Reaction path invariance

Coord.transform. PES representation is changed 2. Use the original calculation of paths as in the first row 3 . Result A different, non-invariant reaction path... 

A minimum on a potential energy surface represents an equilibrium stracture. There will invariably be a number of such local minima, and we can imagine a number of paths on the surface that connect one particular minimum to another. If the highest-energy point on each path is considered, the transition structure can be defined as the lowest of these maxima. The reaction path is the lowest-energy route between two minima. 

T. Komatsuzaki and R. S. Berry, Regularity in chaotic reaction paths III Ar6 local invariances at the reaction bottleneck, J. Chem. Phys. 115, 4105 (2001). 

Equilibrium between simple salts and aqueous solutions is often relatively easily demonstrated in the laboratory when the composition of the solid is invariant, such as occurs in the KCI-H2O system. However, when an additional component which coprecipitates is added to the system, the solid composition is no longer invariant. Very long times may be required to reach equilibrium when the reaction path requires shifts in the composition of both the solution and solid. Equilibrium is not established until the solid composition is homogeneous and the chemical potentials of all components between solid and aqueous phases are equivalent. As a result, equilibrium is rarely demonstrated with a solid solution series. 

I am suggesting that often the applicability of Barkley-Butler type plots, that is, the linear relationship between the entropy and enthalpy of activation in a series may come about because of there being a distribution of reaction paths. Small variations in the importance of low activation energy, low probability paths could then account for the data in Dr. Taube s table. By contrast, transition state theory in its approximate application, invariably leads to diagrams of energy vs. reaction path which, in spite of all protest, one reaction path, whatever it is, one transition state, and one energy. 

However, the theory of symmetry invariants also strikes the redundant coordinate problem when N>4. As an example of the problems encountered, the reaction of Eq. (3.14) requires all 15 atom-atom distances to form a representation of the CNPI group no subset of 3A - 6 = 12 forms a set of irreducible representations. The theory of invariants, as applied to the symmetry of the PES, begins with coordinates that form a set of irreducible representations [191]. Thus we cannot even begin to discuss the symmetry of the PES in terms of as few as 3N-6 atom-atom distances. In addition there is no known way to improve such a PES to arbitrary accuracy, and this approach cannot deal with the existence of multiple reaction paths which are not related by symmetry. A more easily applied method, which can deal with all these difficulties, has recently been developed by our group, and we concentrate here on this interpolation approach [203,204]. 

The results of the fractionation model (Fig. 18.9) differ from the equilibrium model in two principal ways. First, the mineral masses can only increase in the fractionation model, since they are protected from resorption into the fluid. Therefore, the lines in Fig. 18.9 do not assume negative slopes. Second, in the equilibrium calculation the phase rule limits the number of minerals present at any point along the reaction path. In the fractionation calculation, on the other hand, no limit to the number of minerals present exists, since the minerals do not necessarily maintain equilibrium with the fluid. Therefore, the fractionation calculation ends with twelve minerals in the system, whereas the equilibrium calculation reaches an invariant point at which only six minerals are present. 

In chemical terms, normally hyperbolic invariant manifolds play the role of an extension of the concept of transition states. The reason why it is an extension is as follows. As already explained, transition states in the traditional sense are regarded as normally hyperbolic invariant manifolds in phase space. In addition to them, those saddle points with more than two unstable directions can be considered as normally hyperbolic invariant manifolds. Such saddle points are shown to play an important role in the dynamical phase transition of clusters [14]. Furthermore, as is already mentioned, a normally hyperbolic invariant manifold with unstable degrees of freedom along its tangential directions can be constructed as far as instability of its normal directions is stronger than its tangential ones. For either of the above cases, the reaction paths in the phase space correspond to the normal directions of these manifolds and constitute their stable or unstable manifolds. 

The normally hyperbolic invariant manifolds are structurally stable under perturbations. The wider the gap of instability is between the normal and tangential directions, the more stable it is. The existence of this gap can be interpreted as an adiabatic condition between the reaction paths and the rest of the degrees of freedom. 

The first method comes from the idea that the connections among normally hyperbolic invariant manifolds would form a network, which means that one manifold would be connected with multiple manifolds through homoclinic or heteroclinic intersections. Then, a tangency would signify a location in the phase space where their connections change. This idea offers a clue to understand, based on dynamics, those reactions where one transition state is connected with multiple transition states. In these reaction processes, the branching points of the reaction paths and the reaction rates to each of them are important We expect that analysis of the network is the first step toward this direction. 

In Sections IV and V, we discussed these two processes from the viewpoint of chaos. In these discussions, the following points are of importance. As for the barrier crossing, intersection (either homoclinic or heteroclinic) between stable and unstable manifolds offers global information on the reaction paths. It not only includes how transition states are connected with each other, but also reveals how reaction paths bifurcate. Here, the concepts of normally hyperbolic invariant manifolds and crisis are essential. With regard to IVR, the concept of Arnold web is crucial. Then, we suggest that coarse-grained features of the Arnold webs should be studied. In particular, the hierarchy of the web and whether the web is uniformly dense or not would play an important role. 

Our goal is the representation of reaction mechanisms. An individual reaction path has many incidental features, some not compatible with quantum mechanics. Instead of a reaction path, a reaction mechanism can be better represented by a formal reaction itinerary, where the main, invariant features of the journey are relevant. Such a reaction itinerary can be represented by a whole family of similar paths, and it is natural to model a reaction itinerary by a homotopy equivalence class of paths. 

THE INVARIANCE OF THE REACTION PATH DESCRIPTION IN ANY COORDINATE SYSTEM... 

Reaction paths are a widely used concept in theoretical chemistry. It is evident that the invariance problem, which was mathematically solved a long time ago (cf. the report given in Ref. [1 ]), penetrates again and again the discussions in this field (see Ref. [2]). We give both the non-invariant and the invariant definitions with respect to the choice of the particular coordinate system for two important kinds of chemical reaction pathways (RP), namely, steepest descent lines (SDP) and gradient extremal (GE) curves. 

Perhaps, the constant confusion concerning the invariance problem (cf. [2]) comes from the fact that the usual concepts for defining reaction paths use the properties of the PES in a concrete coordinate system. But, again, from a purely mathematical point of view, a change in the coordinate system by means of a definite transformation formula, can always be compensated for by changing the method for the computation of the reaction path through an inverse transformation formula. In Scheme 1 we... 

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