Equivalence properties of the Cremer & Pople puckering coordinates for N-membered rings

Abstract

The Cremer & Pople puckering coordinates are an invaluable tool in the conformational mapping and classification of rings fragments, especially for medium size rings. However, because of the topological symmetries that many chemical fragments exhibit, it is important to be aware of which values of the puckering coordinates should be considered equivalent because they correspond to the same fragments, up to a permutation of the atoms labels or a coordinates inversion. In this paper a precise description of the action of the relevant permutational operators and coordinate inversion is given for a ring of any size. The results are then specialized to the case of medium rings (6–8 atoms) for which on one hand, the symmetry equivalents of the puckering coordinates are obtained and on the other hand, in the case of structurally equivalent atoms, simple algorithms are derived to compute the representative puckering coordinates that fall into the representative asymmetric unit.