Shannon entropies and Fisher information of K-shell electrons of neutral atoms

Abstract

We represent the two K-shell electrons of neutral atoms by Hylleraas-type wave function which fulfils the exact behavior at the electron-electron and electron-nucleus coalescence points and, derive a simple method to construct expressions for single-particle position-and momentum-space charge densities, ρ(r) and γ(p) respectively. We make use of the results for ρ(r) and γ(p) to critically examine the effect of correlation on bare (uncorrelated) values of Shannon information entropies (S) and of Fisher information (F ) for the K-shell electrons of atoms from helium to neon. Due to inter-electronic repulsion the values of the uncorrelated Shannon position-space entropies are augmented while those of the momentum-space entropies are reduced. The corresponding Fisher information are found to exhibit opposite behavior in respect of this. Attempts are made to provide some plausible explanation for the observed response of S and F to electronic correlation.

Calculation of information entropies for the 1s2 state of helium‐like ions

“…[ 48 ] For a wavefunction that includes several terms of r 12 , it is appropriate to employ the Hylleraas coordinates. [ 22,29,32–35,39–42,44–46,49,50 ]…”

“…[48] For a wavefunction that includes several terms of r 12 , it is appropriate to employ the Hylleraas coordinates. [22,29,[32][33][34][35][39][40][41][42][44][45][46]49,50] In the present study, we used the two-parameter wavefunction as suggested by the entangled, or non-separable, wavefunction:…”

“…[ 28–43 ] For two‐electron systems, the literature has used the density functional formalism [ 28 ] for Shannon entropy; many terms of the Hylleraas wavefunction [ 29–33 ] for Shannon, Rényi, and Tsallis entropies and Onicescu information energy; and configuration interaction [ 29,34 ] for linear and von Neumann entanglement entropies. Recently, two and three parameters have been used to calculate the information quantities in two‐electron systems, such as Fisher and Shannon, [ 44–46 ] Rényi, [ 39 ] and Tsallis. [ 40,41 ] Using more than three‐parameter wavefunctions will make the entropy calculation difficult, and the execution can only be done using highly sophisticated computers.…”

“…Shannon, [44][45][46] Rényi, [39] and Tsallis. [40,41] Using more than three-parameter wavefunctions will make the entropy calculation difficult, and the execution can only be done using highly sophisticated computers.…”

Shannon‐information entropy sum in the confined hydrogenic atom

“…As such, it is one measure of the position‐momentum correlation . It has been studied in a variety of models, fermionic and bosonic systems including Bose‐Einstein condensates . Its N‐dependence has also been used to support the theory of gravity as emergent phenomena …”