Quantum similarity measures definition

In previous papers [1,2] the authors have worked out the theoretical foundation of Quantum Similarity (QS). In this paper, several practical results will be listed. All of them came from some of the Quantum Similarity Measures (QSM) defined previously and are particular cases of the general QSM definition obtained in reference [1]. 

Molecular Similarity and QSAR. - In a first contribution on the design of a practical, fast and reliable molecular similarity index Popelier107 proposed a measure operating in an abstract space spanned by properties evaluated at BCPs, called BCP space. Molecules are believed to be represented compactly and reliably in BCP space, as this space extracts the relevant information from the molecular ab initio wave functions. Typical problems of continuous quantum similarity measures are hereby avoided. The practical use of this novel method is adequately illustrated via the Hammett equation for para- and me/a-substituted benzoic acids. On the basis of the author s definition of distances between molecules in BCP space, the experimental sequence of acidities determined by the well-known a constant of a set of substituted congeners is reproduced. Moreover, the approach points out where the common reactive centre of the molecules is. The generality and feasibility of this method will enable predictions in medically related Quantitative Structure Activity Relationships (QSAR). This contribution combines the historically disparate fields of molecular similarity and QSAR. 

Density Functions play a fundamental role in the definition of Quantum Theory, due to this they are the basic materials used in order to define Quantum Objects and from this intermediate step, they constitute the support of Quantum Similarity Measures. Here, the connection of Wavefunctions with Extended Density Functions is analysed. Various products of this preliminary discussion are described, among others the concept of Kinetic Energy Distributions. Another discussed set of concepts, directly related with the present paper, is constituted by the Extended Hilbert Space definition, where their vectors are defined as column structures or diagonal matrices, containing both wavefunctions and their gradients. The shapes of new density distributions are described and analysed. All the steps above summarised are completed and illustrated, when possible, with practical application examples and visualisation pictures. 

The actual discussion has the aim to adopt this previous spirit, but obviously choosing a much more modest point of view, attached to Quantum Similarity Measures (QSM). This work is focused to explore the various possible extensions for the study of DF, the auxiliary building block elements of QSM [16-38]. In order to fulfil such a purpose, this study will start analysing a sound formal basis as a first step to understand the role of momentum operators in computational Quantum Chemistry. From this introductory position, it will be finally obtained a general pattern enveloping the whole area of DF study, beginning at the basic aspects and ending over the final applications of extended DF definitions. 

Appendix A. Definitions leading to Quantum Similarity measures. 

Application of Eq. [86] in the general definition of a molecular quantum similarity measure then gives ... 

After discussing the definition of atomic density functions, quantum similarity measures are introduced and three case studies illustrate that specific influences on the density function of electron correlation and relativity can be quantified in this way. Although no periodic patterns were found in Mendeleev s table, the methodology is particularly successful for quantifying the influence of relativistic effects on the density function. 

A general definition of the Quantum Molecular Similarity Measure is reported. Particular cases of this definition are discussed, drawing special attention to the new definition of Gravitational-like Quantum Molecular Similarity Measures. Applications to the study of fluoromethanes and chloro-methanes, the Carbonic Anhydrase enzyme, and the Hammond postulate are presented. Our calculations fully support the use of Quantum Molecular Similarity Measums as an efficient molecular engineering tool in order to predict physical properties, lMok>gical and pbarraacdogical activities, as well as to interpret complex chemical problems. 

A similarity measure over the unit shell of any VSS can be defined through the description of the mathematical elements, which have been described so far. In the simplest way, an MQSM can be defined knowing the appropriate density function tags of two quantum objects Pa Pb j adapted to some shells of the corresponding VSS, and with as a weight some positive definite operator ff. In that case, the integral measure... 

Vol. 1, R. Carbo-Dorca and P. G. Mezey, Eds., JAI Press, London, 1996, pp. 1-42. Quantum Molecular Similarity Measures Concepts, Definitions, and Applications to Quantitative Structure-Property Relationships. 

In classical mechanics It Is assumed that at each Instant of time a particle is at a definite position x. Review of experiments, however, reveals that each of many measurements of position of Identical particles in identical conditions does not yield the same result. In addition, and more importantly, the result of each measurement is unpredictable. Similar remarks can be made about measurement results of properties, such as energy and momentum, of any system. Close scrutiny of the experimental evidence has ruled out the possibility that the unpredictability of microscopic measurement results are due to either inaccuracies in the prescription of initial conditions or errors in measurement. As a result, it has been concluded that this unpredictability reflects objective characteristics inherent to the nature of matter, and that it can be described only by quantum theory. In this theory, measurement results are predicted probabilistically, namely, with ranges of values and a probability distribution over each range. In constrast to statistics, each set of probabilities of quantum mechanics is associated with a state of matter, including a state of a single particle, and not with a model that describes ignorance or faulty experimentation. 

---