Some Notions Regarding Molecular Shape and Scaling

Some Notions Regarding Molecular Shape and Scaling 

Molecular structure and molecular phenomena can be regarded as exhibiting a hierarchical organization. Depending on the scale used to analyze a 

As a result, we must adapt our point of view to the size of each system, dealing only with properties that are relevant to each scale. This notion applies to the subject in this chapter. In our present case, the molecular shape descriptors (and the notion of molecular shape itself) will change depending on the scale or size of the system considered. 

For an w-atom molecule, configurational space is the set-theoretical quotient between the full coordinate space (9i ) and the rigid translation and rotation operators. This space is sometimes referred to in mathematics as a size-and-shape spaceJ- If only translations are considered, we deal with pre-size-and-shape space. Finally, the term shape space is reserved to the configurations left after translation, rotation, and scaling have been removed. Geometrical descriptors are defined on size-and-shape spaces, whereas topological descriptors work on shape space configurations. 

We shall retain the actual scale when discussing shape features. Even though rotation and translation will not play a role in our analysis, the molecular shape may depend on size. Often, such a dependence takes the form of a power law relation (a scaling law) between the descriptor and the number of atoms in the molecule. 

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