Self-similarity measures

Based on the molecular quantum similarity measmes. Molecular Quantum Self-Similarity Measures (MQS-SM) were proposed as molecular descriptors where each molecule is compared with itself and all the others, and appropriate Hermitian operators 2 are associated to each molecular property [Ponec et al, 1999]. 

Quantum Self-Similarity Measures quantum similarity quantum similarity = quantum similarity Quantum Similarity Indices quantum similarity Quantum Similarity Measures quantum similarity representation molecular descriptors rigidity —> flexibility indices... 

Amat, L., Besalu, E. and Carbo-Dorca, R. (2001) Identification of active molecular sites using quantum-self-similarity measures. J. Chem. Inf. Comput. Sci., 41, 978-991. 

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... 

It is called a molecular quantum similarity measure (MQSM), and it corresponds to the similarity integral between molecules A and B. In the first integral in Eq. [13], a so-called molecular quantum self-similarity measure (MQSSM) is obtained ... 

As we will describe, these self-similarity measures can have several relationships with different physicochemical properties. 

When A equals B, the similarity measure of Eq. [14] is called the molecular quantum self-similarity measure, which corresponds to the square of the Euclidean norm of the density function. In other words. 

Figure 6 Correlation between observed Log P values and self-similarity measures for a series of acetic acid esters, carboxylic acids, and amides. Reprinted by permission of John Wiley and Sons, Inc. from Amat et...

It was found by Sola et al. that a good linear relationship exists between the overlap self-similarity measure and the molar volume for isoelec-tronic molecules when the atoms of the systems being compared belong to the same row of the periodic table. In the same context, it was shown that the... 

Figure 7 Evolution of the ab initio (HF/3-21G) energy (in Hartree) and self-similarity measure versus dihedral angle in ethane in the adiabatic (a) and the non-adiabatic (b) approach.

Molecular Basis of LEER. Modeling of the Electronic Substituent Effect Using Fragment Quantum Self-Similarity Measures. 

The Use of Ab Initio Quantum Molecular Self-Similarity Measures to Analyze Electron Charge Density Distributions. 

Such QS measures permit to construct tensor (or hypermatrix) discrete descriptions of quantum systems, representing them in terms of other known quantum systems. Obviously enough, a triple-density self-similarity measure corresponds to an integral of type Equation 17.3, where the three DEs are the same, for example. 

A quantum-chemical study of the mechanism of the condensation reaction between propanoic acid and aniline has been made. The mechanism of proton-transfer reactions of 3,5-dinitrosalicylic acid (41) has been the subject of a review (50 references). A successful correlation has been obtained between dissociation of a variety of acids in different solvents and quantum self-similarity measures of the CO2H fragment. The substituent effects of the isopropyl group in 2- (42), 3- (43), and 4-isopropylbenzoic acid (44) have been evaluated from their enthalpies of formation, gas-phase acidities, acidities in MeOH and in DMSO, and their IR spectra in tetrachloromethane. Particular attention was given to the influence of variable conformation on the observed steric effect. In contrast to 2-r-butylbenzoic acid and similarly to 2-methylbenzoic acid, 2-isopropylbenzoic acid exists in two planar conformations (42a,b) in equilibrium. Owing to this conformational freedom, the... 

Such transformed elements of any TDQSH will yield a new TDCSI hyper vector (TDCSIH), which will possess unit elements in the corresponding self similarity measure hyper vector sites and values less than the unit elsewhere. 

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