Exploring the conformation space of cycloalkanes by linearized embedding

Abstract

Linearized embedding is a variant on the usual distance geometry methods for finding atomic Cartesian coordinates given constraints on interatomic distances. Instead of dealing primarily with the matrix of interatomic distances, linearized embedding concentrates on properties of the metric matrix, the matrix of inner products between pairs of vectors defining local coordinate systems within the molecule. Here, the approach is used to explore the full conformation space allowed to small cyclic alkanes, given the constraint of exact bond lenghts and bond angles. Useful general tools developed along the way are expressions for rotation matrices in any number of dimensions and a generalization of spherical coordinates to any number of dimensions. Analytical results give some novel views of the conformation spaces of cyclopropane, cyclobutane, cyclopentane, and eyclohexane. A combination of numerical and analytical approaches gives the most comprehensive description to date of the cycloheptane conformation space with fixed bond lengths and angles. In this representation, the pseudorotation paths of cyclohexane and cycloheptane are closed curved lines on the surfaces of spheres.