features are far more significant than some specific geometrical features of our
models, especially, if one considers the Heisenberg uncertainty relation, where no
precise geometrical features are compatible with the typically available momentum
information. The inherent fuzziness of molecular electron density clouds is far more
compatible with a topological analysis than with some rigid geometrical description. In particular, linking classical concepts with quantum chemical reality, and
providing computational, algorithmic approaches as well as algebraic frameworks,
such as the fundamental group of reaction mechanisms discussed here, algebraic
topology and differential topology have an increasingly important role in theoretical
and computational chemistry.
The reaction path model on a potential energy surface is based on an essentially
classical mechanical concept. In reality, the actual displacements of atomic nuclei in
chemical reactions do not follow a formal path, just as electronic rearrangements do
not follow any formal path, either in a molecule when the molecular electron density
changes due to some interaction, or in some change of the electronic state. This
follows from the fact that the Heisenberg uncertainty relation and the wave-particle
duality apply to both electrons and nuclei. Nevertheless, nuclei are certainly more
“particle-like” than electrons within any given molecule. Consequently, the concept
of nuclear positions in a molecule, although not strictly valid, is still a useful
approximate concept, whereas the concept of electronic “positions” within a
molecule is far too crude to have much use beyond very simplistic models. In the
above sense, a formal reaction path, describing some essential aspects of the geometrical displacements of the nuclei in a reaction has a useful role as a practical
approximation in many instances.
It is evident though that a single reaction path cannot faithfully represent the
quantum mechanical process of a chemical reaction, and some broadening of this
concept may serve as an improvement of the approximation. In this contribution a
topological approach is discussed, using certain equivalence classes of formal
reaction paths on potential energy surfaces to describe a quantum chemical concept
of reaction mechanisms. These reaction mechanisms are dependent on an energy
bound A over the actual potential energy surface. The family of all reaction
mechanisms has a group-theoretical algebraic structure, called the “Fundamental
Group of Reaction Mechanisms”, defined as the one-dimensional homotopy group
of the potential energy surface (actually a hypersurface) truncated at some energy
bound A. A brief review is given here for relevant earlier results in this field [3], as
well as their relations to the global and local shape problems of molecules [4–7],
where topological methods also play a dominant role. As a current development,
this model of the complete set of reaction mechanisms on a given potential energy
surface is now combined with some topological features of the Universal Molecule
model [8–11] for the actual common stoichiometry of the family of nuclei associated with the same potential energy surface.
244
P.G. Mezey
models, especially, if one considers the Heisenberg uncertainty relation, where no
precise geometrical features are compatible with the typically available momentum
information. The inherent fuzziness of molecular electron density clouds is far more
compatible with a topological analysis than with some rigid geometrical description. In particular, linking classical concepts with quantum chemical reality, and
providing computational, algorithmic approaches as well as algebraic frameworks,
such as the fundamental group of reaction mechanisms discussed here, algebraic
topology and differential topology have an increasingly important role in theoretical
and computational chemistry.
The reaction path model on a potential energy surface is based on an essentially
classical mechanical concept. In reality, the actual displacements of atomic nuclei in
chemical reactions do not follow a formal path, just as electronic rearrangements do
not follow any formal path, either in a molecule when the molecular electron density
changes due to some interaction, or in some change of the electronic state. This
follows from the fact that the Heisenberg uncertainty relation and the wave-particle
duality apply to both electrons and nuclei. Nevertheless, nuclei are certainly more
“particle-like” than electrons within any given molecule. Consequently, the concept
of nuclear positions in a molecule, although not strictly valid, is still a useful
approximate concept, whereas the concept of electronic “positions” within a
molecule is far too crude to have much use beyond very simplistic models. In the
above sense, a formal reaction path, describing some essential aspects of the geometrical displacements of the nuclei in a reaction has a useful role as a practical
approximation in many instances.
It is evident though that a single reaction path cannot faithfully represent the
quantum mechanical process of a chemical reaction, and some broadening of this
concept may serve as an improvement of the approximation. In this contribution a
topological approach is discussed, using certain equivalence classes of formal
reaction paths on potential energy surfaces to describe a quantum chemical concept
of reaction mechanisms. These reaction mechanisms are dependent on an energy
bound A over the actual potential energy surface. The family of all reaction
mechanisms has a group-theoretical algebraic structure, called the “Fundamental
Group of Reaction Mechanisms”, defined as the one-dimensional homotopy group
of the potential energy surface (actually a hypersurface) truncated at some energy
bound A. A brief review is given here for relevant earlier results in this field [3], as
well as their relations to the global and local shape problems of molecules [4–7],
where topological methods also play a dominant role. As a current development,
this model of the complete set of reaction mechanisms on a given potential energy
surface is now combined with some topological features of the Universal Molecule
model [8–11] for the actual common stoichiometry of the family of nuclei associated with the same potential energy surface.
244
P.G. Mezey