  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.

Numerical Analysis;Materials Science