A unified formalism for collinear and non-collinear approaches in the four-component Dirac-Kohn-Sham theory based on G-spinors
Non-collinear density functional theories were developed to extend the use of established collinear exchange-correlation functionals to systems with unpaired electrons in the presence of significant spin-orbit coupling. A comparison of different approaches and implementations is not straightforward, as the methods are often formulated using different fundamental variables and numerical approximations. A consistent review of the formal and numerical aspects of collinear and non-collinear schemes has recently been reported (Desmarais et al., J. Chem. Phys. 154, 204110 (2021)) in the context of two-component methods. In this work, we present an initial effort towards a unified formulation of collinear and non-collinear approximations, encompassing both canonical and Scalmani-Frisch schemes, within the relativistic four-component DKS formalism based on G-spinor basis sets. Our preliminary implementation of the collinear and canonical non-collinear formulations in the DKS module of the \texttt{BERTHA} code extends its applicability and provides a benchmark for a series of simple open-shell hydride molecules (namely, H$_2$X$^+$, with X = O, S, Se, Te, and Po). Finally, we show that incorporating the magnetisation vector into the reformulated non-collinear canonical LDA approach enables a description of H$_2$ dissociation - and open-shell systems more broadly - that closely parallels unrestricted non-relativistic approaches, notably without explicitly imposing the broken symmetry solution as is often required in non-relativistic collinear calculations. This unified formulation forms the basis for a rigorous comparison between different numerical approximations, which will be essential for obtaining stable results for the non-collinear GGA exchange-correlation functionals.