Generalized Jensen divergence analysis of atomic electron densities in conjugated spaces

Angulo, J. C.; López-Rosa, S.; Antolín, J.; Esquivel, Rodolfo O.

doi:10.1002/qua.22728

Cited by 8 publications

(12 citation statements)

References 50 publications

(56 reference statements)

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“…Let us mention the relevant role played by the exchange and kinetic terms in the expansion of the functional for the total energy, 1 the Shannon entropy 25 (as a limiting case having its roots in information theory, statistical mechanics, and thermodynamics), or the disequilibrium 33 in the study of organization patterns in statistical mechanics, to mention a few of them. Additionally, let us remark that some of the previous functionals are essential ingredients in defining a variety of further information-theoretic concepts (e.g., complexity, 34 similarity, 37 and divergence [38][39][40] ).…”

Section: Upper Bounds On Frequency Moments In Terms Of Radial Exmentioning

confidence: 99%

“…II), they have been considered as essential tools in a wide variety of fields, including the study of threedimensional (d = 3) many-electron systems and physical processes. Many of those studies have emphasized the role played by the order "q" of the aforementioned density functionals, as also done in more recent applications based on the concepts of Jensen-Rényi, 39 Jensen-Tsallis, 38 and Kullback-Leibler 45 divergences, the quantum similarity index 46,47 and its generalization 48 or the generalized shape complexity, 49 among others. In most of these applications, considering low enough values (usually below unity) of the order q appears very important in order to get non-trivial information, based on the features of the one-particle density in position space, regarding a variety of relevant properties of, e.g., atomic systems.…”

Section: Particular Cases Of Physical Interestmentioning

confidence: 99%

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Uncertainty inequalities among frequency moments and radial expectation values: Applications to atomic systems

Angulo

Bouvrie

Antolín

2012

Journal of Mathematical Physics

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Rigorous and universal bounds on frequency moments of one-particle densities in terms of radial expectation values in the conjugate space are obtained. The results, valid for any d-dimensional quantum-mechanical system, are derived by using Rényi-like position-momentum inequalities in an information-theoretical framework. Especially interesting are the upper bounds on the Dirac exchange and Thomas-Fermi kinetic energies, as well as the disequilibrium or self-similarity of both position and momentum distributions. A variety of bounds for these functionals in a given space are known, but most usually in terms of quantities defined within the same space. Very few results including a density functional on one space, and expectation values on the conjugate one, are found in the literature. A pioneering bound on the disequilibrium in terms of the kinetic energy is improved in this work. A numerical study of the aforementioned relationships is carried out for atomic systems in their ground state. Some results are given in terms of relevant physical quantities, including the kinetic and electron-nucleus attraction energies, the diamagnetic susceptibility and the height of the peak of the Compton profile, among others. C 2012 American Institute of Physics. [http://dx.are also known. Although these relationships are usually applied in three-dimensional systems (i.e., with vectors of three components r and p), all of them are valid for arbitrary dimensionality. 16,17 Such uncertainty relations are physically relevant, not only because of their importance in a theoretical quantum-mechanical framework, 18-20 but also in the development of quantum information and computation. 21,22 In this sense, the studies of entropic uncertainty relations 23 and their connection with entanglement 24 are also remarkable.The aim of this study is to present uncertainty inequalities, in the form of bounds on a frequency moment (quantity defined in Sec. II) in a given space (position or momentum) in terms of two radial expectation values in the conjugate space. Such inequalities can handle radial expectation values of positive or negative orders. The bounds provided here are of universal validity (i.e., for any d-dimensional quantum mechanical system). By way of example, we carry out a numerical study for selected inequalities of physical interest in atomic systems, and the results are interpreted taking into account that some radial expectation values for atomic densities, in both position and momentum spaces, are physically relevant and/or experimentally accessible.The paper is structured as follows: Section II is devoted to the definition of the one-particle densities from the wave function, as well as the main quantities we will deal with (radial expectation values, frequency moments, Rényi entropies). In Sec. III, the uncertainty relations associated to those quantities are provided, as universal bounds on frequency moments in terms of radial expectation values in the conjugate space. Particular cases of physical interest are detailed in Se...

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Section: Upper Bounds On Frequency Moments In Terms Of Radial Exmentioning

confidence: 99%

Section: Particular Cases Of Physical Interestmentioning

confidence: 99%

Uncertainty inequalities among frequency moments and radial expectation values: Applications to atomic systems

Angulo

Bouvrie

Antolín

2012

Journal of Mathematical Physics

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“…Generalizations of this divergence have been provided very recently [35] by considering mean values for an arbitrary number of distributions and their associated weights:…”

Section: Geometric Rényi Divergencementioning

confidence: 99%

“…Since then, many other measures of divergence have been proposed and studied, in particular by using both local (Fisher) [32,33] and global (Shannon) [31] theoreticinformation magnitudes. Some generalized entropies such as the Rényi one [34] have also been used to formulate deeper or more precise measures of discrepancy or distinguishability [35]. Applications of similarity and divergence measures to the study of atomic systems have been carried out in recent years, including the use of the quantum similarity index (QSI) [18] and the Jensen-Shannon divergence (JSD) [36].…”

Section: Introductionmentioning

confidence: 99%

Geometric Rényi divergence: A comparative measure with applications to atomic densities

2011

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An alternative one-parameter measure of divergence is proposed, quantifying the discrepancy among general probability densities. Its main mathematical properties include (i) comparison among an arbitrary number of functions, (ii) the possibility of assigning different weights to each function according to its relevance on the comparative procedure, and (iii) ability to modify the relative contribution of different regions within the domain. Applications to the study of atomic density functions, in both conjugated spaces, show the versatility and universality of this divergence.

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“…In general, it quantifies the question of how different or similar one density is to another. Such distances between electronic charge densities have been related to differences in physical properties between different species, among other applications . KL has also been used in molecular dynamics simulations .…”

Section: Introductionmentioning

confidence: 99%

Entropic Kullback‐Leibler type distance measures for quantum distributions

Laguna

Salazar

Sagar

2019

Int J of Quantum Chemistry

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Entropic distance measures for quantum mechanical probability distributions, which are characterized by nodal structure and symmetry holes, are considered. We illustrate how the Kullback-Leibler (KL) distance is not well defined in some instances and propose instead the use of the cumulative residual Kullback-Leibler (CRKL) distance.The KL and CRKL measures are compared and contrasted for some representative quantum mechanical systems: The particle in an infinite well, the harmonic oscillator, and hydrogenic systems. We present cases where CRKL can be used to obtain distances whereas KL cannot be used, and also highlight examples where the KL and CRKL measures yield different behaviors and interpretations. An extension of the CRKL definition is obtained for application to harmonic oscillator systems defined over [−∞, ∞]. Distance measures for two-variable (particle) distributions are also considered to address generalizations of the mutual information correlation measure.The use of CRKL in measuring distances between orbitals is also discussed. K E Y W O R D S cumulative residual Kullback-Leibler entropy, Kullback-Leibler, quantum distributions, relative entropy

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Generalized Jensen divergence analysis of atomic electron densities in conjugated spaces

Cited by 8 publications

References 50 publications

Uncertainty inequalities among frequency moments and radial expectation values: Applications to atomic systems

Uncertainty inequalities among frequency moments and radial expectation values: Applications to atomic systems

Geometric Rényi divergence: A comparative measure with applications to atomic densities

Entropic Kullback‐Leibler type distance measures for quantum distributions

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