General definition of ring puckering coordinates

TLDR: In this article, a unique mean plane is defined for a general monocyclic puckered ring, which is described by amplitude and phase coordinates which are generalizations of those introduced for cyclopentane by Kilpatrick, Pitzer, and Spitzer.

Abstract: A unique mean plane is defined for a general monocyclic puckered ring. The geometry of the puckering relative to this plane is described by amplitude and phase coordinates which are generalizations of those introduced for cyclopentane by Kilpatrick, Pitzer, and Spitzer. Unlike earlier treatments based on torsion angles, no mathematical approximations are involved. A short treatment of the four-, five-, and six-membered ring demonstrates the usefulness of this concept. Finally, an example is given of the analysis of crystallographic structural data in terms of these coordinates. Although the nonplanar character of closed rings in many cyclic compounds has been widely recognized for many years, there remain some difficulties in its quantitative specification. An important first step was taken by Kilpatrick, Pitzer, and Spitzer in their 1947 discussion of the molecular structure of cyclopentane.' Starting with the normal modes of out-of-plane motions of a planar regular pentagon,* they pointed out that displacement of the j t h carbon atom perpendicular to the plane could be written 2 112 zj = (/'SI 4 COS (2+ + 4 n ( j 11/51 (11 where q is a puckering amplitude and $ is a phase angle describing various kinds of puckering. By considering changes in an empirical potential energy for displacements perpendicular to the original planar form, they gave reasons to believe that the lowest energy was obtained for a nonzero value of q (finite puckering) but that this minimum was largely independent of $. Motion involving a change in fi at constant q was described as pseudorotation. Subsequent refinement of this work has involved models in which constraints to require constant bond lengths are imposed3q4 and extensions to larger rings5-' and some heterocyclic systems are considered.* Although the correctness of the model of Kilpatrick, et a f . , I and the utility of the (q. $) coordinate system is generally accepted, application to a general five-membered ring with unequal bond lengths and angles is not straightforward. Given the Cartesian coordinates for the five atoms (as from a crystal structure), determination of puckering displacements z, requires specification of the plane z = 0. A least-squares choice (minimization of Zz i2) is one possibility, but the five displacements relative to this plane cannot generally be expressed in terms of two parameters q and $ according to eq 1. An attempt to define a generalized set of puckering cordinates which avoids these difficulties was made by Geise, Altona, Romers, and S~ndara l ingam.~l ' Their quantitative description of puckering in five-membered rings involves the five torsion angles 0, rather than displacements perpendicular to some plane. These torsion angles are directly derivable from the atomic coordinates and are all zero in the planar form. They proposed a relationship of the form\