Logical Models of Molecular Shapes and Their Families

Abstract

A new discrete mathematical model of molecular shape is proposed, making use of the partition property of a representation of molecular shape. According to its geometrical and topological structure, a molecular surface can be partitioned into unbounded two-dimensional subsets (domains) and some common subsets of closures of two or more domains. The sets of these domains as a base of a finite topology, containing the Boolean n-cube as a lower Boolean sub-lattice of this topology, defines the domain of the proposed logical model. A logical function can be obtained that reflects the properties of the topological domains as well as the interrelations on the set of domains. Based on classical or quantum-chemical representations of molecular shape, these models allow one the implementation of methods of logical diagnostics in chemistry, and the definition of a metric on the set of molecular shape equivalence classes. The families of molecular shapes can be considered as sets of logical models. The proposed model is unified in the sense that the structures of differentiable and non-differentiable surfaces can be represented in the same mathematical framework. These logical models will also work for interpenetrations of the above types of surfaces.