Quantum similarity

Second, information is obtained on the nature of the nuclei in the molecule from the cusp condition. Third, the Hohenberg-Kohn theorems point out that besides determining the number of electrons, the density also determines the external potential that is present in the molecular Hamiltonian. Once the number of electrons is known from Eq. [6] and because the external potential is determined by the electron density, the Hamiltonian is completely determined. Once the electronic Hamiltonian is determined, we can solve Schrodinger s equation for the wave function, subsequently determining all observable properties of the system. 

Having established that the electron density is the basic molecular descriptor, and that a theoretical justification exists for its selection, the theory of molecular quantum similarity can now be developed. 

In 1980, Carbo, Arnau and Leyda were the first to use molecular quantum similarity. As an anecdote, in the submitted version of the manuscript, the title was How far is one molecule from another After a reviewer s comment, this title was changed to How similar is one molecule to another The revised title has a much more obvious reference to similarity. In a sense, both titles are descriptive, because in that manuscript, the first degree of molecular similarity with a distance measure was presented. More precisely, a distance measure was introduced as 

In this equation, A and B denote two different molecules for which the electron densities are represented as p (r) and pg(r) respectively. Equation [7] is an Euclidean distance between the electron densities of both molecules. Euclidean 

In the case of k = 2, Eq. [8] corresponds to the well-known Euclidean distance of which Eq. [7] is an integral version. In the case of k = 1, we find the Manhattan or city-block distance. The choice of Euclidean distance is a computationally interesting choice, but it is by no means the only one possible. In this context, it is appropriate to mention the four requirements that should be associated with a true distance measure  

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Carbb, R., Calabuig, B., Vera, L. and Besalu, E. (1994) Molecular quantum similarity theoretical framework, ordering principles, and visualization techniques. In Advances in Quantum Chemistry, Vol. 25, Lowdin, P.-O., Sabin, J.R. and Zemer, M.C. (Eds.), Academic Press, New York. 

Besalu, E., Carbo, R., Meslres, J. and Sola, M. Foundations and Recent Developments on Molecular Quantum Similarity. 173, 31-62 (1995). 

Carbo, R., B. Calabuig, L. Vera, andE. Basalu. 1994. Molecular Quantum Similarity Theoretical Framework, Ordering principles, and Visualization Techniques. 25, 253. 

Two objects are similar and have similar properties to the extent that they have similar distributions of charge in real space. Thus chemical similarity should be defined and determined using the atoms of QTAIM whose properties are directly determined by their spatial charge distributions [32]. Current measures of molecular similarity are couched in terms of Carbo s molecular quantum similarity measure (MQSM) [33-35], a procedure that requires maximization of the spatial integration of the overlap of the density distributions of two molecules the similarity of which is to be determined, and where the product of the density distributions can be weighted by some operator [36]. The MQSM method has several difficulties associated with its implementation [31] ... 

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],... 

Introducing the notion of a molecular quantum similarity measure (MQSM) ZAB as... 

Having established the most important concepts for MQS, the next step is to actually compute the numerical values associated with the quantum similarity measures. Electron densities can naturally be obtained from many quantum chemical methods such as DFT, Hartree-Fock, configuration interaction, and many more, even from experiment. 

By its size, this chapter fails to address the entire background of MQS and for more information, the reader is referred to several reviews that have been published on the topic. Also it could not address many related approaches, such as the density matrix similarity ideas of Ciosloswki and Fleischmann [79,80], the work of Leherte et al. [81-83] describing simplified alignment algorithms based on quantum similarity or the empirical procedure of Popelier et al. on using only a reduced number of points of the density function to express similarity [84-88]. It is worth noting that MQS is not restricted to the most commonly used electron density in position space. Many concepts and theoretical developments in the theory can be extended to momentum space where one deals with the three components of linear momentum... 

The book covers a gamut of related topics such as methods for determining atoms-in-molecuies, population analysis, electrostatic potential, molecular quantum similarity, aromaticity, and biological activity. It also discusses the role of reactivity concepts in industrial and other practical applications. Whether you are searching for new products or new research projects, this is the ultimate guide for understanding chemical reactivity. 

Patrick Bultinck, Xavier Girones and Ramon Carbo-Dorca, Molecular Quantum Similarity Theory and Applications. 

Amat, L. and Carbo-Dorca, R. (1999) Fitted electronic density functions from H to Rn for use in quantum similarity measures cis-diamminedichloroplatinum(II) complex as an application example. 

A. Gallegos, D. Robert, X. Girones, R. Carbo-Dorca, Structure-Toxicity Relationships of Polycyclic Aromatic Hydrocarbons Using Molecular Quantum Similarity ,. /. Corn-put.-Aided Mol. Des. 2001, 15, 67 - 80. 

R. Modeling antimalarial activity application of kinetic energy density quantum similarity measures as descriptors in QSAR./. Chem. Inf. Comput. Sci. 2000, 40, 1400—1407. 

Girones, X. and Carbo-Dorca, R. (2002) Molecular quantum similarity-based QSARs for binding affinities of several steroid sets../. Chem. Inf. Comvut. Sci. 42, 1185-1193. 

Foundations and Recent Developments on Molecular Quantum Similarity... 

R. Carbo, Molecular Quantum Similarity in QSAR and Drug Design, Springer-Verlag, New York, 2000. 

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