Systematic Study of Hard-Wall 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 either not been systematic, having only looked at select atoms and/or select electron configurations, or they have not used robust numerical methods. To remedy the situation, we perform in this work a methodical study of atoms in hard-wall confinement with the HelFEM program, which employs the finite element method that trivially implements the hard-wall potential, guarantees variational results, and allows for easily finding the numerically exact solution. Our fully numerical calculations are based on nonrelativistic density functional theory and spherically averaged densities. We consider three levels of density functional approximations: the local density approximation employing the Perdew-Wang (PW92) functional, the generalized-gradient approximation (GGA) employing the Perdew-Burke-Ernzerhof (PBE) functional, and the meta-GGA approximation employing the r²SCAN functional. Importantly, the completely dissimilar density functional approximations are in excellent agreement, suggesting that the observed results are not artifacts of the employed level of theory. We systematically examine low-lying configurations of the H-Xe atoms and their monocations and investigate how the configurations─especially the ground-state configuration─behave as a function of the position of the hard-wall boundary. We perform calculations with 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.