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Physica Scripta

Rényi, Fisher, Shannon, and their electron correlation tools for two-electron series

Ibraheem M. Nasser, Mostafa Zeama, Afaf Abdel‐Hady

Published Jul 28, 2020• 13 citations
Publisher

Abstract

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.

DOI
10.1088/1402-4896/abaa09

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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.
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.
Fisher information is calculated for the ground state of He-isoelectronic series, in position ([Formula: see text]) space. The results are given and discussed as a function of the nuclear charge (Z) and the screening parameter (λ) in the case study of Yukawa potential. Simple and explicit one-, two-, and three-correlated terms of Hylleraas wave function are used to focus on extracting the most characteristic physical features of the results. The numerical values of Fisher information are given in 1- and 2-electron charge densities, and their ratio of 2- to 1-electron densities results are defined and analyzed. To enable a comparison with others, the Fisher–Shannon information products, which measure the electron–electron correlation strength, are calculated in 1-electron density. The calculations of Fisher information, the ratio, and the Shannon-information products for two-electron systems in the presence of Yukawa potential are carried out for the first time in this study.
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 This work presents analytical and numerical 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 and the exponential‐cosine‐screened Coulomb potential. 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 performed using the one‐electron charge density of entangled two‐parameter wavefunction. On one hand, the results that are presented for 10 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 formulas 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.
In this paper, we have presented an analytical model of two electron systems consisting of a many particle correlated wave function with some variational parameters α, λ and µ and used it to quantify the electron-electron correlation described by the wave function containing explicitly r12 (inter atomic distance between two electrons) dependent term. The single particle wave functions and the charge densities have been extracted from the said correlated wave function both for the uncorrelated and correlated systems in coordinate space and its momentum analogs have been obtained by taking the Fourier transform of the coordinate analogs. We have computed and presented the results of the numerical values of the theoretic information entropies of the Shannon entropy, Fisher information entropy, Shannon power and the FisherShannon product. The numerical values are consistently found to satisfy the Beckner, Bialynicki-Birula and Mycielski (BBM ) inequality relation; Stam-Cramer-Rao inequalities or Fisher based uncertainty relation and Fisher-Shannon product relation for the uncorrelated and correlated systems in both the coordinate and momentum spaces.