Self-Consistent <i>GW</i> via Conservation of Spectral Moments

We expand on a recently introduced alternate framework for <i>GW</i> simulation of charged excitations [Scott, C. J. C. <i>J. Chem. Phys.</i> 2023, 158, 124102], based around the conservation of directly computed spectral moments of the <i>GW</i> self-energy. Featuring a number of desirable formal properties over other implementations, we also detail efficiency improvements and a parallelism strategy, resulting in an implementation with a demonstrably similar scaling to an established Hartree-Fock code, with only an order of magnitude increase in cost. We also detail the applicability of a range of self-consistent <i>GW</i> variants within this framework, including a scheme for full self-consistency of all dynamical variables, while avoiding the Matsubara axis or analytic continuation, allowing formal convergence at zero temperature. By investigating a range of self-consistency protocols over the <i>GW</i>100 molecular test set, we find a little-explored self-consistent variant based around a simpler coupled chemical potential and Fock matrix optimization to be the most accurate self-consistent <i>GW</i> approach. Additionally, we validate recently observed evidence that Tamm-Dancoff-based screening approximations within <i>GW</i> lead to higher accuracy than traditional random phase approximation screening over these molecular test cases. Finally, we consider the Chlorophyll A molecule, finding agreement with experiment within the experimental uncertainty, and a description of the full-frequency spectrum of charged excitations.