Molecular Face-Rotating Polyhedra: Chiral Cages Inspired by Mathematics

Abstract

Molecular polyhedral cages, notable for their enclosed inner cavities, can possess varying degrees of symmetry, spanning from regular Platonic polyhedra to lower symmetry forms that may display chirality. Crafting chiral molecular cages typically involves using building blocks containing stereogenic elements or arranging achiral components in a manner that lacks mirror and inversion symmetries. Achieving precise control over their chirality poses both significance and challenges.

In this Account, we present an overview of our research endeavors in the realm of chiral molecular polyhedral cages, drawing inspiration from Buckminster Fuller’s “Face-Rotating Polyhedra (FRP)”. Mathematically, FRP introduce a unique form of chirality distinguished by a rotating pattern around the center of each face, setting it apart from regular polyhedra.

Molecular FRP can be constructed using two types of facial building blocks. The first includes rigid, planar molecules such as truxene and triazatruxene, which exhibit either clockwise or counterclockwise rotations in two dimensions. The second category involves propeller-like molecules, e.g., tetraphenylethylene, 1,2,3,4,5-penta(4-phenylaldehyde)pyrrole, and tridurylborane, displaying dynamic stereochemistry.

The synthesis of FRP may potentially yield a diverse array of stereoisomers. Achieving high stereoselectivity becomes feasible through the selection of building blocks with specific substitution patterns and rigidity. Prominent noncovalent repulsive forces within the resulting cages often play a pivotal role in the dynamic covalent assembly process, ultimately leading to the formation of thermodynamically stable FRP products.

The capacity to generate a multitude of stereoisomers, combined with the integration of chiral vertices, has facilitated investigations into phenomena such as chiral self-sorting and the “sergeant and soldiers” chiral amplification effect in FRP. Even the inclusion of one chiral vertex significantly impacts the stereochemical configuration of the entire cage. While many facial building blocks establish a stable rotational pattern in FRP, other units, such as tridurylborane, can dynamically transition between P and M configurations within the cage structures. The kinetic characteristics of such stereolabile FRP can be elucidated through physicochemical investigations.

Our research extends beyond the FRP concept to encompass mathematical analysis of these structures. Graph theory, particularly the coloring problem, sheds light on the intricate facial patterns exhibited by various FRP stereoisomers and serves as an efficient tool to facilitate the discovery of novel FRP structures. This approach offers a fresh paradigm for designing and analyzing chiral molecular polyhedral cages, showcasing in our work the synergy between mathematics and molecular design.