Relativistic two-component formulation of time-dependent current-density functional theory:Application to the linear response of solids

In this paper we derive the relativistic two-component formulation of time-dependent current-density-functional theory. To arrive at a two-component current-density formulation we apply a Foldy-Wouthuysen-type transformation to the time-dependent four-component Dirac-Kohn-Sham equations of relativistic density-functional theory. The two-component Hamiltonian is obtained as a regular expansion which is gauge invariant at each order of approximation, and to zeroth order it represents the time-dependent version of the relativistic zeroth order regular Hamiltonian obtained by van Lenthe , for the ground state [J. Chem. Phys.99, 4597 (1993)]. The corresponding zeroth order regular expression for the density is unchanged, whereas the current-density operator now comprises a paramagnetic, a diamagnetic, and a spin contribution, similar to the Gordon decomposition of the Dirac four current. The zeroth order current density is directly related to the mean velocity corresponding to the zeroth order Hamiltonian. These density and current density operators satisfy the continuity equation. This zeroth order approximation is therefore consistent and physically realistic. By combining this formalism with the formulation of the linear response of solids within time-dependent current-density functional theory [Romaniello and de Boeij, Phys. Rev. B71, 155108 (2005)], we derive a method that can treat orbital and spin contributions to the response in a unified way. The effect of spin-orbit coupling can now be taken into account. As first test we apply the method to calculate the relativistic effects in the linear response of several metals and nonmetals to a macroscopic electric field. Treatment of spin-orbit coupling yields visible changes in the spectra: a smooth onset of the interband transitions in the absorption spectrum of Au, a sharp onset with peak at about 0.46 eV in the absorption spectrum of W, and a low-frequency doublet structure in the absorption spectra of ZnTe, CdTe, and HgTe in agreement with experimental results.