Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2102.13571 · Quantum Reports

Shannon Entropy in Confined He-Like Ions within a Density Functional Formalism

Sangita Majumdar, Amlan K. Roy

Published Feb 10, 2020• 32 citations
arXivHTMLPDF

Abstract

Shannon entropy in position ( S r ) and momentum ( S p ) spaces, along with their sum ( S t ) are presented for unit-normalized densities of He, Li + and Be 2 + ions, spatially confined at the center of an impenetrable spherical enclosure defined by a radius r c . Both ground, as well as some selected low-lying singly excited states, viz., 1sns (n = 2–4) 3S, 1snp (n = 2–3) 3P, 1s3d 3D, are considered within a density functional methodology that makes use of a work function-based exchange potential along with two correlation potentials (local Wigner-type parametrized functional, as well as the more involved non-linear gradient- and Laplacian-dependent Lee-Yang-Parr functional). The radial Kohn-Sham (KS) equation is solved using an optimal spatial discretization scheme via the generalized pseudospectral (GPS) method. A detailed systematic analysis of the confined system (relative to the corresponding free system) is performed for these quantities with respect to r c in tabular and graphical forms, with and without electron correlation. Due to compression, the pattern of entropy in the aforementioned states becomes characterized by various crossovers at intermediate and lower r c regions. The impact of electron correlation is more pronounced in the weaker confinement limit and appears to decay with the rise in confinement strength. The exchange-only results are quite good to provide a decent qualitative discussion. The lower bounds provided by the entropic uncertainty relation hold well in all cases. Several other new interesting features are observed.

DOI
10.3390/quantum2010012

Related papers

Structure and information measures of few-electron systems under a spherically symmetric Gaussian potential within a density functional approach

Raveena Arya, Santanu Mondal, Amlan K. Roy

Energies of H, He-like ($Z=2-18$) ions, Li, and Be are investigated under a spherically symmetric Gaussian potential through a density functional formalism. The radial Kohn-Sham equation has been solved by invoking a work function-based exchange potential. The effect of electron correlation is analyzed by incorporating two functionals: a local parameterized Wigner functional and a non-linear gradient- and Laplacian-dependent Lee-Yang-Parr (LYP) functional. The generalized pseudospectral method is employed to provide accurate numerical eigenfunctions and eigenvalues. This allows nonuniform, optimal spatial discretization fulfilling the Dirichlet boundary conditions. This work demonstrates a possible manipulation of energy by controlling dot parameters. Apart from ground states, exploratory results are also reported for low-lying excited state $1s2s$ ($^{1,3}S$) of He atom. Companion calculations are also performed for various information-theoretic measures, such as Shannon entropy in position ($S_{r}$), momentum ($S_{p}$) spaces, and Fisher information in position space ($I_{r}$). The behavior of correlation functionals in presence of Gaussian potential is examined critically. We find that energy increases, $S_{r}$ exhibits minima, while $S_{p}$, $I_{r}$ attain maxima for a decrease in the width of potential, whereas an increase in potential depth further amplifies these effects across all properties. The Fisher-Shannon plane reveals a progressive localization as well as the compression of electronic density, and thereby indicates a weakening of relative electron-correlation effects. In the Collin's conjecture, it gives rise to a non-linear loop-like feature. Much of the results are presented here for the first time.

quant-pharXiv:2606.23538•Jun 23, 2026

Characteristic features of Shannon information entropy of confined atoms

K. D. Sen

The Shannon information entropy of 1-normalized electron density in position and momentum space Sr and Sp, and the sum ST, respectively, are reported for the ground-state H, He+, Li2+, H-, He, Li+, Li, and B atoms confined inside an impenetrable spherical boundary defined by radius R. We find new characteristic features in ST denoted by well-defined minimum and maximum as a function of confinement. The results are analyzed in the background of the irreducible lower bound stipulated by the entropy uncertainty principle [I. Bialynicki-Birula and J. Mycielski, Commun. Math. Phys. 44, 129 (1975)]. The spherical confinement model leads to the ST values which satisfy the lower bound up to the limits of extreme confinements with the interesting new result displaying regions over which a set of upper and lower bounds to the information entropy sum can be locally prescribed. Similar calculations on the H atom in 2s excited states are presented and their novel characteristics are discussed.

The Journal of Chemical Physics•Aug 15, 2005•193 citations

Benchmark values of S hannon entropy for spherically confined hydrogen atom

Li Guang Jiao, Li Rong Zan, Yong Zhi Zhang +1 more

Abstract The information‐theoretic measure of confined hydrogen atom has been investigated extensively in the literature. However, most of them were focused on the ground state and accurate values of information entropies, such as Shannon entropy, for confined hydrogen are still not determined. In this work, we establish the benchmark results of the Shannon entropy for confined hydrogen atom in a spherical impenetrable sphere, in both position and momentum spaces. This is done by examining the bound state energies, the normalization of wave functions, and the scaling property with respect to isoelectronic hydrogenic ions. The angular and radial parts of Shannon entropy in two conjugate spaces are provided in detail for both free and confined hydrogen atom in ground and several excited states. The entropies in position space decrease logarithmically with decreasing the size of confinement, while those in momentum space increase logarithmically. The Shannon entropy sum, however, approaches to finite values when the confinement radius closes to zero. It is also found that the Shannon entropy sum shares same trend for states with similar density distributions. Variations of entropy for nodeless bound states are significantly distinct form those owning nodes when changing the confinement radius.

International Journal of Quantum Chemistry•Mar 21, 2017•81 citations

On the high-momentum oscillatory distribution and related quantum information theoretical measures of confined alkali systems (H, Li)

Koustav D. Chakladar, Santanu Mondal, Sukhamoy Bhattacharyya +2 more

Abstract The effect of impenetrable spherical confinement on alkali systems (H, Li) has been thoroughly examined, emphasizing the distinctive high-momentum oscillatory behavior of the momentum space radial density. We have utilized the Ritz variational framework with a Slater-type basis set to derive the position-space wavefunction and Fourier–Dirac transformation of the former to find the momentum-space wavefunction, analytically. The derived momentum-space density has been analyzed in four asymptotic limits ( R → ∞ , R → 0 , p → ∞ , and p → 0 ; R being the confinement radius and p being the radial component of linear momentum) and its oscillatory behavior in strong spatial confinement region is critically investigated. Oscillatory behavior is also noticed in the Compton profile of the compressed atomic system. Furthermore, the effects of confinement on Shannon information entropy in both position and momentum spaces are investigated, offering insights into the Bialynicki–Birula–Mycielski inequality and the local variations of information measures in terms of Shannon entropy density.

Journal of Physics B Atomic Molecular and Optical Physics•Mar 25, 2025•4 citations

Shannon and Fisher Entropies for the 2P and 3D States of the Hydrogen Atom Confined in a Penetrable Spherical Box

Andrei Solorzano Perez, José Manuel Nieto-Jalil

A straightforward way to study the electronic structure and certain physical properties of atoms and molecules under high pressures is through the confinement (caging) model, in which the atom is placed at the center of a spherical box with impenetrable walls. The hydrogen atom confined inside penetrable and impenetrable spherical cavities is perhaps the most studied compressed system, and exact solutions exist for both types of confinement. The physical properties of the hydrogen atom confined within impenetrable-wall cavities (AHCI) have been widely analyzed; however, the same system embedded in penetrable spherical cavities (AHCP) has received less attention. A more physical model suggests using a spherical box with penetrable walls. Therefore, in this work, we compute via the variational method, the Shannon and Fisher information entropy in position space,$S_{r}$and$I_{r}$, for the$2 \mathbf{p}$and 3d states of a hydrogen atom subjected to spatial confinement by penetrable and impenetrable spherical cavities. Additionally, we calculate the tunneling probability for the penetrable confinement and seek to relate it to the overlaps of the radial probability densities as a function of the cavity radius and the height of the confining barrier. To model the penetrable confinement, we employ a logistic potential which, unlike earlier reports that used a discontinuous step potential, is a continuous function. The wavefunction is expanded in terms of Slater functions that satisfy appropriate boundary conditions; in some cases, we find notable differences with respect to the impenetrable potential.

Mexican International Conference on Computer Science•Nov 10, 2025•0 citations

Analysis of Shannon Entropy and Quantum States of a Confined Hydrogen Atom Screened by the Hellmann Potential

Kirtee Kumar, Vinod Prasad

Abstract This study investigates the impact of spatial confinement and the Hellmann potential on the Shannon entropy of a hydrogenic atom. A hydrogenic atom screened by the Hellmann potential and confined within an impenetrable spherical potential is analyzed. The Schrödinger equation is solved numerically using the finite difference method to determine energy eigenvalues and wavefunctions. These wavefunctions are examined in both position and momentum spaces to calculate the Shannon entropy in position space , the Shannon entropy in momentum space , and the total Shannon entropy . How the confinement radius and the screening parameter affect these entropies is investigated, and the results are compared to those of unconstrained systems. The findings are contrasted with previously reported results and discussed in relation to the Beckner–Bialynicki–Birula–Mycielski (BBM) inequality, offering insights into the behavior of quantum states under confinement. Significant variations in Shannon entropy and energy levels are observed with changes in confinement and screening parameters, providing a deeper understanding of quantum mechanics in confined systems. The study reveals a critical screening parameter at which the behavior of entropy transitions, shedding light on the interplay between spatial confinement, screening effects, and quantum uncertainties.

Advanced Theory and Simulations•Dec 20, 2024•4 citations
Shannon entropy in position ( S r ) and momentum ( S p ) spaces, along with their sum ( S t ) are presented for unit-normalized densities of He, Li + and Be 2 + ions, spatially confined at the center of an impenetrable spherical enclosure defined by a radius r c . Both ground, as well as some selected low-lying singly excited states, viz., 1sns (n = 2–4) 3S, 1snp (n = 2–3) 3P, 1s3d 3D, are considered within a density functional methodology that makes use of a work function-based exchange potential along with two correlation potentials (local Wigner-type parametrized functional, as well as the more involved non-linear gradient- and Laplacian-dependent Lee-Yang-Parr functional). The radial Kohn-Sham (KS) equation is solved using an optimal spatial discretization scheme via the generalized pseudospectral (GPS) method. A detailed systematic analysis of the confined system (relative to the corresponding free system) is performed for these quantities with respect to r c in tabular and graphical forms, with and without electron correlation. Due to compression, the pattern of entropy in the aforementioned states becomes characterized by various crossovers at intermediate and lower r c regions. The impact of electron correlation is more pronounced in the weaker confinement limit and appears to decay with the rise in confinement strength. The exchange-only results are quite good to provide a decent qualitative discussion. The lower bounds provided by the entropic uncertainty relation hold well in all cases. Several other new interesting features are observed.
Energies of H, He-like (Z=2−18Z=2-18Z=2−18) ions, Li, and Be are investigated under a spherically symmetric Gaussian potential through a density functional formalism. The radial Kohn-Sham equation has been solved by invoking a work function-based exchange potential. The effect of electron correlation is analyzed by incorporating two functionals: a local parameterized Wigner functional and a non-linear gradient- and Laplacian-dependent Lee-Yang-Parr (LYP) functional. The generalized pseudospectral method is employed to provide accurate numerical eigenfunctions and eigenvalues. This allows nonuniform, optimal spatial discretization fulfilling the Dirichlet boundary conditions. This work demonstrates a possible manipulation of energy by controlling dot parameters. Apart from ground states, exploratory results are also reported for low-lying excited state 1s2s1s2s1s2s (1,3S^{1,3}S1,3S) of He atom. Companion calculations are also performed for various information-theoretic measures, such as Shannon entropy in position (SrS_{r}Sr​), momentum (SpS_{p}Sp​) spaces, and Fisher information in position space (IrI_{r}Ir​). The behavior of correlation functionals in presence of Gaussian potential is examined critically. We find that energy increases, SrS_{r}Sr​ exhibits minima, while SpS_{p}Sp​, IrI_{r}Ir​ attain maxima for a decrease in the width of potential, whereas an increase in potential depth further amplifies these effects across all properties. The Fisher-Shannon plane reveals a progressive localization as well as the compression of electronic density, and thereby indicates a weakening of relative electron-correlation effects. In the Collin's conjecture, it gives rise to a non-linear loop-like feature. Much of the results are presented here for the first time.
The Shannon information entropy of 1-normalized electron density in position and momentum space Sr and Sp, and the sum ST, respectively, are reported for the ground-state H, He+, Li2+, H-, He, Li+, Li, and B atoms confined inside an impenetrable spherical boundary defined by radius R. We find new characteristic features in ST denoted by well-defined minimum and maximum as a function of confinement. The results are analyzed in the background of the irreducible lower bound stipulated by the entropy uncertainty principle [I. Bialynicki-Birula and J. Mycielski, Commun. Math. Phys. 44, 129 (1975)]. The spherical confinement model leads to the ST values which satisfy the lower bound up to the limits of extreme confinements with the interesting new result displaying regions over which a set of upper and lower bounds to the information entropy sum can be locally prescribed. Similar calculations on the H atom in 2s excited states are presented and their novel characteristics are discussed.
Abstract The information‐theoretic measure of confined hydrogen atom has been investigated extensively in the literature. However, most of them were focused on the ground state and accurate values of information entropies, such as Shannon entropy, for confined hydrogen are still not determined. In this work, we establish the benchmark results of the Shannon entropy for confined hydrogen atom in a spherical impenetrable sphere, in both position and momentum spaces. This is done by examining the bound state energies, the normalization of wave functions, and the scaling property with respect to isoelectronic hydrogenic ions. The angular and radial parts of Shannon entropy in two conjugate spaces are provided in detail for both free and confined hydrogen atom in ground and several excited states. The entropies in position space decrease logarithmically with decreasing the size of confinement, while those in momentum space increase logarithmically. The Shannon entropy sum, however, approaches to finite values when the confinement radius closes to zero. It is also found that the Shannon entropy sum shares same trend for states with similar density distributions. Variations of entropy for nodeless bound states are significantly distinct form those owning nodes when changing the confinement radius.
Abstract The effect of impenetrable spherical confinement on alkali systems (H, Li) has been thoroughly examined, emphasizing the distinctive high-momentum oscillatory behavior of the momentum space radial density. We have utilized the Ritz variational framework with a Slater-type basis set to derive the position-space wavefunction and Fourier–Dirac transformation of the former to find the momentum-space wavefunction, analytically. The derived momentum-space density has been analyzed in four asymptotic limits ( R → ∞ , R → 0 , p → ∞ , and p → 0 ; R being the confinement radius and p being the radial component of linear momentum) and its oscillatory behavior in strong spatial confinement region is critically investigated. Oscillatory behavior is also noticed in the Compton profile of the compressed atomic system. Furthermore, the effects of confinement on Shannon information entropy in both position and momentum spaces are investigated, offering insights into the Bialynicki–Birula–Mycielski inequality and the local variations of information measures in terms of Shannon entropy density.
A straightforward way to study the electronic structure and certain physical properties of atoms and molecules under high pressures is through the confinement (caging) model, in which the atom is placed at the center of a spherical box with impenetrable walls. The hydrogen atom confined inside penetrable and impenetrable spherical cavities is perhaps the most studied compressed system, and exact solutions exist for both types of confinement. The physical properties of the hydrogen atom confined within impenetrable-wall cavities (AHCI) have been widely analyzed; however, the same system embedded in penetrable spherical cavities (AHCP) has received less attention. A more physical model suggests using a spherical box with penetrable walls. Therefore, in this work, we compute via the variational method, the Shannon and Fisher information entropy in position space,SrS_{r}Sr​andIrI_{r}Ir​, for the2p2 \mathbf{p}2pand 3d states of a hydrogen atom subjected to spatial confinement by penetrable and impenetrable spherical cavities. Additionally, we calculate the tunneling probability for the penetrable confinement and seek to relate it to the overlaps of the radial probability densities as a function of the cavity radius and the height of the confining barrier. To model the penetrable confinement, we employ a logistic potential which, unlike earlier reports that used a discontinuous step potential, is a continuous function. The wavefunction is expanded in terms of Slater functions that satisfy appropriate boundary conditions; in some cases, we find notable differences with respect to the impenetrable potential.
Abstract This study investigates the impact of spatial confinement and the Hellmann potential on the Shannon entropy of a hydrogenic atom. A hydrogenic atom screened by the Hellmann potential and confined within an impenetrable spherical potential is analyzed. The Schrödinger equation is solved numerically using the finite difference method to determine energy eigenvalues and wavefunctions. These wavefunctions are examined in both position and momentum spaces to calculate the Shannon entropy in position space , the Shannon entropy in momentum space , and the total Shannon entropy . How the confinement radius and the screening parameter affect these entropies is investigated, and the results are compared to those of unconstrained systems. The findings are contrasted with previously reported results and discussed in relation to the Beckner–Bialynicki–Birula–Mycielski (BBM) inequality, offering insights into the behavior of quantum states under confinement. Significant variations in Shannon entropy and energy levels are observed with changes in confinement and screening parameters, providing a deeper understanding of quantum mechanics in confined systems. The study reveals a critical screening parameter at which the behavior of entropy transitions, shedding light on the interplay between spatial confinement, screening effects, and quantum uncertainties.