Probing chirality across the electromagnetic spectrum with the full semi-classical light--matter interaction

We present a formulation and implementation of anisotropic and isotropic electronic circular dichroism (ECD) using the full semi-classical light–matter interaction operator within a four-component relativistic framework. Our treatment uniquely accounts for both beyond-first-order light–matter interactions and relativistic effects, enabling us to investigate the ECD response across the electromagnetic spectrum from optical to x-ray wavelengths where relativistic selection rules and spatial field variations gain increasing importance. We consider the isotropic and oriented ECD across the valence transition and sulfur L- and K-edge transitions in the simplest disulfides, H₂S₂ and (CH₃S)₂, and evaluate the influence of the full interaction by comparing to a traditional truncated formulation in the Coulomb gauge (velocity representation). Additionally, we demonstrate that in the relativistic formalism, it is possible to work in the velocity representation, hence keeping order-by-order gauge-origin invariance, contrary to the multipolar gauge, yet being able to distinguish electric and magnetic multipole contributions. Going beyond a first-order treatment in the wave vector is mandatory in the higher-energy end of the soft x-ray region and beyond where the consequent intensity redistribution becomes significant. While the sulfur K-edge absorption spectrum is essentially unaffected by this redistribution, the signed differential counterpart is not: At least third-order contributions are required to describe the differential absorption profile that is otherwise overestimated by a factor of about two. The first-order description deteriorates at higher transition energies (beyond ∼1000 eV) where it may even fail to predict the sign of individual differential oscillator strengths.