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The Journal of Chemical Physics

Characteristic features of Shannon information entropy of confined atoms

K. D. Sen

Published Aug 15, 2005• 193 citations
Publisher

Abstract

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.

DOI
10.1063/1.2008212

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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.
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.
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.
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 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.
In this work, we study the Shannon information entropies [Formula: see text] and [Formula: see text] of an infinite spherical well. The Shannon entropy [Formula: see text] is calculated numerically in terms of the analytical result of the wave function in momentum space. Some typical features of the position and momentum probability densities [Formula: see text] and [Formula: see text] as well as the information entropy densities [Formula: see text] and [Formula: see text] are demonstrated. We find that the position entropy [Formula: see text] increases with the radius a of the spherical well for given quantum numbers l, m and n. It is interesting to note that the position entropy [Formula: see text] decreases with the quantum numbers l and n for a fixed radius a and quantum number m. The position entropy [Formula: see text] is almost independent of the quantum numbers l, m and n. The momentum entropy [Formula: see text] first increases and then decreases with respect to the radius a. We also note that the [Formula: see text] increases with the radius a and finally arrives at a constant. In addition, the Bialynicki–Birula–Mycielski (BBM) inequality is verified and also hold for this confined system.
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.