Molecular Geometry: A New Challenge and Opportunity for Geometers

Abstract

Summary form only given. Molecular structure determines molecular function and the geometry is one of the most fundamental aspects of the molecular structure regardless it is organic or inorganic. Despite of its importance, the theory for understanding the geometry of molecules has not been sufficiently developed. In this talk, we will present a unified theory of molecular geometry (MG) as a new discipline and demonstrate how the theory can be used for "accurately", "efficiently", and "conveniently" solving all molecular problems related on structure.The MG theory is based on the beta-complex which is a derived structure from the Voronoi diagram of atoms and its dual structure called the quasi-triangulation. Voronoi diagrams are everywhere in nature and are useful for understanding the spatial structure among generators. Unlike the well-known ordinary Voronoi diagram of points, the Voronoi diagram of spherical atoms has been known to be difficult to compute and to possess a few anomaly cases. Once computed, however, it nicely defines the proximity among the atoms in molecules.This talk will discuss the quasi-triangulation, the dual structure of the Voronoi diagram of atoms, and the beta-complex in the three-dimensional space. It turns out that the beta-complex, together with the Voronoi diagram and quasi-triangulation, can be used to accurately, efficiently, and conveniently solve seemingly unrelated geometry and topology problems for molecules within a single theoretical and computational framework. Among many application areas which will be explained, structural molecular biology and noble material design are the most immediate application area. In this talk, we will also demonstrate our molecular modeling and analysis software, BetaMol in 3D and BetaConcept in 2D, which are entirely based on the beta-complex and the Voronoi diagram. Programs are freely available at the Voronoi Diagram Research Center (VDRC, http://voronoi.hanyang.ac.kr/).