Quantum similarity and QSPR in Euclidean-, and Minkowskian–Banach spaces

Abstract

This paper describes first how Euclidian- and Minkowskian–Banach spaces are related via the definition of a metric or signature vector. Also, it is discussed later on how these spaces can be generated using homothecies of the unit sphere or shell. Such possibility allows for proposing a process aiming at the dimension condensation in such spaces. The condensation of dimensions permits the account of the incompleteness of classical QSPR procedures, independently of whether the algorithm used is statistical bound or AI-neural network related. Next, a quantum QSPR framework within Minkowskian vector spaces is discussed. Then, a well-defined set of general isometric vectors is proposed, and connected to the set of molecular density functions generating the quantum similarity metric matrix. A convenient quantum QSPR algorithm emerges from this Minkowskian mathematical structure and isometry.

Spaces of mathematical chemistry

“…There are other approaches to the chemical space that characterise molecules in terms of their quantum chemistry descriptions, often electronic densities. In this case the notion of nearness is given by the resemblance of those quantum chemical descriptions [19,20]. A more recent account of the chemical space, although the most traditional in the history of chemistry, is that of a set of chemicals related by chemical reactions [21,22].…”

“…There are different ontological levels for chemistry, which range from bulk substances to quasi-molecular species [63] and chemical annotation, over the history, has spanned all these levels. 20 Hence, before the acceptance of molecular structural theory, substances were labelled by their composition, then by the binary notation following the dualistic theory by Berzelius [59]. Today the ontology of chemistry revolves around molecular structures represented as labelled graphs and encoded via SMILES or InChI structures.…”

“…In Figure 3 the three most populated frameworks are depicted. The finding that chemists do not populate the chemical space with a large diversity of molecular frameworks is presumably related to the conservatism in the selection of starting materials to undergo chemical synthesis 20 Examples of quasi-molecular species include molecular clusters, van der Waals complexes and other chemical species [64]. [22,68], a conservatism that has been also found for the subspace of pharmaceutical chemicals [69].…”

Classification Maps: A New Mathematical Tool Supporting the Diagnosis of Age-Related Macular Degeneration

“…The descriptors are applied in methods used in the theory of molecular similarity. The assumption "The molecules that are similar in certain aspects have similar properties" is a cornerstone of the methods of Quantitative Structure-Activity/Property Relationships (QSAR/QSPR) [52][53][54]. Such techniques are applied to predict the activity, reactivity or properties of new molecules.…”