressible by a graph precisely identical with a Kekulean diagram or
chemicograph”. The invariant and co-variant are here assignable to a set of atoms of
given valences and a set of bonds, respectively.
1
The concept of infinite graph applicable to non-covalent molecular materials being less directly
fruitful because of the ambiguity in the definition of the eigenvalue spectrum.
2
B. Silvi et al.