the variable interaction between the atoms. The most general molecular graph is
therefore an edge- and vertex-weighted version of the complete graph of G, with b
values possibly tending to zero for pairs of “almost non-bonded” atoms. This
generalization allowed the definition of the adiabatic TRE versus the usual TRE
referred to as the vertical TRE [35].
Finally, graph theory not only remains a powerful analysis tool for direct
interpretation of molecular properties, but is also a systematic synthesis tool for the
description of molecular structures. In the same way the electronic properties are
accurately described by a weighted superposition of mesomeric forms (or valence
bond structures), the underlying structure is completely described by a series of line
graphs. Following Estrada, indeed [36], whereas the covalent connectivity (Lewis
structure) is described by a vertex-weighted graph G, the 2D-geometry (bond
distances) described by the corresponding edge-weighted graph is equivalently
described by the corresponding vertex-weighted line graph L
1
ðGÞ. Pursuing in the
same spirit, the local 3D-geometry (bond angles) is described by the
vertex-weighted second line graph L
2
ðGÞ, and the global 3D-geometry (torsion
angles) by the vertex-weighted third line graph L
3
ðGÞ. A molecule is therefore fully
described by the series of its vertex-weighted line graphs L
n
ðGÞ.
1.4 Continuous Chemical Topology: Chemical Bonding
and Functional Analysis
1.4.1 Intuitive Bonding Chemical Concepts and Quantum
Mechanics
For almost a century Quantum Mechanics is the physical theory which describes the
interaction between particles, such as electrons and nuclei in molecules, and
therefore enables the quantitative exact predictions of observables. The reduction of
chemistry to physics corresponds to the mechanistic working program of Dirac
[37]:
The underlying physical laws necessary for the mathematical theory of a large part of
physics and the whole of chemistry are thus completely known, and the difficulty is only
that the exact application of these laws leads to equations much too complicated to be
soluble.
Although the predictive power of Quantum Mechanics is unquestionable, its
ability to provide explanations has been questioned from an epistemological point
of view by Thom [38]. As neither atoms in molecules nor bonding and non bonding
pairs are defined in terms of quantum mechanical observables, the bridges linking
the intuitive chemical approach to Quantum Mechanics are built either by interpreting the approximate molecular wave functions or on the basis of the statistical
interpretation of Quantum Mechanics. In the latter approach, the analysis of
8
B. Silvi et al.
therefore an edge- and vertex-weighted version of the complete graph of G, with b
values possibly tending to zero for pairs of “almost non-bonded” atoms. This
generalization allowed the definition of the adiabatic TRE versus the usual TRE
referred to as the vertical TRE [35].
Finally, graph theory not only remains a powerful analysis tool for direct
interpretation of molecular properties, but is also a systematic synthesis tool for the
description of molecular structures. In the same way the electronic properties are
accurately described by a weighted superposition of mesomeric forms (or valence
bond structures), the underlying structure is completely described by a series of line
graphs. Following Estrada, indeed [36], whereas the covalent connectivity (Lewis
structure) is described by a vertex-weighted graph G, the 2D-geometry (bond
distances) described by the corresponding edge-weighted graph is equivalently
described by the corresponding vertex-weighted line graph L
1
ðGÞ. Pursuing in the
same spirit, the local 3D-geometry (bond angles) is described by the
vertex-weighted second line graph L
2
ðGÞ, and the global 3D-geometry (torsion
angles) by the vertex-weighted third line graph L
3
ðGÞ. A molecule is therefore fully
described by the series of its vertex-weighted line graphs L
n
ðGÞ.
1.4 Continuous Chemical Topology: Chemical Bonding
and Functional Analysis
1.4.1 Intuitive Bonding Chemical Concepts and Quantum
Mechanics
For almost a century Quantum Mechanics is the physical theory which describes the
interaction between particles, such as electrons and nuclei in molecules, and
therefore enables the quantitative exact predictions of observables. The reduction of
chemistry to physics corresponds to the mechanistic working program of Dirac
[37]:
The underlying physical laws necessary for the mathematical theory of a large part of
physics and the whole of chemistry are thus completely known, and the difficulty is only
that the exact application of these laws leads to equations much too complicated to be
soluble.
Although the predictive power of Quantum Mechanics is unquestionable, its
ability to provide explanations has been questioned from an epistemological point
of view by Thom [38]. As neither atoms in molecules nor bonding and non bonding
pairs are defined in terms of quantum mechanical observables, the bridges linking
the intuitive chemical approach to Quantum Mechanics are built either by interpreting the approximate molecular wave functions or on the basis of the statistical
interpretation of Quantum Mechanics. In the latter approach, the analysis of
8
B. Silvi et al.