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Shannon entropy in confined He-like ions within a density functional formalism

Shannon entropy in position ($S_{\rvec}$) and momentum ($S_{\pvec}$) 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, \emph{viz.,} 1sns (n $=$ 2-4) $^3$S, 1snp (n $=$ 2-3) $^3$P, 1s3d $^3$D 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 corresponding free system) has been performed for these quantities with respect to $r_c$ in tabular and graphical forms, \emph{with and without} electron correlation. Due to compression, the pattern of entropy in aforementioned states gets characterized by various crossovers at intermediate and lower $r_c$ regions. The impact of electron correlation is more pronounced in weaker confinement limit, and appears to decay with rise in confinement strength. The exchange-only results are quite good to provide a decent qualitative discussion. The lower-bounds provided by entropic uncertainty relation holds good in all cases. Several other new interesting features are observed.