In the terminology of algebraic topology, the lower index 1 in the notation
π 1 (F(A)) refers to dimension, and such a group is called the one-dimensional
homotopy group of the given F(A) set (since the objects related to one another by
continuous deformations are one-dimensional paths, as opposed to deformations of
two-dimensional sheets, or higher-dimensional objects).
Alternatively, such a group is called the Fundamental Group of the relevant set,
in our case, π 1 (F(A)) is the fundamental group of the F(A) level set of the metric
nuclear configuration space M, with respect to the actual potential energy surface E
(K). Since the elements of this group π 1 (F(A)) are equivalence classes of loop
reaction paths, which can be regarded as the relevant circular (loop) reaction
mechanisms on F(A), and since these circular reaction mechanisms contain, as
parts, all non-circular reaction mechanisms as well, these groups have been named
the Fundamental Groups of Reaction Mechanisms [3].
9.3 Generalizations of Transformations Between
Molecules: The Universal Molecule Model
A chemical reaction can be regarded as a transformation between molecules: typically, a change of the nuclear coordinates serves as an indication of this transformation, and the nuclear configuration space and potential energy surface models
with the associated reaction path and reaction mechanism approaches provide a
useful description.
Of course, in the process of a chemical reaction other important changes, most
importantly, changes of the bonding pattern and the associated changes in the shape
of the electron density [4–7] also occur, where the latter changes are those which
are most directly detected by other, neighboring molecules. Even local changes of
molecular electron densities encode important information: based on the holographic electron density theorem [7], any small positive volume part of the
ground-state molecular electron density cloud contains the complete information
about the entire molecule.
If some common trends can be found in a family of molecules, then those trends
can be exploited in a predictive manner for any additional molecules which may fit
some aspects of this trend. In fact, such models involve some, often abstract “interpolations” and “extrapolations” among molecules, although in some instances,
the actual variables along which these, often inexact transformations occur, are not
necessarily clearly defined.
One model, the “Universal Molecule” model explicitly allows for such transformations: all parameters describing the molecules are considered as abstract, continuous variables, even those, which in the physical reality are restricted to be integers,
such as the case for the nuclear charges [8–11]. Using nuclear charges as examples, a
model where these integer values are replaced by continuous variables describes
reality only in specific cases, when the variables become integers. However, just as
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π 1 (F(A)) refers to dimension, and such a group is called the one-dimensional
homotopy group of the given F(A) set (since the objects related to one another by
continuous deformations are one-dimensional paths, as opposed to deformations of
two-dimensional sheets, or higher-dimensional objects).
Alternatively, such a group is called the Fundamental Group of the relevant set,
in our case, π 1 (F(A)) is the fundamental group of the F(A) level set of the metric
nuclear configuration space M, with respect to the actual potential energy surface E
(K). Since the elements of this group π 1 (F(A)) are equivalence classes of loop
reaction paths, which can be regarded as the relevant circular (loop) reaction
mechanisms on F(A), and since these circular reaction mechanisms contain, as
parts, all non-circular reaction mechanisms as well, these groups have been named
the Fundamental Groups of Reaction Mechanisms [3].
9.3 Generalizations of Transformations Between
Molecules: The Universal Molecule Model
A chemical reaction can be regarded as a transformation between molecules: typically, a change of the nuclear coordinates serves as an indication of this transformation, and the nuclear configuration space and potential energy surface models
with the associated reaction path and reaction mechanism approaches provide a
useful description.
Of course, in the process of a chemical reaction other important changes, most
importantly, changes of the bonding pattern and the associated changes in the shape
of the electron density [4–7] also occur, where the latter changes are those which
are most directly detected by other, neighboring molecules. Even local changes of
molecular electron densities encode important information: based on the holographic electron density theorem [7], any small positive volume part of the
ground-state molecular electron density cloud contains the complete information
about the entire molecule.
If some common trends can be found in a family of molecules, then those trends
can be exploited in a predictive manner for any additional molecules which may fit
some aspects of this trend. In fact, such models involve some, often abstract “interpolations” and “extrapolations” among molecules, although in some instances,
the actual variables along which these, often inexact transformations occur, are not
necessarily clearly defined.
One model, the “Universal Molecule” model explicitly allows for such transformations: all parameters describing the molecules are considered as abstract, continuous variables, even those, which in the physical reality are restricted to be integers,
such as the case for the nuclear charges [8–11]. Using nuclear charges as examples, a
model where these integer values are replaced by continuous variables describes
reality only in specific cases, when the variables become integers. However, just as
9 Topological Tools for the Study of Families of Reaction …
251