Real-space pseudopotential method for spin-orbit coupling within density functional theory

We present a formalism and implementation for real-space calculations that incorporate relativistic effects, including spin-orbit coupling. We demonstrate the validity of the method using the test cases of AuH and the ${\mathrm{Au}}_2$ dimer. The proposed approach differs from nonrelativistic real-space calculations in the addition of nonlocal pseudopotential projectors, which are translated to small rank-1 matrix “stencils” operating on the discretized wave functions. This formalism retains all the usual benefits of the real-space approach, especially with respect to massive parallelization. We expect it to be readily applicable for computational studies of large systems exhibiting spin-orbit coupling effects.

Figure 1

(Color online) Electronic structure of the AuH (left) and ${\mathrm{Au}}_2$ (right) molecules calculated using PARSEC with scalar-relativistic (SR) pseudopotentials, with spin-orbit coupling as a perturbation (PRT), and with a fully self-consistent spin-orbit (SO) calculation, compared with SR and SO calculations using VASP. Solid and dotted lines denote filled and empty states, respectively.

Figure 2

(Color online) (a) Formation energy of the ${\mathrm{Au}}_2$ dimer as a function of bond length. Circles denote computed values, and the dashed line denotes a sixth-order polynomial fit of the computed results. (b) Interatomic force for the ${\mathrm{Au}}_2$ dimer as a function of bond length. Diamonds denote forces computed directly using the Hellmann-Feynman theorem, and the dashed line denotes forces computed from the analytical derivative of the above polynomial fit.