Quantum mechanical origin of QSAR: theory and applications

Abstract In this paper, search is carried out on how to develop the formalism where quantum similarity measures (QSM) become a natural product of the theoretical framework. This fact is later used to establish a fundamental connection between quantum theory and QSAR, which is analysed in turn within the realm of discrete quantum chemistry. In order to perform such a task, several theoretical tools are revised in a previous step. The first section is devoted to construct the concept of tagged set . Next, the definition of quantum object (QO) is clarified by means of ideas from a quantum theory background and the previous tagged set formalism. In the definition of QO, density functions (DF) play a fundamental role and a possible simplified mathematical picture is presented. On the road to prepare the problem solving tools, convex sets result to be prominent, and the notion of vector semispace appears as a consequence. The transformation rule , a device to connect wavefunctions with DF, is defined in a new step. Various products of this preliminary discussion are described, among others the concept of kinetic energy and other observable distributions. QSM as a source of discrete representation of molecular structures is made evident in this context. Further theoretical development intends to study discretisation, the transformation of infinite dimensional functional spaces into n -dimensional ones. This result adds new perspectives to the discrete representation of QO, because it (a) provides a source of new QO descriptors, like a generalisation of scalar product and new similarity indices, (b) describes the QSPR theoretical background enabling the construction of the adequate mathematical tools in order to discuss connected problems (limitations of linear models, tuned-QSAR, p -valued classification of QO, among others), (c) allows the construction of sound and general alternatives of Hammet's σ or log  P parameters. All the steps above summarised are completed and illustrated, when possible, with practical application examples and visualisation pictures.

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