We present a formulation of relativistic linear response time-dependent density functional theory for calculation of electronic excitation energies in the framework of the four-component Dirac-Coulomb Hamiltonian. This approach is based on the noncollinear ansatz originally developed by Scalmani and Frisch [J. Chem. Theory Comp. 8, 2193 (2012)], and improves upon past treatment of the limit cases in which the spin density approaches zero. As a result of these improvements, the presented approach is capable of treating both closed-and open-shell reference states. Robust convergence of the Davidson-Olsen eigenproblem algorithm for open-shell reference states was achieved through the use of a solver which considers both left and right eigenvectors. The applicability of the present methodology on both closed-and open-shell reference states is demonstrated on calculations of low-lying excitation energies for Group 3 atomic systems (Sc 3+-Ac 3+) with non-degenerate ground states, as well as for Group 11 atomic systems (Cu-Rg) and octahedral actinide complexes (PaCl 2− 6 , UCl − 6 , and NpF 6) with effective doublet ground states. Major developments in the field of relativistic LR-TDDFT methods featuring variational inclusion of SOC began with the independent works of Gao et al. 29 in the four-component framework, and Wang and Ziegler 30 in the two-component framework. Both works combine noncollinear DFT theory with the ALDA approximation.

Approaches and programs for calculations of the EPR g-tensor in the framework of the two-and fourcomponent methods are still very rare. There are three main reasons for this: the wider community's unawareness of the importance of second-and higher order spin-orbit effects on the g-tensor, the methodological problems associated with performing such calculations and the lack of understanding of these problems. This paper reports on the implementation of a method for calculation of the g-tensor in the framework of the relativistic unrestricted two-and four-component Hartree-Fock and density functional theory approaches based on the Kramers pair formalism. This implementation allows us to analyse problems which arise when the g-tensor is calculated via Kramers pairs in the unrestricted framework.