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

Sangita Majumdar,Amlan K Roy

doi:10.3390/quantum2010012

Sangita Majumdar, Amlan K Roy

Open Access

https://doi.org/10.3390/quantum2010012

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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.

Highlights

A particle in an impenetrable box of infinite height has served the role of a simple, elegant pedagogical tool to illustrate the effects of boundary condition on the energy spectrum of a quantum system
Ground and excited states were studied via a simple density functional theory (DFT) method, by solving the radial KS equation through a generalized Legendre pseudospectral method
As the X-only entropies were comparable to their HF counterparts in the free atom limit, it was expected that this would hold in the confined case as well

Summary

Introduction

A particle in an impenetrable box of infinite height has served the role of a simple, elegant pedagogical tool to illustrate the effects of boundary condition on the energy spectrum of a quantum system Understanding of such a system in some sub-region Ω of space (in contrast to the “whole” space available in a free system) offers new insights to simulate realistic situations in highly inhomogeneous media or in an external field. Matter constricted under such an extreme pressure environment gives rise to a wide range of novel changes (from the respective free counterpart) in energy spectra, electronic structure, chemical reactivity, ionization potential, polarizability etc., depending on the geometrical forms of the cavity and dimensions. The interested reader may refer to the following reviews [1,2,3,4,5] and the references therein

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