For catchment regions of energy minima, that is, for (3N−6)-dimensional
catchment regions of nuclear configuration space M with respect to a given
potential energy surface E(K), an intuitively attractive analogy can be drawn with
actual watersheds taken as the set of all locations from where rain is collecting in a
common sinkhole [3].
Critical points on a potential energy surface all have vanishing energy gradients,
and are characterized by the second derivatives of the energy function E(K)
according to the local nuclear coordinates. The Hessian matrix of second energy
derivatives provides curvature information, and the eigenvalues of the Hessian
matrix are important clues concerning the importance of these critical points. For
simplicity in the discussion, here we ignore the cases of degenerate critical points,
where the Hessian matrix has one or more zero eigenvalues; these special cases are
discussed in [3], and will not modify the essential conclusions in this section.
Index λ is the number of negative eigenvalues of the Hessian matrix at the given
critical point, where λ = 0 corresponds to energy minima, whereas λ = 1 corresponds to saddle points of transition structures (often referred to as transition
“states”). Other critical points of higher index, λ > 1, are usually avoided by
minimum energy paths, that is, by the “most likely” ideal reaction paths (which are,
of course, strictly speaking, unrealistic, even classically, showing no vibrational
contributions).
This preference for critical points of indices 0 and 1 suggests a modification of
the fundamental group approach for reaction mechanisms described in the previous
section.
From the level set F(A) of all parts of the given potential energy surface below
energy bound A, one may consider to eliminate all points of most lowerdimensional catchments regions, except the points of those catchment regions
which belong to energy minima and transition structures, that is, to identitypreserving distortions of energy minima, and identity-preserving distortions of
transition structures of chemical reactions. In other words, one may decide to
eliminate all those points K where the energy E(K) does not fall below the energy
bound A, and also all points which fall within a catchment region of index 2 or
higher.
This λ—constrained level set, denoted by
F k¼0;1 A
ð Þ;
can then replace the original level set F(A) in the derivation of the conditions and
properties of the fundamental group of reaction mechanisms, and a new, somewhat
more distinguishing and more revealing algebraic structure is obtained. All formal
steps of the development of the fundamental group of reaction mechanisms on F(A)
can be repeated for this new set F λ=0,1 (A), and the resulting fundamental group,
9 Topological Tools for the Study of Families of Reaction …
253
catchment regions of nuclear configuration space M with respect to a given
potential energy surface E(K), an intuitively attractive analogy can be drawn with
actual watersheds taken as the set of all locations from where rain is collecting in a
common sinkhole [3].
Critical points on a potential energy surface all have vanishing energy gradients,
and are characterized by the second derivatives of the energy function E(K)
according to the local nuclear coordinates. The Hessian matrix of second energy
derivatives provides curvature information, and the eigenvalues of the Hessian
matrix are important clues concerning the importance of these critical points. For
simplicity in the discussion, here we ignore the cases of degenerate critical points,
where the Hessian matrix has one or more zero eigenvalues; these special cases are
discussed in [3], and will not modify the essential conclusions in this section.
Index λ is the number of negative eigenvalues of the Hessian matrix at the given
critical point, where λ = 0 corresponds to energy minima, whereas λ = 1 corresponds to saddle points of transition structures (often referred to as transition
“states”). Other critical points of higher index, λ > 1, are usually avoided by
minimum energy paths, that is, by the “most likely” ideal reaction paths (which are,
of course, strictly speaking, unrealistic, even classically, showing no vibrational
contributions).
This preference for critical points of indices 0 and 1 suggests a modification of
the fundamental group approach for reaction mechanisms described in the previous
section.
From the level set F(A) of all parts of the given potential energy surface below
energy bound A, one may consider to eliminate all points of most lowerdimensional catchments regions, except the points of those catchment regions
which belong to energy minima and transition structures, that is, to identitypreserving distortions of energy minima, and identity-preserving distortions of
transition structures of chemical reactions. In other words, one may decide to
eliminate all those points K where the energy E(K) does not fall below the energy
bound A, and also all points which fall within a catchment region of index 2 or
higher.
This λ—constrained level set, denoted by
F k¼0;1 A
ð Þ;
can then replace the original level set F(A) in the derivation of the conditions and
properties of the fundamental group of reaction mechanisms, and a new, somewhat
more distinguishing and more revealing algebraic structure is obtained. All formal
steps of the development of the fundamental group of reaction mechanisms on F(A)
can be repeated for this new set F λ=0,1 (A), and the resulting fundamental group,
9 Topological Tools for the Study of Families of Reaction …
253