Spin-orbit coupling with ultrasoft pseudopotentials: Application to Au and Pt

We show how to include the spin-orbit coupling in electronic structure calculations of crystalline solids based on plane waves and ultrasoft pseudopotentials. The spin-orbit split electronic band structures of fcc-Au and of fcc-Pt are calculated by fully relativistic pseudopotentials derived from an atomic Dirac-like equation. The results are compared with accurate all-electron calculations available in the literature.

Figure 1

Theoretical electronic band structure of fcc-Pt, calculated at the experimental lattice constant $a_0=7.40\mathrm{a.u.}$ The energy eigenvalues at $\Gamma$, $X$, $W$, $L$, and $K$ high-symmetry points are reported in Table .

Figure 2

Theoretical electronic band structure of fcc-Au, calculated at the experimental lattice constant $a_0=7.67\mathrm{a.u.}$ The energy eigenvalues at $\Gamma$, $X$, $W$, $L$, and $K$ high-symmetry points are reported in Table .

Figure 3

Theoretical electronic band structure of fcc-Pt and fcc-Au, calculated at the experimental lattice constant $a_0=7.67\mathrm{a.u.}$ (Pt) and $a_0=7.40\mathrm{a.u.}$ (Au) along the $\Delta$ line. Continuous lines are the results obtained with fully relativistic PPs, while dotted lines indicate the results of scalar relativistic PPs. The numbers refer to the labeling of the states according to the single group for the scalar relativistic bands and to the double group for the fully relativistic bands.