  We present three schemes to go beyond the electric-dipole approximation in
X-ray absorption spectroscopy calculations within a four-component relativistic
framework. The first is based on the full semi-classical light-matter
interaction operator, and the two others on a truncated interaction within
Coulomb gauge (velocity representation) and multipolar gauge (length
representation). We generalize the derivation of multipolar gauge to an
arbitrary expansion point and show that the potentials corresponding to
different expansion point are related by a gauge transformation, provided the
expansion is not truncated. This suggests that the observed gauge-origin
dependence in multipolar gauge is more than just a finite-basis set effect. The
simplicity of the relativistic formalism enables arbitrary-order
implementations of the truncated interactions, with and without rotational
averaging, allowing us to test their convergence behavior numerically by
comparison to the full formulation. We confirm the observation that the
oscillator strength of the electric-dipole allowed ligand K-edge transition of
TiCl$_4$, when calculated to second order in the wave vector, become negative,
but also show that inclusion of higher-order contributions allows convergence
to the result obtained using the full light-matter interaction. However, at
higher energies, the slow convergence of such expansions becomes dramatic and
renders such approaches at best impractical. When going beyond the
electric-dipole approximation, we therefore recommend the use of the full
light-matter interaction.
