Chapter 9
Topological Tools for the Study
of Families of Reaction Mechanisms:
The Fundamental Groups of Potential
Surfaces in the Universal Molecule
Context
Paul G. Mezey
Abstract Two types of the main topological properties of potential energy surfaces
are compared, where the first types are related to the chemical processes, conformational changes and chemical reactions along the potential energy surface, and
where the second types are describing the presence, interrelations, structural variability, and shape variations of identifiable chemical species associated with the
potential surface. Some new relations are obtained when the families of topologically equivalent reaction paths representing reaction mechanisms at some energy
bound, and the algebraic structure of the fundamental group of reaction mechanisms
for a given collection of atoms (that is, for a given stoichiometry) are constrained by
the collection of “catchment regions” of the potential surface, representing chemical
species. These relations, providing additional detail when they are compared to the
more traditional, unconstrained cases, are phrased in terms of potential energy
surface level set relations and the originally integer, but “unquantized” continuous
variables of the Universal Molecule model.
9.1 Introduction
Topological methods, especially those of algebraic and differential topology [1, 2],
provide very powerful tools for the description of chemical problems, far beyond
the “skeletal models” provided by graph theory. Molecules are better described by
topology than by geometry, since a whole range of possible geometries of a given
molecule preserves the chemical identity of the molecule, that is, the topological
P.G. Mezey (&)
Department of Chemistry, and Department of Physics and Physical Oceanography,
Memorial University of Newfoundland, 283 Prince Philip Drive,
St. John’s, NL A1B 3X7, Canada
e-mail: pmezey@mun.ca; paul.mezey@gmail.com
URL: http://www.mun.ca/research/chairs/mezey.php
© Springer International Publishing Switzerland 2016
R. Chauvin et al. (eds.), Applications of Topological Methods
in Molecular Chemistry, Challenges and Advances in Computational
Chemistry and Physics 22, DOI 10.1007/978-3-319-29022-5_9
243
Topological Tools for the Study
of Families of Reaction Mechanisms:
The Fundamental Groups of Potential
Surfaces in the Universal Molecule
Context
Paul G. Mezey
Abstract Two types of the main topological properties of potential energy surfaces
are compared, where the first types are related to the chemical processes, conformational changes and chemical reactions along the potential energy surface, and
where the second types are describing the presence, interrelations, structural variability, and shape variations of identifiable chemical species associated with the
potential surface. Some new relations are obtained when the families of topologically equivalent reaction paths representing reaction mechanisms at some energy
bound, and the algebraic structure of the fundamental group of reaction mechanisms
for a given collection of atoms (that is, for a given stoichiometry) are constrained by
the collection of “catchment regions” of the potential surface, representing chemical
species. These relations, providing additional detail when they are compared to the
more traditional, unconstrained cases, are phrased in terms of potential energy
surface level set relations and the originally integer, but “unquantized” continuous
variables of the Universal Molecule model.
9.1 Introduction
Topological methods, especially those of algebraic and differential topology [1, 2],
provide very powerful tools for the description of chemical problems, far beyond
the “skeletal models” provided by graph theory. Molecules are better described by
topology than by geometry, since a whole range of possible geometries of a given
molecule preserves the chemical identity of the molecule, that is, the topological
P.G. Mezey (&)
Department of Chemistry, and Department of Physics and Physical Oceanography,
Memorial University of Newfoundland, 283 Prince Philip Drive,
St. John’s, NL A1B 3X7, Canada
e-mail: pmezey@mun.ca; paul.mezey@gmail.com
URL: http://www.mun.ca/research/chairs/mezey.php
© Springer International Publishing Switzerland 2016
R. Chauvin et al. (eds.), Applications of Topological Methods
in Molecular Chemistry, Challenges and Advances in Computational
Chemistry and Physics 22, DOI 10.1007/978-3-319-29022-5_9
243