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We present three schemes to go beyond the electric-dipole approximation inX-ray absorption spectroscopy calculations within a four-component relativisticframework. The first is based on the full semi-classical light-matterinteraction operator, and the two others on a truncated interaction withinCoulomb gauge (velocity representation) and multipolar gauge (lengthrepresentation). We generalize the derivation of multipolar gauge to anarbitrary expansion point and show that the potentials corresponding todifferent expansion point are related by a gauge transformation, provided theexpansion is not truncated. This suggests that the observed gauge-origindependence in multipolar gauge is more than just a finite-basis set effect. Thesimplicity of the relativistic formalism enables arbitrary-orderimplementations of the truncated interactions, with and without rotationalaveraging, allowing us to test their convergence behavior numerically bycomparison to the full formulation. We confirm the observation that theoscillator strength of the electric-dipole allowed ligand K-edge transition ofTiCl$_4$, when calculated to second order in the wave vector, become negative,but also show that inclusion of higher-order contributions allows convergenceto the result obtained using the full light-matter interaction. However, athigher energies, the slow convergence of such expansions becomes dramatic andrenders such approaches at best impractical. When going beyond theelectric-dipole approximation, we therefore recommend the use of the fulllight-matter interaction.

Beyond the electric-dipole approximation in simulations of X-ray absorption spectroscopy: Lessons from relativistic theory