Quantum similarity matrix

Bultinck, P. and Carbo-Dorca, R. (2003) Molecular quantum similarity matrix based clustering of molecules using dendrograms. J. Chem. Inf. Comput. Sci., 43, 170-177. 

We have already used similarity matrices to cluster molecules. As such, they provide the necessary data to investigate the construction of a molecular set taxonomy. The most common techniques to do so include the molecular point clouds previously described. There, the columns of the molecular quantum similarity matrix yielded coordinates of the molecules in the N-dimensional space. Often, the N dimensionality cannot yet be used for a graphical representation. However, several techniques exist to reduce the dimensionality of the data, which allow it to be represented graphically on common devices like a computer screen or a plotter. In addition to these plots, several instances have involved Kruskal trees and other algorithms. °... 

Another matter is the consistency of the molecular quantum similarity matrix Z. The MQSM produced by a specific alignment technique for a given molecular pair of the set of molecules that construct Z should not be contradictory with the computed MQSM for the other pairs of molecules. To illustrate this point, consider the Euclidean distance, as defined by the square root of Eq. [13] ... 

Collecting all MQSMs computed between the element pairs of a given QOS, a so-called quantum similarity matrix is obtained, having been constructed according to the definition [141] by means of Z = / . Because of the structure of the quantum similarity matrix elements, the matrix can be considered as an element of a VSS of some appropriate dimension. The similarity matrix Z is a symmetric matrix with positive definite elements, whose columns zj (or rows) are also elements of some N—dimensional VSS. As such, a real symmetric matrix X exists that, in general, generates Z as... 

However, the column stochastic similarity matrix [145] appears to be no longer symmetric as is its originating quantum similarity matrix Z. The columns of the stochastic matrix [145], as defined in Eq. [146], can substitute the QOS density function tag elements, which have previously generated them. A new kind of discrete QOS can be constructed in this way. 

Quantum Similarity Matrix Based Clustering of Molecules Using Dendrograms. 

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... 

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... 

Even if other methodologies may provide better results, it must be stated that the methodology presented in this work, and that includes descriptor generation, similarity matrix transformation, and statistical procedure, has not been altered in any way to take into account the nature of the studied system. In this way, the exposed QSAR protocol is potentially capable of handling and characterizing different molecular biological activities from diverse molecular sets without introducing further information than those provided by quantum similarity, which is based on electronic... 

Once an operator has been chosen for the calculation of the MQSM for a set of N molecules, one can calculate all MQSMs between every two molecules, which gives rise to the whole N x N array of MQSM. This symmetrical matrix is called the molecular quantum similarity measure matrix (MQSMM), denoted Z. 

Another important concept in molecular quantum similarity is associated with convex conditions. The idea underlying convex conditions, associated with a numerical set, a vector, a matrix, or a function, has been described previously in the initial work on VSS and related issues.Convex conditions correspond to several properties of some mathematical objects. The symbol X(x) means that the conditions (x) = 1 A x V(R ) hold simultaneously for a given mathematical object x, which is present as an argument in the convex conditions symbol. Convex conditions become the same as considering the object as a vector belonging to the unit shell of some VSS. For such kind of elements,... 

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. 

As with the uncoupled case, one solution involves diagonalizing the Liouville matrix, iL+R+K. If U is the matrix with the eigenvectors as cohmms, and A is the diagonal matrix with the eigenvalues down the diagonal, then (B2.4.32) can be written as (B2.4.33). This is similar to other eigenvalue problems in quantum mechanics, such as the transfonnation to nonnal co-ordinates in vibrational spectroscopy. 

Our theorem permits the following inference. The statistical matrix of every pure case in quantum mechanics is equivalent to an elementary matrix and can be transformed into it by a similarity transformation. Because p is hermitian, the transforming matrix is unitary. A mixture can, therefore, always be written in the diagonal form Eq. (7-92). 

Sodium Acetate-Sodium Chloride Mixtures. Ramasamy and Hurtubise (12) obtained RTF and RTF quantum yields, triplet formation efficiency, and phosphorescence lifetime values for the anion of p-aminobenzoic acid adsorbed on sodium acetate and on several sodium acetate-sodium chloride mixtures. Rate constants were calculated for phosphorescence and for radiationless transition from the triplet state. The results showed that several factors were important for maximum RTF from the anion of p-aminobenzoic acid. One of the most important of these was how efficiently the matrix was packed with sodium acetate molecules. A similar conclusion was found for RTF however, the RTF quantum yield increased more dramatically than the RTF quantum yield. 

Section II discusses the real wave packet propagation method we have found useful for the description of several three- and four-atom problems. As with many other wave packet or time-dependent quantum mechanical methods, as well as iterative diagonalization procedures for time-independent problems, repeated actions of a Hamiltonian matrix on a vector represent the major computational bottleneck of the method. Section III discusses relevant issues concerning the efficient numerical representation of the wave packet and the action of the Hamiltonian matrix on a vector in four-atom dynamics problems. Similar considerations apply to problems with fewer or more atoms. Problems involving four or more atoms can be computationally very taxing. Modern (parallel) computer architectures can be exploited to reduce the physical time to solution and Section IV discusses some parallel algorithms we have developed. Section V presents our concluding remarks. 

In equilibrium statistical mechanics involving quantum effects, we need to know the density matrix in order to calculate averages of the quantities of interest. This density matrix is the quantum analog of the classical Boltzmann factor. It can be obtained by solving a differential equation very similar to the time-dependent Schrodinger equation... 

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