Hamann has shown how to construct a norm-conserving pseudopotential for the unbound 4f state in the Ba atom. This pseudopotential which was calculated at -1.877 eV, fails to yield the sharp 4f resonance found at +1.501 eV in the Dirac atom. We show that this is a consequence not of the energy at which the pseudopotential is calculated but rather of the cutoff radius used. We calculate the pseudopotential at -1.877 eV using a cutoff radius intermediate to those used by Hamann for the bound 4f state of the ${\mathrm{Ba}}^{+}$ ion and the unbound 4f state of Ba, and obtain the positive-energy resonance. When the [${\partial }^2$(ψ’/ψ)/∂$E^2$]-preserving form of the norm-conserving pseudopotential recently developed by Shirley et al. is used, the resonance occurs at 1.507 eV and is actually larger than that of the Dirac atom. We also discuss the application of these new ideas to the separable form of the norm-conserving pseudopotential.

• Received 13 September 1989

DOI:https://doi.org/10.1103/PhysRevB.41.907