The linear space of ring conformations

Abstract

The description of ring conformation in terms i ' a set of pocketing coordinates relative to a mean plane is shown to be equivalent to the group theoretic definition of the conformation of a puckered ring in terms of out-of-plane displacements of a planar polygon. A description of the conformation of a general N-membered ring, based on crystallographic coordinates, is provided in terms of the one-dimensional displacement modes of the regular polygon of symmetry. The set of puckered forms therefore represent a linear space. The out-of-plane displacement modes of the irreducible representations provide a natural basis set. Two linearly independent modes equivalent to the two orthogonal modes of each two-dimensional representation, and a one-dimensional mode for an even-membered ring, form a (N-3)-dimensional basis. The linear coefficients are independent of the puckering amplitude and of the ring numbering scheme. The linear combination of primitive forms provides a simple algorithm to identify classical forms and a quantitative description of conformations, intermediate between the classical forms. The one-dimensional model describes the conformation of large rings. Conformational analysis of nine-membered rings is completed by projection of the conformational space onto a three-dimensional surface defined by the puckering parameters. Intermediate forms are expressed as a linear combination of six primitive forms. The conformation of larger rings is characterized by the linear coefficients, interpreted graphically. A nomenclature for any symmetrical conformation is proposed.