Some Interrelations between Symmetry and Automorphism Groups

Apart from the identity operation E, the elements of a space group are geometrically well-defined subspaces of the 3-dimensional space, i.e. points (C), straight lines (C _n , S _n ) and planes (σ _h , σ _v , σ _d ). Thus the operations associated with these elements are realized in each point of the space. On the other hand, each point of the space and, thus, each center of the molecule is transformed by such an operation, provided the point does not belong to the symmetry element considered. As a final consequence of these circumstances, the molecule treated by means of symmetry groups is considered to be rigid. This means that each atom of the molecule is associated with a triple of coordinates characterizing its mean position with absolute precision. Vibrations of the atoms about their mean positions are discussed in terms of elongation vectors (see paragraph 8.4.3). This concept begins to break down when an internal degree of freedom (e.g. the torsion of a methyl group) becomes fully excited. As the shall show later, automorphism groups of the molecular graph are well-suited to treat the symmetry in non-rigid molecules.