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An efficient first principles approach to calculate X-ray magnetic circular dichroism (XMCD) and X-ray natural circular dichroism (XNCD) is developed and applied in the near edge region at the K-and L1-edges in solids. Computation of circular dichroism requires precise calculations of X-ray absorption spectra (XAS) for circularly polarized light. For the derivation of the XAS cross section, we used a relativistic description of the photon-electron interaction that results in an additional term in the cross-section that couples the electric dipole operator with an operator σ · (epsilon× r) that we name spin-position. The numerical method relies on pseudopotentials, on the gauge including projected augmented wave method and on a collinear spin relativistic description of the electronic structure. We apply the method to the calculations of K-edge XMCD spectra of ferromagnetic iron, cobalt and nickel and of I L1-edge XNCD spectra of α-LiIO3, a compound with broken inversion symmetry. For XMCD spectra we find that, even if the electric dipole term is the dominant one, the electric quadrupole term is not negligible (8% in amplitude in the case of iron). The term coupling the electric dipole operator with the spin-position operator is significant (28% in amplitude in the case of iron). We obtain a sum-rule relating this new term to the spin magnetic moment of the p-states. In α-LiIO3 we recover the expected angular dependence of the XNCD spectra.

A numerically efficient methodology that allows for relativistic calculations including spin-orbit coupling for spatially anisotropic potentials is employed to study the energetics and electronic properties of In/Si(111)($4\ifmmode\times\else\texttimes\fi{}1$)/($8\ifmmode\times\else\texttimes\fi{}2$) nanowires within density-functional theory. For the subtle total energy balance between the metallic In zigzag chain structure observed at room temperature and the insulating In hexagons that form below the critical temperature, scalar-relativistic corrections to the kinetic energy are clearly far more important than the spin-orbit coupling. On the other hand, a relativistic treatment including spin-orbit coupling is required to describe correctly the electronic band structure that shows a large, strongly anisotropic $k$-point-dependent spin splitting of the In-related states at the $X$ point of the surface Brillouin zone. Suitably combined with the metal-insulator transition, the resulting quasi-1D Rashba effect may be used for spin filtering.