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.