Aspects of Quantum Similarity

A continuing effort has been maintained by the scientific community to provide a firm theoretical and mathematical basis for molecular quantum similarity. According to Carbo-Dorca et al., the basis of molecular quantum similarity and of quantum QSAR (as described later) is fovmded in the concepts of tagged sets and vector semispaces. To make quantum similarity understandable to the novice, the following explanatory paragraphs provide first the required mathematical basis and second an extensive discussion of some useful aspects of vector semispaces. 

As mentioned, the application of molecular descriptors with quantum mechanical origins was proposed several years ago, ° but the first ideas about quantum similarity (QS) and QS measures (QSM) were published aroimd 1980. However, it has been not until recently that the mathematical and physical foundations of QS have been developed in a series of publica- 

In those articles, several new theoretical definitions, related to old concepts, were described. The first idea described was the so-called tagged set concept Tagged sets not only seemed to be essential to QS theory, but they also constituted a generaHzation of the well-known fuzzy set theoretical setup.  

A tagged set Z is defined as the Cartesian product of two sets Z = O X T, where O, the object set, contains as elements the so-called objects and T, the tag set, contains as elements the so-called tags. Thus, any element of Z is constructed by an ordered pair made by an object and a tag. That is, V0eZ = OxT 3coeOA3TeT 0 = (( x). 

After the seminal structure building of the QS formalism, several additional studies appeared over time, which developed new theoretical details. Especially noteworthy is the concept of vector semispace (VSS). This mathematical construction will be shown to be the main platform on which several QS ideas are built, related in turn, to probability distributions and hence to quantum mechanical probability density functions. Such quantum mechanical density distributions form a characteristic functional set, which can be easily connected to VSS properties. Construction of the so-called quantum objects (QO) and their collections the QO sets (QOS) (see, for example, Carbo-Dorca ), easily permit the interpretation of the nature of quantum similarity measures for relationships between such quantum mechanically originated elements. Within quantum similarity context, QOS appear as a particular kind of tagged sets, where objects are submicroscopic systems and their density functions become tags. 

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The last 15 years witnessed a multitude of studies on various aspects of quantum similarity of molecules (the use of different separation operators [37], the replacement of the density by more appropriate reactivity oriented functions [38, 39] within the context of conceptual DFT [40], the treatment of enantiomers [31,41 3]). With the exception of two papers by Carbo and co-workers, the study of isolated atoms remained surprisingly unexplored. In the first paper [44] atomic self-similarity was studied, whereas the second one [45] contains a relatively short study on atomic and nuclear similarity, leading to the conclusion that atoms bear the highest resemblance to their neighbors in the Periodic Table. 

The next section, entitled Basic Aspects of Molecular Similarity, gives a general overview of molecular similarity and the usual vocabulary used by chemists in this field. In the section entitled The Electron Density as Molecular Descriptor, some elements of quantum chemistry will be presented. These sections should not be expected by the reader to offer a rigorous discussion of all aspects of such broad fields, and therefore, only the most important definitions and concepts will be introduced. The following sections will then address extensively the subject of molecular quantum similarity in both theoretical and practical aspects. An ample references list will help the interested reader to look up the more specialized literature. 

In fact, one of the objectives of the book is to introduce nonexpert readers to modem computational spectroscopy approaches. In this respect, the essential basic background of the described theoretical models is provided, but for the extended description of concepts related to theory of molecular spectra readers are referred to the widely available specialized volumes. Similarly, although computational spectroscopy studies rely on quantum mechanical computations, only necessary aspects of quantum theory related directly to spectroscopy will be presented. Additionally, we have chosen to analyze only those physical-chemical effects which are important for molecular systems containing atoms from the first three rows of the periodic table, while we wiU not discuss in detail effects and computational models specifically related to transition metals or heavier elements. Particular attention has been devoted to the description of computational tools which can be effectively applied to the analysis and understanding of complex spectroscopy data. In this respect, several illustrative examples are provided along with discussions about the most appropriate computational models for specific problems. 

A Brief Review of the QSAR Technique. Most of the 2D QSAR methods employ graph theoretic indices to characterize molecular structures, which have been extensively studied by Radic, Kier, and Hall [see 23]. Although these structural indices represent different aspects of the molecular structures, their physicochemical meaning is unclear. The successful applications of these topological indices combined with MLR analysis have been summarized recently. Similarly, the ADAPT system employs topological indices as well as other structural parameters (e.g., steric and quantum mechanical parameters) coupled with MLR method for QSAR analysis [24]. It has been extensively applied to QSAR/QSPR studies in analytical chemistry, toxicity analysis, and other biological activity prediction. On the other hand, parameters derived from various experiments through chemometric methods have also been used in the study of peptide QSAR, where partial least-squares (PLS) analysis has been employed [25]. 

Such dependence is naturally not acceptable if one wants to put similarity between quantum systems in a theoretical framework. As will be shown below, the so-called theory of molecular quantum similarity (MQS) does offer a solid basis. The aim of the present chapter is to introduce the basic aspects of the theory and to allow the reader to follow the literature. For applications and a more in-depth presentation of the mathematical aspects, the reader is referred to the review by Bultinck et al. [4],... 

The spherical pendulum, which consists of a mass attached by a massless rigid rod to a frictionless universal joint, exhibits complicated motion combining vertical oscillations similar to those of the simple pendulum, whose motion is constrained to a vertical plane, with rotation in a horizontal plane. Chaos in this system was first observed over 100 years ago by Webster [2] and the details of the motion discussed at length by Whittaker [3] and Pars [4]. All aspects of its possible motion are covered by the case, when the mass is projected with a horizontal speed V in a horizontal direction perpendicular to the vertical plane containing the initial position of the pendulum when it makes some acute angle with the downward vertical direction. In many respects, the motion is similar to that of the symmetric top with one point fixed, which has been studied ad nauseum by many of the early heroes of quantum mechanics [5]. 

Since many of these developments reach into the molecular domain, the understanding of nano-structured functional materials equally necessitates fundamental aspects of molecular physics, chemistry, and biology. The elementary energy and charge transfer processes bear much similarity to the molecular phenomena that have been revealed in unprecedented detail by ultrafast optical spectroscopies. Indeed, these spectroscopies, which were initially developed and applied for the study of small molecular species, have already evolved into an invaluable tool to monitor ultrafast dynamics in complex biological and materials systems. The molecular-level phenomena in question are often of intrinsically quantum mechanical character, and involve tunneling, non-Born-Oppenheimer effects, and quantum-mechanical phase coherence. Many of the advances that were made over recent years in the understanding of complex molecular systems can therefore be transposed and extended to the study of... 

CPL and CD are based upon similar aspects of molecular structure. It is important to realize, however, that, even if the same states are involved, these measurements do not usually supply redundant information. From the Franck-Condon principle, CPL is a probe of excited state geometry, and CD is a probe of ground state geometry. CPL measurements have some advantages over the measurement of CD, as well as some inherent limitations. The most serious limitation is, quite obviously, that the optically active molecule under study must contain a luminescent chromophore with a reasonable quantum yield. Although this severely limits the range of possible applications of CPL, it does result in a specificity and selectivity that is not present in CD or absorption experiments. 

A quantum chemical approach is proposed for the representation of functional groups in chemistry. The approach is based on a simple density domain condition and on the additive, fuzzy electron density fragmenation method that also serves for the rapid caculation of ab initio quality electron densities of large molecules. Several aspects of the approach are described, including methods for similarity and complementarity analysis of functional groups. 

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