Symmorphy transformation

The shape of a continuum A can be characterized by a subfamily Gsph(/1) of Gpo , where the subfamily Gsp, (/4) contain all those homeomorphisms S from G on, that bring the transformed object Sv4 into an arrangement that is indistinguishable from A. Since for such homeomorphisms S the morphologies of the transformed object and the original object A are indistinguishable, S is called a symmorphy transformation of object A. 

The set Gjph( ) of all symmorphy transformations S of the object A is a subgroup of Gp, . Clearly, if neither S, and S, alters the appearance of the morphology of A, then the transformations S3, defined as transformation Sj followed by S2 and denoted as the formal product... 

In the special case of selecting the three-dimensional object as a molecular charge density function p(r), the analogies between point symmetry and symmorphy are rather clear. By analogy with the point symmetry of nuclear arrangements, the molecular charge density function pit) can provide a criterion for selecting the symmorphy transformations of p(r) from the infinite family G, of homeomorphisms of the three-dimensional space. A homeomorphism 5 is a symmorphy transformation... 

For the molecular charge density function p(r), all those homeomor-phisms S of family Ghom are symmorphy transformations for which... 

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. 

These special transformations are the symmorphy transformations [43,1081 of the given "object", for example, of the electronic charge density function p(r). The terminology is justified by the analogy with symmetry in symmetry the metric properties are preserved, in symmorphy the morphology, the appearance of shape, is preserved. 

In symmorphy transformations the shape of the object is invariant in our example, the shape of the continuous charge density p(r) remains the same. Nevertheless, the metric properties, such as the distance between two points may change. 

A simple example, taken from reference [108], illustrates this point. Assume that the object undergoing a symmorphy transformation has a circular cross-section of radius equal to 1, where the circle is parametrized by an angle variable a, taken from the interval 0 a 2k. If a selected transformation t of the space has the effect of leaving all points of this circle on the same circle, but shifting points along the circle in a nonuniform manner, then the appearance of the circle will not change. For example, this is the case if t transforms the parameter a into the angle variable (i, where... 

The family of all symmorphy transformations of the given object p(r) form a subgroup gp of the group G of all homeomorphic transformations of the 3D space. This subgroup gp is, in fact, defined by the shape properties of the 3D object, and it provides a complete characterization of its shape, in our case, the shape of the molecular charge density function p(r). 

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. 

These two observations serve as the basis for the generalization of symmetry to symmorphy. For a general point set configuration K, it is K that selects a subset of special reflections, rotations, and inversions from the set of infinitely many such operations of the space. The conditions for selection is the indistinguishability of the original and transformed configurations. 

Point symmetry groups provide at least a partial characterization of molecular shapes. This characterization can be improved considerably by extending the family of point symmetry operators to a much larger family of continuous transformations. Symmorphy is a particular extension of the point symmetry group concept of finite point sets, such as a collection of atomic nuclei, to a complete algebraic shape characterization of continua, such as a three-dimensional electron distribution of a molecule. 

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... 

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