Relativistic Effects in Magnetic Circular Dichroism: Restricted Magnetic Balance and Temperature Dependence

Magnetic circular dichroism of transition metal complexes and open-shell systems are challenging to simulate and analyze, mainly due to the interplay of spin-orbit couplings and finite-magnetic-field induced Zeeman effects with the complex selection rules dictated by the circularly polarized light. In this work, we introduce an ab initio relativistic two-component formalism based on the restricted magnetic-balanced Hamiltonian for simulating MCD spectra. Both homogeneous finite magnetic field and relativistic effects are included variationally in the ground state reference. Finite-field London orbitals are used to enforce the constrained gauge-origin independence in the calculation using localized atomic orbitals. Through benchmark studies of AuCl₄^-, Pt(CN)₄^2-, and Mo(CN)₈^3-, we discuss how relativistic effects are manifested in MCD for both closed-shell and open-shell molecular complexes and how the interplay between spin-orbit coupling and magnetic field modulates the MCD selection rules. Finally, an investigation on temperature-dependent MCD is carried out and compared to experiment.