Barycentric placement of molecules in space

1 Definition (The barycenter of a molecule) Assume that the molecule is placed in 3D space and that the coordinate triples of the atoms are the triples 

This vector x consists of the coordinates of the barycenter of the molecule. Subtracting it from the coordinate vectors, we get the vectors 

The corresponding coordinate triples y, now considered as points in 3D space, describe a new placement obtained by a translation of the molecule by x. This new placement of the molecule has its barycenter at the origin. For this reason we call such a placement a barycentricplacement. 

We shall apply this in the definition of point group (see Definition 3.2), which requires 

It is also known as dot product. The scalar product yields the Euclidean metric based on the Euclidean distance. A metric on a vector space is a mapping d with d(x, y) 0, and d(x,y) = 0 if and only if x = y. Moreover, it must be symmetric, d(x, y) = d(y,x) and obey the triangle inequality  

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