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

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.