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
