Two-Component Calculations: Cubic Scaling Implementation and Comparison of Vertex-Corrected and Partially Self-Consistent Variants.

We report an all-electron, atomic orbital (AO)-based, two-component (2C) implementation of the approximation (GWA) for closed-shell molecules. Our algorithm is based on the space-time formulation of the GWA and uses analytical continuation (AC) of the self-energy, and pair-atomic density fitting (PADF) to switch between AO and auxiliary basis. By calculating the dynamical contribution to the self-energy at a quasi-one-component level, our 2C- algorithm is only about a factor of 2-3 slower than in the scalar relativistic case. Additionally, we present a 2C implementation of the simplest vertex correction to the self-energy, the statically screened 32 correction. Comparison of first ionization potentials (IPs) of a set of 67 molecules with heavy elements (a subset of the SOC81 set) calculated with our implementation against results from the WEST code reveals mean absolute deviations (MAD) of around 70 meV for @PBE and @PBE0. We check the accuracy of our AC treatment by comparison to full-frequency calculations, which shows that in the absence of multisolution cases, the errors due to AC are only minor. This implies that the main sources of the observed deviations between both implementations are the different single-particle bases and the pseudopotential approximation in the WEST code. Finally, we assess the performance of some (partially self-consistent) variants of the GWA for the calculation of first IPs by comparison to vertical experimental reference values. @PBE0 (25% exact exchange) and @BHLYP (50% exact exchange) perform best with mean absolute deviations (MAD) of about 200 meV. Explicit treatment of spin-orbit effects at the 2C level is crucial for systematic agreement with experiment. On the other hand, eigenvalue-only self-consistent (ev) and quasi-particle self-consistent (qs) significantly overestimate the IPs. Perturbative 32 corrections increase the IPs and therefore improve the agreement with experiment in cases where alone underestimates the IPs. With a MAD of only 140 meV, 2C-@PBE0 + 32 is in best agreement with the experimental reference values.