Systematic study of confinement induced effects on atomic electronic structure

We point out that although a litany of studies have been published on atoms in hard-wall confinement, they have not been systematic or have not used robust numerical methods. We report a methodical study of atoms in hard-wall confinement employing a robust finite element method (FEM) in HelFEM that guarantees variational results and allows easily finding the numerically exact solution. Our fully numerical calculations are non-relativistic and are carried out at three levels of density functional theory with spherically averaged densities: the PW92, PBE, and r$^2$SCAN functionals. The three are in excellent agreement, confirming the physicality of our results. We systematically examine low lying configurations of the H-Xe atoms and their monocations, and investigate how the configurations - especially the ground state - behave as a function of the position of the hard-wall boundary. We consider both spin-polarized as well as spin-restricted densities, and demonstrate that spin-polarization effects are significant in open shell configurations, even though some previous studies have only considered the spin-restricted model. We demonstrate the importance of considering ground state changes for confined atoms by computing the ionization radii for the H-Xe atoms and observe significant differences to earlier studies. Confirming previous observations, we identify electron shifts on the outermost shells for a majority of the elements: valence $s$ electrons are highly unfavored under strong confinement, and the high-lying $3d$ and $4f$ orbitals become occupied in atoms of periods 2-3 and 3-4, respectively. We also comment on deficiencies of a commonly used density based estimate for the van der Waals (vdW) radius of atoms, and propose a better behaved variant in terms of the number of electrons outside the vdW radius that we expect will prove useful in future studies.