The interrelation between the endocyclic torsion angles phi(j) (j = 0-5) in a six-membered ring is given by a truncated Fourier series, phi(j) = PHI-2 cos (P2 + 4-pi-j/6) + PHI-3 cos (phi-j). It is deduced from a set of 8451 experimentally determined six-membered ring conformations extracted from the Cambridge Structural Database that this equation reproduces the observed endocyclic torsions within 1-degrees, except for sulfur-containing rings where the margins appear to be slightly larger. These findings are corroborated by an analysis of six-membered rings generated by molecular mechanics. The ring puckering coordinates PHI-2, P2, and PHI-3 map out the conformations attainable by six-membered rings on the surface of a sphere. Thus, a convenient and pictorial description of conformational space accessible to six-membered rings is obtained. A comparison is made with the well-known Cremer-Pople ring puckering formalism. It is shown that, especially in the case of nonequilateral rings, the present method is more consistent with internal angular characteristics (e.g., local flattening) displayed by six-membered rings than the Cremer-Pople formalism.
