Quantum self similarity

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. 

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 ... 

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. 

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

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... 

In the previous subsection, we considered the held emitted by one atom and then absorbed by another atom as a superposition of outgoing and incoming spherical waves of multipole photons. This wave picture completely eliminates an inquiry concerning the trajectory of photons between the atoms. In fact, the path of a particle in quantum mechanics is not a well-defined notion. The most that we can state about the path of a quantum particle in many cases is that it is represented by a nondifferentiable, statistically self-similar curve [93]. For example, the path of a tunneling electron and time spending in the barrier are not still defined unambiguously [94]. Moreover, some experiments on photonic tunneling and transmission of information show the possibility of superluminal motion of photons inside an opaque barrier [95]. 

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... 

Quantum similarities and self-similarities D for neutral atoms were computed for nuclear charges Z = 1-54 only in the position space [47,48], but afterwards a more complete analysis including Z = 1-103 neutral systems and singly charged ions has been done in position and momentum spaces [49]. 

Even without knowing the cause of galactic redshifts their quantum nature rules out the Doppler interpretation and conhrms that galaxies have structures self-similar to that of the solar system. The major source of galactic light is stars on characteristically quantized orbits around the galactic... 

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. 

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