Symmorphy equivalence

If two transformations 11 and t2 are related to each another as t is related to t] in conditions (8.19) and (8.20), then ti and t2 are said to be symmorphy equivalent, or in short, symmorphic to one another with respect to the given object p(r). Symmorphy equivalence is denoted by... 

The family of all symmorphy equivalence classes of a given object p(r) form the symmorphy group hp of the object p(r). This group hp is formally defined [43,108] as the quotient group of group gp with respect to the symmorphy equivalence nip. The product of two symmorphy classes Tj and T2 in the symmorphy group hp is defined as the class T3,... 

Some simplifications are possible if equivalence classes of symmorphy transformations can be defined where operations S from the same class transform the space occupied by the object A the same way and differ only in parts of the space where A is not present. Furthermore, using the Brouwer fixed point theorem, a subgroup structure of symmorphy groups Gjph( ) provides a more detailed characterization of molecular shape. These aspects will not be reviewed here. 

However, the group gp is much too complicated for practical purposes of molecular shape characterization. Fortunately, the behavior of transformations t of family gp far away from the object p(r) is of little importance, and one can introduce some simplifications. Let us assume that the 3D function considered [e.g., an approximate electron density function p(r)], becomes identically zero outside a sphere S of a sufficiently large radius. As long as two symmorphy transformations tj and t2 have the same effect within this sphere, the differences between these transformations have no relevance to the shape of p(r), even if they have different effects in some domains outside the sphere. All such transformations t of equivalent effects within the relevant part of the 3D space can be collected into equivalence classes. In the symmorphy approach to the analysis of molecular shape, these classes are taken as the actual tools of shape characterization. 

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