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 $L_1$ 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 $\mathbf{\sigma }·(\mathbf{\epsilon }\times \mathbf{r})$ that we call the spin position operator. 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 calculations of $K$-edge XMCD spectra of ferromagnetic iron, cobalt, and nickel and of I $L_1$-edge XNCD spectra of $\alpha \text{−}{\mathrm{LiIO}}_3$, 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 term to the spin magnetic moment of the $p$ states. In $\alpha \text{−}{\mathrm{LiIO}}_3$ we recover the expected angular dependence of the XNCD spectra.

• Received 4 April 2017

DOI:https://doi.org/10.1103/PhysRevB.96.085123