Generalized Turnstile Rotation: Formulation, Visualization, Workflow Implementation, and Application for Modeling Polytopal Rearrangements

Abstract

Turnstile rotation is a well-known polytopal rearrangement mechanism in coordination compounds, yet its computational description is often hindered by the challenges of conventional internal coordinates to represent such collective, large-amplitude motions. In this work, we present a mathematical formulation for generalized $N$-arm turnstile rotation, a dedicated PyMOL plugin (gTA) for intuitive structure visualization and a command-line utility (gTA-cli) for structure manipulation. On this basis, we develop a practical computational workflow that combines turnstile rotation-driven relaxed scans with two downstream routes to locate the transition states. The generality of the approach is demonstrated through five chemically diverse case studies, ranging from classical rearrangements in SF${}_4$, IF${}_7$, and [Co(en)${}_3$]${}^{3+}$ to previously unexplored fluxional processes in bismuth and nickel complexes. In all cases, the method enables the identification of transition states associated with pronounced polytopal rearrangements that are difficult to access using standard coordinate schemes. These results establish generalized turnstile rotation as a fundamental molecular motion and highlight its potential as a broadly applicable strategy for studying fluxionality and dynamic stereochemistry in coordination chemistry. The PyMOL plugin gTA and the Python utility gTA-cli with the associated relaxed scan workflows are open-source and freely available on Github at https://github.com/smutao/gTA-plugin and https://github.com/smutao/gTA-workflow, respectively.