Optimizing Generalized Norm-Conserving Pseudopotentials

The ``generalized'' method permits the construction of norm-conserving pseudopotentials at energies that do not correspond to bound atomic states, giving added flexibility in the treatment of angular-momentum channels for which no bound states exist.\footnote{D. R. Hamann, Phys. Rev. B \textbf{40}, 2980 (1989).} An effective method for optimizing the convergence of pseudopotential calculations with plane-wave-basis cutoff energy requires atomic wave functions with decaying tails, and has not been applicable to such ``generalized'' states.\footnote{A. M. Rappe,\textit{ et al.}, Phys. Rev. B \textbf{41}, 1227 (1990).} By introducing a potential well outside the core radius for selected angular-momenta, an artificial decaying tail can be produced for positive-energy states. This permits the application of the optimization method, and we find convergence behavior comparable to that for ordinary bound states. In practice, we terminate the positive-energy all-electron wave function smoothly with an exponential or Gaussian tail, and never need to treat the implied well potential explicitly. The projectors to form fully-nonlocal operators\footnote{L. Kleinman and D. M. Bylander, Phys. Rev. Lett. \textbf{48}, 1425 (1982).} can be terminated at the core radii as usual, despite differences of the semi-local potentials outside the well radii.