5 – Fuzzy Measures of Molecular Shape and Size

This chapter provides the basic concepts of fuzzy sets. Fuzzy set methods have been developed for a variety of applications, initially mostly in the engineering and technology; many applications in the natural sciences quickly followed. The Heisenberg relationship and many other aspects of quantum mechanics can be interpreted in terms of fuzzy sets. The theory of fuzzy sets has experienced an extremely rapid development; more recent introduction to the fundamentals and some of the more advanced topics of fuzzy set theory have been given from the dual perspectives of systematic theory and applications. The chapter reviews the specific notations and the fuzzy set concepts that are most relevant to the molecular shape problem, followed by a simple proof for a special fuzzy set generalization of the Hausdorff distance, motivated by the quantum chemical properties of fuzzy electronic densities of molecules. All aspects of molecular shape and size are fully reflected by the molecular electron density distribution. A molecule is an arrangement of atomic nuclei surrounded by a fuzzy electron density cloud. Fuzzy electron density modeling of large molecules has been improved to a level which is comparable to that for small molecules. For the description of shape differences between fuzzy objects—such as molecular electron density clouds—it is useful to generalize the Hausdorff metric for fuzzy sets.

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