Spin-orbit coupling from a two-component self-consistent approach. II. Non-collinear density functional theories

We revise formal and numerical aspects of collinear and non-collinear density functional theories in the context of a two-component self-consistent treatment of spin-orbit coupling. Theoretical and numerical analyses of the non-collinear approaches confirm their ability to yield the proper collinear limit and provide rotational invariance of the total energy for functionals in the local-density or generalized-gradient approximations (GGAs). Calculations on simple molecules corroborate the formal considerations and highlight the importance of an effective screening algorithm to provide the sufficient level of numerical stability required for a rotationally invariant implementation of non-collinear GGA functionals. The illustrative calculations provide a first numerical comparison of both previously proposed non-collinear formulations for GGA functionals. The proposed screening procedure allows us to effectively deal with points of small magnetization, which would otherwise be problematic for the evaluation of the exchange-correlation energy and/or potential for non-collinear GGA functionals. Both previously suggested formulations for the non-collinear GGA are confirmed to be adequate for total energy calculations, provided that the screening is achieved on a sufficiently fine grid. All methods are implemented in the Crystal program.