Communications on quantum similarity (4): collective distances computed by means of similarity matrices, as generators of intrinsic ordering among quantum multimolecular polyhedra

This study generalizes the notion of distance via defining an axiomatic collective distance, between arbitrary vector sets. A first part discusses conceptual tools, which can be later useful for general mathematical practice or as computational quantum similarity indices. After preliminary definitions, two elements, which can be associated with arbitrary sets of a vector space, are described: the centroid and the variance vectors. The Minkowski norm of the variance vector is shown to comply with the axioms of a collective distance. The role of the Gram matrix, associated with a vector set, is linked to the definition of numerical variance. Several simple application examples involving linear algebra and N-dimensional geometry are given. In a second part, all previous definitions are applied to quantum multimolecular polyhedra (QMP), where a set of molecular quantum mechanical density functions act as vertices. The numerical Minkowski norm of the variance vector in any QMP could be considered as a superposition of molecular contributions, corresponding to a new set of quantum similarity indices, which can generate intrinsic ordering among QMP vertices. In this way, the role of quantum similarity matrix elements is evidenced. Application to collections of molecular structures is analyzed as an illustrative practical exercise. The connection of the QMP framework with classical and quantum quantitative structure-properties relation (QSPR) becomes evident with the aid of numerical examples computed over several molecular sets acting as QMP. (C) 2015 JohnWiley & Sons, Ltd.