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

Abstract

Information entropies, such as Rényi, Fisher, and Shannon, are calculated for the 1s2-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.

Coulomb correlation and information entropies in confined helium-like atoms

“…The study of quantum systems in information theory context already has a series of published works, for instance, analyses involving the one-dimensional systems [13][14][15][16][17] and confined hydrogen atom [18][19][20][21][22][23]. Studies on the free or confined helium-like atoms also have received attention in the informational context [24][25][26][27][28][29][30][31][32][33].…”

Shannon, Rényi entropies, and Fisher information calculations of the Li1+ and Be2+ ions screened by the ion-sphere plasma model

“…Fisher-Shannon [54] and Fisher-Rényi [55] information products have been introduced to measure the spatial localization/delocalization of the electron density and are considered as an appropriate measure of electronelectron correlations as well as the information entropies themselves. Using the one-electron density method and independent wave function, see equation (5) with β = γ = 0, the product b P IR is found to be independent of α and R c , and can take the numeric values of 1.237333, 0.669819, and 0.502364 for β = 1, 2, and 3; respectively [42]. It is worth to mention that the Rényi entropy is a generalization of S, with one parameter β, in which…”

“…These quantities indicate the localization or delocalization of the density where the Shannon entropy is a global measurement and the Fisher information detect local changes. Furthermore, the Rényi entropy has been introduced in the last century to study the electron density [37][38][39][40], electron correlation [41,42], and relate it with the complexity of a distribution [43,44]. However, those entropy quantities have not been tested under the effect of the ion-sphere model potential.…”