their full 3D extent as so-called interatomic surfaces. These surfaces are the sharp
boundaries between atoms inside a molecule. Because a molecule is simply the
union of its topological atoms the boundary between molecules is also
sharp. Hence, a molecule in condensed matter is fully bounded by interatomic
surfaces, which are parameter free. The use of molecular contour surfaces of
constant electron density (e.g. ρ = 0.001 a.u.) is artificial and exists for practical (i.e.
visualisation) purposes only. Note that a molecule in the gas phase, alone in the
Universe, is a fiction: sooner or later one will find another molecule far away that
still shares a topological boundary with the original “isolated” molecule.
The hydrogen cyanide molecule, H–C≡N, can be isomerised to hydrogen isocyanide, H–N
+
≡C
– , by tilting the hydrogen over the carbon and gradually rotating it
to the right. Eventually, this hydrogen ends up at the right hand side of a new linear
arrangement, which can also be written as C
–
≡ N
+
–H for easy comparison with H–
C≡N. At some sharp transition point during the rotation of hydrogen, the atomic
interaction line flips over: where it originally connected H and C, it then connects
H and N. This means that the connectivity of the atoms suddenly changes, which is
a topological feature. This is an example of a so-called conflict mechanism.
We now ask in which way(s) the name topology is appropriate for what we have
discovered. Topology is the mathematical study of shapes and topological spaces. It
is an area of mathematics concerned with the properties of space that are preserved
under continuous deformations including stretching and bending. Avoiding precise
mathematical terms, one can define topology as the study of qualitative properties
of certain objects that are invariant under continuous transformations. For example,
Euler’s work on the Königsberg bridge problem was one of the earliest topological
studies. He showed it was impossible to find a route through the city of Königsberg
that crosses each of the seven bridges exactly once. This solution only depended on
which bridges are connected to which islands and riverbanks. In other words, only
connectivity mattered, not the length of the bridges or the distances between them.
This work also marked the beginning of graph theory and thereby establishes a link
between topology and graph theory. In general, the motivation behind topology is
that some geometric problems do not depend on the exact shape of the objects
involved, but rather on the way they are put together. How does this way of
thinking apply to the analysis of the electron density discussed above?
The keyword “topology” was first used [15] in the expression “quantum
topology” by Bader et al. in 1979. Unfortunately, this name is already taken by a
branch of mathematics that connects quantum mechanics with low-dimensional
topology, which has little to do with Bader et al. developed and which culminated
in QCT. In any event, the name “quantum topology” was inspired by a paper [16]
by Collard and Hall published in 1977. These authors were the first to use the
Poincaré-Hopf theorem, which links the respective numbers of each of the four
possible types of critical points to the so-called Euler characteristic. The latter is a
purely topological concept and hence justifies the name topology. The Euler
characteristic is an application of algebraic topology, one of the four branches of
topology, which uses tools from abstract algebra to study topological spaces.
Collard and Hall also pointed out that the analysis of the discontinuous change in
2 On Quantum Chemical Topology
31
boundaries between atoms inside a molecule. Because a molecule is simply the
union of its topological atoms the boundary between molecules is also
sharp. Hence, a molecule in condensed matter is fully bounded by interatomic
surfaces, which are parameter free. The use of molecular contour surfaces of
constant electron density (e.g. ρ = 0.001 a.u.) is artificial and exists for practical (i.e.
visualisation) purposes only. Note that a molecule in the gas phase, alone in the
Universe, is a fiction: sooner or later one will find another molecule far away that
still shares a topological boundary with the original “isolated” molecule.
The hydrogen cyanide molecule, H–C≡N, can be isomerised to hydrogen isocyanide, H–N
+
≡C
– , by tilting the hydrogen over the carbon and gradually rotating it
to the right. Eventually, this hydrogen ends up at the right hand side of a new linear
arrangement, which can also be written as C
–
≡ N
+
–H for easy comparison with H–
C≡N. At some sharp transition point during the rotation of hydrogen, the atomic
interaction line flips over: where it originally connected H and C, it then connects
H and N. This means that the connectivity of the atoms suddenly changes, which is
a topological feature. This is an example of a so-called conflict mechanism.
We now ask in which way(s) the name topology is appropriate for what we have
discovered. Topology is the mathematical study of shapes and topological spaces. It
is an area of mathematics concerned with the properties of space that are preserved
under continuous deformations including stretching and bending. Avoiding precise
mathematical terms, one can define topology as the study of qualitative properties
of certain objects that are invariant under continuous transformations. For example,
Euler’s work on the Königsberg bridge problem was one of the earliest topological
studies. He showed it was impossible to find a route through the city of Königsberg
that crosses each of the seven bridges exactly once. This solution only depended on
which bridges are connected to which islands and riverbanks. In other words, only
connectivity mattered, not the length of the bridges or the distances between them.
This work also marked the beginning of graph theory and thereby establishes a link
between topology and graph theory. In general, the motivation behind topology is
that some geometric problems do not depend on the exact shape of the objects
involved, but rather on the way they are put together. How does this way of
thinking apply to the analysis of the electron density discussed above?
The keyword “topology” was first used [15] in the expression “quantum
topology” by Bader et al. in 1979. Unfortunately, this name is already taken by a
branch of mathematics that connects quantum mechanics with low-dimensional
topology, which has little to do with Bader et al. developed and which culminated
in QCT. In any event, the name “quantum topology” was inspired by a paper [16]
by Collard and Hall published in 1977. These authors were the first to use the
Poincaré-Hopf theorem, which links the respective numbers of each of the four
possible types of critical points to the so-called Euler characteristic. The latter is a
purely topological concept and hence justifies the name topology. The Euler
characteristic is an application of algebraic topology, one of the four branches of
topology, which uses tools from abstract algebra to study topological spaces.
Collard and Hall also pointed out that the analysis of the discontinuous change in
2 On Quantum Chemical Topology
31