9.2 From Reaction Path to Reaction Mechanism
The collection of all possible arrangements of a given set of N atomic nuclei forms
the relevant nuclear configuration space, that can be chosen as a metric space M,
with a well-defined distance function d(K, K′) for any pair of nuclear configurations
K and K′, where the dimension of this space is 3N − 6 for N > 2 (diatomic and
monoatomic cases are special with respect to internal coordinates).
What is less well-known, although it can be shown easily [3], that a nuclear
configuration space M of internal coordinates can never be a vector space, and this
fact is a frequent source of Euclidean-geometry-based misinterpretations of
potential energy surface (in fact, potential energy hypersurface) problems.
Nevertheless, M being a metric space, it allows the use of many tools of abstract
geometry as well as topology, and a reasonably detailed description of formal
reaction paths and reaction mechanisms is possible [3].
The potential energy surface E(K) is an energy function where the variables are
the internal nuclear coordinates, collected into the symbol K of the nuclear configuration. Note, that different electronic states are associated to different potential
energy surfaces.
If energy E is regarded as one additional variable beyond the 3N−6 internal
coordinates of the nuclear configurations (the dimension of space M), than the total
dimension is 3N−5, and the energy function is in fact a (3N−6)-dimensional
hypersurface; an object that has one dimension less than the dimension of the
complete space. Note, however, that for brevity, the term “potential energy surface”
is used more often.
It is customary to think of a reaction path as a line in configuration space M, for
example, one leading from some “reactant configuration” K reactant to some “product
configuration” K product , through some intermediate K configurations. One may
associate the energy value of each configuration to the corresponding point along
the path in configuration space, and by taking these values measured along an extra
“energy” dimension, another line is obtained, a “relief path”, or “relief reaction
path” along the potential energy surface. It is also customary to regard energy E as a
“vertical” dimension, and one may think that the “relief reaction path” on the
potential energy surface E(K) runs “above” the reaction path within the configuration space M. One may also think that the reaction path in the configuration space
M is the “shadow at high noon” of the relief reaction path along the potential energy
surface E(K).
For the purposes of a consistent mathematical treatment in the following topological description, one may regard a path not as the collection of points in some
space, but as a formal, continuous function p(u), assigning points in the actual space
M to values u of the unit interval, [0,1]. In our case, a path p(u) is regarded as a
continuous mapping from the closed interval [0,1] to space M, describing a continuous change of nuclear configurations K, represented by a displacement in M.
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The collection of all possible arrangements of a given set of N atomic nuclei forms
the relevant nuclear configuration space, that can be chosen as a metric space M,
with a well-defined distance function d(K, K′) for any pair of nuclear configurations
K and K′, where the dimension of this space is 3N − 6 for N > 2 (diatomic and
monoatomic cases are special with respect to internal coordinates).
What is less well-known, although it can be shown easily [3], that a nuclear
configuration space M of internal coordinates can never be a vector space, and this
fact is a frequent source of Euclidean-geometry-based misinterpretations of
potential energy surface (in fact, potential energy hypersurface) problems.
Nevertheless, M being a metric space, it allows the use of many tools of abstract
geometry as well as topology, and a reasonably detailed description of formal
reaction paths and reaction mechanisms is possible [3].
The potential energy surface E(K) is an energy function where the variables are
the internal nuclear coordinates, collected into the symbol K of the nuclear configuration. Note, that different electronic states are associated to different potential
energy surfaces.
If energy E is regarded as one additional variable beyond the 3N−6 internal
coordinates of the nuclear configurations (the dimension of space M), than the total
dimension is 3N−5, and the energy function is in fact a (3N−6)-dimensional
hypersurface; an object that has one dimension less than the dimension of the
complete space. Note, however, that for brevity, the term “potential energy surface”
is used more often.
It is customary to think of a reaction path as a line in configuration space M, for
example, one leading from some “reactant configuration” K reactant to some “product
configuration” K product , through some intermediate K configurations. One may
associate the energy value of each configuration to the corresponding point along
the path in configuration space, and by taking these values measured along an extra
“energy” dimension, another line is obtained, a “relief path”, or “relief reaction
path” along the potential energy surface. It is also customary to regard energy E as a
“vertical” dimension, and one may think that the “relief reaction path” on the
potential energy surface E(K) runs “above” the reaction path within the configuration space M. One may also think that the reaction path in the configuration space
M is the “shadow at high noon” of the relief reaction path along the potential energy
surface E(K).
For the purposes of a consistent mathematical treatment in the following topological description, one may regard a path not as the collection of points in some
space, but as a formal, continuous function p(u), assigning points in the actual space
M to values u of the unit interval, [0,1]. In our case, a path p(u) is regarded as a
continuous mapping from the closed interval [0,1] to space M, describing a continuous change of nuclear configurations K, represented by a displacement in M.
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