Foundation of Quantum Similarity Measures and Their Relationship to QSPR: Density Function Structure, Approximations, and Application Examples RAMON CARBO ´ -DORCA, XAVIER GIRONE ´ S Institute of Computational Chemistry, University of Girona, Girona 17071, Catalonia, Spain Received 6 September 2002; accepted 26 March 2004 Published online 13 August 2004 in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/qua.20191 ABSTRACT: This work presents a schematic description of the theoretical foundations of quantum similarity measures and the varied usefulness of the enveloping mathematical structure. The study starts with the definition of tagged sets, continuing with inward matrix products, matrix signatures, and vector semispaces. From there, the construction and structure of quantum density functions become clear and facilitate entry into the description of quantum object sets, as well as into the construction of atomic shell approximations (ASA). An application of the ASA is presented, consisting of the density surfaces of a protein structure. Based on this previous background, quantum similarity measures are naturally constructed, and similarity matrices, composed of all the quantum similarity measures on a quantum object set, along with the quantum mechanical concept of expectation value of an operator, allow the setup of a fundamental quantitative structure– activity relationship (QSPR) equation based on quantum descriptors. An application example is presented based on the inhibition of photosynthesis produced by some naphthyridinone derivatives, which makes them good herbicide candidates. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem 101: 8 –20, 2005 Key words: tagged sets; inward matrix product; matrix signature; vector semispaces; -shells; convex conditions; generating rules; density functions; quantum objects; quantum object sets; molecular quantum similarity measures; similarity matrices; discrete quantum object sets; QSPR Correspondence to: R. Carbo ´-Dorca; e-mail: quantum@iqc.udg.es Contract grant sponsor: Foundation M. F. de Roviralta. Contract grant sponsor: Centro de Investigacio ´n Cientifica y Tecnolo ´ gica (CICYT). Contract grant number: SAF2000-223. International Journal of Quantum Chemistry, Vol 101, 8 –20 (2005) © 2004 Wiley Periodicals, Inc.