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

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.