Catchment regions

Mezey, P. G. Catchment region partitioning of energy hypersurfaces,I., Theoret.chim.Acta(Berl.), 58 (1981), 390-330... 

The catchment region model provides precise conditions for chemical identity and for limitations on molecular distortions which preserve chemical identity. It also provides an approach to our second question What are the allowed shape variations which may accompany these identity-preserving molecular deformations ... 

The family of 3D shapes available to a given molecule is precisely the family of 3D shapes occurring within its catchment region C(, i) [158]. These are the very shapes attainable by the molecule while undergoing limited deformations preserving chemical identity. 

This neighbor relation is similar to the "symmetric strong neighbor relation" between some potential surface catchment regions of reaction topology, used in the analysis of reaction mechanisms [106,343-345]. 

Take a subclass Pi(C(A,iXC(A, i )) of class Pi, define by the following condition Pi(C(A,i), C(A, i )) is the family of all paths from class Pi which start at the catchment region C(A,i), end at the catchment region C(A, i ), and are homotopic to one another (continuously deformable into one another) while preserving these properties. Evidently the above conditions correspond to an equivalence relation among paths, and, consequently, Pi(C(A,i), C(A, i )) is an equivalence class. Such equivalence classes Pi(C(A,i), C(A, i )) represent formal reaction mechanisms defined in terms of the shape of Density Domains (D-shape). 

Catchment Regions and Symmetry Domains of Nuclear Configuration Spaces... 

The catchment regions can be associated with chemical concepts [1,18]. If the critical point K(, i) has no negative canonical curvatures, X=0, that is, if K(A,i) is a minimum, then the catchment region C(X,i) represents a stable chemical species. 

If the critical point K(X,i) has more than one negative canonical curvatures (A,>1), then its catchment region C(X,i) represents an unstable family of formal configurations (a formal "species") of little direct chemical importance. 

In order to prove this result, we shall show that the catchment regions C(X,i) and C (X,, i ) overlap, and we shall apply the catchment region point symmetry theorem to C (X, i ) and to a point K from the overlapping region. 

First we deal with the special case when the critical point K(X,i) is a maximum. In this case the catchment region is its own boundary, and one may take... 

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