Quantum similarity measures computation

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. 

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

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Robert, D. and Carb6-Dorca, R. (1998a). A Formal Comparison Between Molecular Quantum Similarity Measures and Indices. J.Chem.Inf.Comput.Sci., 38, 469 75. 

Robert, D. and Carbo-Dorca, R. (1998b) Analyzing the triple density molecular quantum similarity measures with the INDSCAL model. J. Chem. Inf. Comput. Sci., 38, 620-623. 

Amat L, Robert D, Besalu E, Carbo-Dorca R. Molecular quantum similarity measures tuned 3D QSAR an antitumoral family validation study. J Chem Inf Comput Sci 1998 38 624-631. 

Robert D, Amat L, Carbo-Dorca R. Three-dimensional quantitative structure-activity relationships from tuned molecular quantum similarity measures prediction of the corticosteroid-binding globulin binding affinity for a steroid family. J Chem Inf Comput Sci 1999 39 333-344. 

Amat L, Carbo-Dorca R, Ponec R. Molecular quantum similarity measures as an alternative to log P values in QSAR studies. J Comput Chem 1998 19 1575-1583. 

Amat L, Carbo-Dorca R. Quantum similarity measures under atomic shell approximation first-order density fitting using elementary Jacobi rotations. J Comput Chem 1997 18 2023-2039. 

Amat L, Carbo-Dorca R. Fitted electronic density functions from H to Rn for use in quantum similarity measures Cis-diamminedichloroplatinum(II) complex as an application example. J Comput Chem 1999 20 911-920. 

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. 

A firm theoretical basis has been estabHshed for molecular quantum similarity, and many computational tools have been developed that allow for the evaluation and quantification of molecular quantum similarity measures among sets of molecules or atoms. Molecular quantum similarity is also the basis of quantum QSAR, another active field of research. 

Applied Sciences and Engineering (ECCOMAS 2000), CDROM edited by Facultat d ln-formatica de Barcelona (FIB)— Universitat Politecnica de Catalunya (UPC)—International Centre for Numerical Methods in Engineering (CIMNE) Barcelona, 2000, Computational Chemistry Section, Chapter 12. Quantum Quantitative Structure-Activity Relationships (QQSAR) A Comprehensive Discussion Based on Inward Matrix Products, Employed as a Tool to Find Approximate Solutions of Strictly Positive Linear Systems and Providing QSAR-Quantum Similarity Measures Connections. 

Field-based similarities are usually evaluated by the cosine or correlation function similarity measure employed initially by Carbo and co-workers (67) to compute molecular similarities based upon quantum mechanical wavefunctions. Such a measure, which is usually called a Carbo similarity index, is given by... 

In the next two subsections, we describe collections of calculations that have been used to probe the physical accuracy of plane-wave DFT calculations. An important feature of plane-wave calculations is that they can be applied to bulk materials and other situations where the localized basis set approaches of molecular quantum chemistry are computationally impractical. To develop benchmarks for the performance of plane-wave methods for these properties, they must be compared with accurate experimental data. One of the reasons that benchmarking efforts for molecular quantum chemistry have been so successful is that very large collections of high-precision experimental data are available for small molecules. Data sets of similar size are not always available for the properties of interest in plane-wave DFT calculations, and this has limited the number of studies that have been performed with the aim of comparing predictions from plane-wave DFT with quantitative experimental information from a large number of materials. There are, of course, many hundreds of comparisons that have been made with individual experimental measurements. If you follow our advice and become familiar with the state-of-the-art literature in your particular area of interest, you will find examples of this kind. Below, we collect a number of examples where efforts have been made to compare the accuracy of plane-wave DFT calculations against systematic collections of experimental data. 

The algebraic and differential topological similarity measures required much simpler mathematical and computational apparatus than the direct comparisons of the original, complex quantum mechanical objects. 

Given a set of N quantum objects, there is always the possibility of computing the whole array of QSM between quantum object pairs, producing a symmetric (N x N) matrix Z = Zrj, the so-called similarity matrix (SM) of the quantum object set. Such a matrix is illustrated below. The self-similarity measures are the diagonal elements of... 

Although QS has started within such similarity-dissimilarity index premises, essentially the fact is that the elementary QS computational element building block reduces to the well-known scalar product of two DFs, a so-called similarity measure. Indeed, given two quantum systans, say [A,B), the familiar quantum mechanical theoretical basis permits to obtain their attached wavefunctions via solving the respective Schrbdinger equations. From the system wavefunctions, a pair of associated DF p,4(r),pg(r) can be simply set up, with the vector r representing some number of particle coordinates. In molecular QS studies, the usual DF chosen is the first-order one thus, vector r = (x,y,z) corresponds to one-electron position coordinate only. Then, the similarity measure between the system pair of DF is simply defined as the overlap similarity integral ... 

Combined Quantum Mechanical and Molecular Mechanical Potentials Hyperconjugation M0ller-Plesset Perturbation Theory Ifatural Bond Orbital Methods Rotational Barriers and Molecular Mechanics Corrections Rotational Barriers Ab Initio Computations Spectroscopy Computational Methods Structural Similarity Measures for Database Searching. 

Finally, in the last chapter (Chapter 12) of Part II of this book, Ramon has studied the molecular quantum similarity (QS) measures involving three density functions. The necessary algorithms have been described here. General theory and computational feasibility of a h3q)ermatricial or tensorial representation of molecular structures associated to any molecular quantum object set (MQOS) have been nicely explained in this chapter. Secondly, generalized Carbo similarity indices (CSI) have also been studied. The theoretical and computational approaches have been supported by various suitable applicative examples. 

A term that is nearly synonymous with complex numbers or functions is their phase. The rising preoccupation with the wave function phase in the last few decades is beyond doubt, to the extent that the importance of phases has of late become comparable to that of the moduli. (We use Dirac s terminology [7], which writes a wave function by a set of coefficients, the amplitudes, each expressible in terms of its absolute value, its modulus, and its phase. ) There is a related growth of literatm e on interference effects, associated with Aharonov-Bohm and Berry phases [8-14], In parallel, one has witnessed in recent years a trend to construct selectively and to manipulate wave functions. The necessary techifiques to achieve these are also anchored in the phases of the wave function components. This bend is manifest in such diverse areas as coherent or squeezed states [15,16], elecbon bansport in mesoscopic systems [17], sculpting of Rydberg-atom wavepackets [18,19], repeated and nondemolition quantum measurements [20], wavepacket collapse [21], and quantum computations [22,23], Experimentally, the determination of phases frequently utilizes measurement of Ramsey fringes [24] or similar" methods [25]. 

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