Exact exchange in relativistic spin-density-functional theory: Exchange splitting versus spin-orbit coupling

Abstract

Recently, the concept of the exact orbital-dependent exchange was introduced into relativistic spin-density-functional theory (RSDFT) within the collinear limit [D. K\"odderitzsch et al., Phys. Rev. B 77, 045101 (2008)]. In this contribution we further expand this exact exchange (EXX) formalism by (i) extending the basic equations to the general noncollinear form of RSDFT and (ii) discussing in detail the solution of the coupled integral equations resulting from orbital-dependent functionals in the framework of RSDFT. The EXX scheme is then applied to open-shell atoms in order to study (i) the relative importance of exchange splitting and spin-orbit coupling, (ii) the consequences of the exact exchange for atomic hyperfine constants, and (iii) the relative stability of the $3{d}^{n\ensuremath{-}1}4{s}^{2}$ and $3{d}^{n}4{s}^{1}$ configurations in case of the $3d$ transition-metal elements. In particular, it is demonstrated that the exact exchange, when combined with the orbital-dependent random-phase approximation for correlation, yields $s\text{\ensuremath{-}}d$-transfer energies which are clearly superior to the values obtained with conventional density functionals.