Relativistic Optimized Norm-Conserving Vanderbilt Pseudopotentials

Two-projector fully non-local pseudopotentials obeying the generalized norm-conserving condition\footnote{D. Vanderbilt, Phys. Rev. B \textbf{41}, 7892 (1990).} and incorporating systematic convergence optimization\footnote{A. M. Rappe\textit{ et al.}, Phys. Rev. B \textbf{41}, 1227 (1990).} have been shown to accurately reproduce all-electron results with high computational efficiency.\footnote{D. R. Hamann, Phys. Rev. B \textbf{88}, 085117 (2013).} The generalized norm-conservation theorem guarantees exact reproduction of all-electron norms, radial log-derivatives, and first energy derivatives of radial log derivatives at several energies, as well as the hermiticity of the non-local pseudopotential operator. This theorem is exact only for non-relativistic all-electron wave functions.\footnote{Vanderbilt} Averaging out small asymmetries of the non-local operators generated using scalar-relativistic Schr\"{o}dinger equation solutions preserves agreement of these quantities to order 10$^{-4}$, and yields excellent results for solids.\footnote{Hamann} I show that fully-relativistic Dirac-equation solutions can be treated in the same manner, with comparably small errors. Spin-orbit band splittings as well as other properties of several solids calculated with these pseudopotentials will be compared to fully-relativistic all-electron results.