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Journal of Physics Conference Series

Comparative study of the scaling behavior of the Rényi entropy for He-like atoms

Mohsen M. Farid, Afaf Abdel‐Hady, Ibraheem M. Nasser

Published Jul 1, 2017• 5 citations
Publisher

Abstract

We solve the Schrödinger equation in the spherical or Hylleraas-coordinate systems, and within the framework of the Ritz's variational principle. The eigenvalues, and the eigenfunctions ψ ( r ) in r - or Hylleraas-space for the 1s 2 -state of the He-like atoms as a function of two variational parameters are calculated. Using a simple scaling procedure, we calculate the scaled wavefunction as a function of the nuclear charge Z . Given the density of states, ρ ( r ), the scaling behavior of the information entropies, e.g., Fisher, Shannon and Rényi's entropies, with their powers and products, as functions of Z are calculated. Scaled wavefunctions for the 1s 2 -state of the He-like atoms, with exchange, have been used to study the scaling behavior. Our results agree with the published results. Furthermore, we present a simple logarithmic equation that shows the dependence of information entropies on Z for He-like atoms. The formulation enhances the computational efficiency of the entropies and other related quantities.

DOI
10.1088/1742-6596/869/1/012011

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We solve the Schrödinger equation in the spherical or Hylleraas-coordinate systems, and within the framework of the Ritz's variational principle. The eigenvalues, and the eigenfunctions ψ ( r ) in r - or Hylleraas-space for the 1s 2 -state of the He-like atoms as a function of two variational parameters are calculated. Using a simple scaling procedure, we calculate the scaled wavefunction as a function of the nuclear charge Z . Given the density of states, ρ ( r ), the scaling behavior of the information entropies, e.g., Fisher, Shannon and Rényi's entropies, with their powers and products, as functions of Z are calculated. Scaled wavefunctions for the 1s 2 -state of the He-like atoms, with exchange, have been used to study the scaling behavior. Our results agree with the published results. Furthermore, we present a simple logarithmic equation that shows the dependence of information entropies on Z for He-like atoms. The formulation enhances the computational efficiency of the entropies and other related quantities.
A comparative study of the atomic Rényi entropies for the exponential-cosine screened Coulomb potential (ECSCP), has been discussed. The study has been done by solving the Schrödinger equation in the Hylleraas - space, and within the framework of Ritz’s variational principle. Different wave functions for the 1s 2 -state of the helium isoelectronic series, in the Hylleraas-space with few variational parameters, have been used to calculate the Shannon and Rényi entropies. They are used to create a comparative study and discuss the scaling behavior of different physical quantities close to the ionization limit. The 3-dimension visualization of the Shannon and Rényi charge densities, as a function of the nuclear charge Z and the screening parameter μ , will be explained and discussed. Comparison with literature results is made, whenever possible.
This work presents analytical and numerical results for the position- and momentum-space information entropies, of the 1s 2 -state of helium-like ions, using different interaction potentials. The potentials that we used are the Yukawa potential (YP), and the exponential-cosine-screened Coulomb potential (ECSCP). The investigated studies allow us to relate the position-space information with the momentum-space information of Shannon and Fisher, as well as Shannon entropy power, and the Fisher-Shannon information product, through different famous relations. The calculation is done using the one-electron charge density of entangled two-parameter wave function. On one hand, the results that are presented for ten members in the helium isoelectronic sequence demonstrate with precision the effect of correlation on bare charge distributions. On the other hand, it leads to some very important results for both the correlated and uncorrelated values of the informatic entropies. Analytical formula for the momentum-space information entropies are given. The effect of the nuclear charge and the screening parameter on the information expressions has been studied for both potentials. Detailed computational and numerical values and characteristics of these information quantities, as a function of the screening parameter, are reported here for the first time. New inequality has been proposed with Fisher’s total value to measure the correlation of two electrons.
This work present analytical results for the position- and momentum-space information entropies, of the 1s2-state of helium-like ions, using different interaction potentials. The potentials that we used are the Yukawa potential (YP), and the exponential-cosine-screened Coulomb potential (ECSCP). The investigated studies allow us to relate the position-space information with the momentum-space information of Shannon and Fisher, as well as Shannon entropy power, and the Fisher-Shannon information product, through different famous relations. The calculation is done using one-electron charge density of entangled two-parameter wave function. On one hand, the results that are presented for fifteen members in the helium iso-electronic sequence demonstrate with precision the effect of correlation on bare charge distributions. On the other hand, it leads to some very important results for both the correlated and uncorrelated values of the informatic entropies. Analytical formula for the momentum-space information entropies are given. The effect of the nuclear charge and the screening parameter on the informatic expressions has been studied for both potentials. Detailed computational and numerical values and characteristics of these quantities are reported here for the first time. New inequality has been proposed with Fisher’s total value to measure the correlation of two electrons.
Abstract Information entropies, such as Rényi, Fisher, and Shannon, are calculated for the 1s 2 -state of the two-electron series, in position space. It is considered that the exponential cosine screened Coulomb potential (ECSCP) describes the medium. Explicit correlated Hylleraas functions with 3-parameter are used to focus on getting the most distinctive physical features of the inter-electronic repulsion results. In one- and two-electron charge densities, the numerical values of the information entropies and their ratios are calculated. Fisher-Shannon, and Fisher-Rényi information products that measure the electron-electron correlation strength, are calculated and discussed. Also, we used the contour maps and 3D-graphs to visualize how the interplay between inter-electronic repulsion and electrons-nuclear charge interaction determines the variation of information entropies as a function of the nuclear charge and the screening parameter. In this work, an expression for the divergence in Hylleraas coordinates has been derived and the results of the information quantities for two-electron systems with ECSCP have been calculated, in one-and two-electron densities, for the first time.
The approximate solutions of the Schrödinger equation with the central generalized Hulthén and Yukawa potential are investigated within the framework of the functional method. The obtained wave function and the energy levels are used to study the Shannon entropy, Renyi entropy, Fisher information, Shannon Fisher complexity, Shannon power and Fisher-Shannon product in both position and momentum spaces for the ground and first excited states. The theoretic information theories such as Shannon entropies Sr,Sp, Fisher information Ip,Ip and Renyi entropies Rβ as well as the information entropy probability densities ρ(r),ϕ(p) are illustrated graphically. The Bialynicki-Birula – Mycielski (BBM) inequality for the Shannon entropies and the Cramer-Rao inequality for the Fisher information are satisfied for the potential model.
The Rényi and Shannon entropies are information‐theoretic measures which have enabled to formulate the position‐momentum uncertainty principle in a much more adequate and stringent way than the (variance‐based) Heisenberg‐like relation. Moreover, they are closely related to various energetic density‐functionals of quantum systems. Here we find sharp upper bounds to these quantities in terms of the second order moment 〈r2〉 for general spherically symmetric potentials, which substantially improve previous results of this type, by means of the Rényi maximization procedure with a covariance constraint due to Costa, Hero and Vignat [1]. The contributions to these bounds coming from the radial and angular parts of the physical wavefunctions are explicitly given.