Projector augmented-wave method: an analysis in a one-dimensional setting

TL;DR

This paper provides a rigorous numerical analysis of the projector augmented-wave (PAW) method in a one-dimensional periodic setting, offering error estimates for eigenvalues when using Dirac potentials.

Contribution

It introduces the first error estimates for the lowest PAW eigenvalue in a 1D periodic Schrödinger operator with Dirac potentials, enhancing understanding of PAW accuracy.

Findings

Error bounds for PAW eigenvalues are established.

The analysis applies to 1D periodic Schrödinger operators with Dirac potentials.

Provides theoretical foundation for PAW method accuracy in simplified models.

Abstract

In this article, a numerical analysis of the projector augmented-wave (PAW) method is presented, restricted to the case of dimension one with Dirac potentials modeling the nuclei in a periodic setting. The PAW method is widely used in electronic ab initio calculations, in conjunction with pseudopotentials. It consists in replacing the original electronic Hamiltonian $H$ by a pseudo-Hamiltonian $H^{PAW}$ via the PAW transformation acting in balls around each nuclei. Formally, the new eigenvalue problem has the same eigenvalues as $H$ and smoother eigenfunctions. In practice, the pseudo-Hamiltonian $H^{PAW}$ has to be truncated, introducing an error that is rarely analyzed. In this paper, error estimates on the lowest PAW eigenvalue are proved for the one-dimensional periodic Schr\"odinger operator with double Dirac potentials.