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. The account of both beyond-first-order light--matter interactions and relativistic effects allows 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, 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 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 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.