Atomic Confinement Potentials and the Generation of Numerical Atomic Orbitals

TL;DR

This paper develops and evaluates various confinement potentials for generating numerical atomic orbitals, demonstrating their insensitivity to potential form and analyzing their effects on atomic orbitals and basis set errors.

Contribution

It introduces a systematic study of multiple confinement potentials for NAO generation and assesses their impact on atomic orbitals and basis set accuracy.

Findings

Ground-state orbitals are insensitive to the form of confinement potential.

Orbitals decay rapidly under confinement.

Steep potentials approach the hard-wall limit and basis set errors are analyzed.

Abstract

We aim to develop novel reusable open source infrastructure [Lehtola, J. Chem. Phys. 159, 180901 (2023)] for numerical atomic orbitals (NAOs). Soft confinement potentials are typically used to force the NAO radial basis functions ${\psi}_{nl}(r)$ to vanish smoothly in increasing $r$ and to generate localized unoccupied states; we review such potentials and other commonly-used techniques in NAO generation as a follow-up to our recent study on atoms in hard-wall confinement [{\AA}str\"om and Lehtola, J. Phys. Chem. A 129, 2791 (2025)]. In addition to NAO generation, confinement potentials are also employed to simulate environmental effects in other research areas, such as studies of (i) atoms in solids, (ii) quantum dots, and (iii) high-pressure chemistry. As in our earlier work, we perform fully numerical density functional calculations with spherically averaged densities, as is usual in NAO studies. Our calculations employ the the finite element method (FEM) implemented in the HelFEM program, yielding variational energies and enabling the use of various boundary conditions. We consider four families of potentials to study the Mg and Ca atoms, which are textbook examples of extended electronic structures. We show that the resulting ground-state orbitals are surprisingly insensitive to the employed form of the confinement potential, and that the orbitals decay quickly under confinement. We study increasingly steep potentials and examine how they approach the hard-wall limit. Finally, we assess NAO basis set truncation errors for types of singular potentials that are now broadly used in the NAO literature.