A general method for constructing and searching conformations in molecular rings: From Cremer–Pople coordinates to 3D geometries

We present a general framework to construct and systematically search for ring conformations based on Cremer-Pople (CP) coordinates. To our knowledge, this is the first algorithm that provides reasonable ring geometries across different ring sizes from arbitrary CP coordinates. A two-stage ring reconstruction algorithm is proposed: (i) projecting the molecule in the xy plane and enforcing ring closure while preserving bond lengths, and (ii) redistributing angular distortions across the ring via constrained minimization to achieve chemically viable conformations. The approach is extended to rings with rigid (multiple) bonds through a rigorous ring-reduction scheme that lowers puckering dimensionality and incorporates local stiffness via adjustable parameters, allowing for straightforward recovery of the full structure. In addition, the positions of the substituents attached to the ring are deduced from invariant local references. To systematically explore the conformational space, we formulate a preconditioned sampling of CP amplitudes as a function of hyperspherical coordinates. In addition, the concept of basis conformations is examined to ensure the unambiguous identification of conformers. Applications to saturated and unsaturated rings with up to eight atoms at the ωB97X-D/def2-TZVPP level demonstrate accurate recovery and classification of known conformers, including correct symmetry multiplicities. Overall, this framework offers a robust and general route to generate high-quality starting geometries for subsequent electronic structure optimizations, while facilitating efficient exploration of puckering space.