Simple eigenvalue-self-consistent Δ ¯ G W 0 .

We show that a rigid scissors-like GW self-consistency approach, labeled here Δ ¯ G W 0 , can be trivially implemented at zero additional cost for large scale one-shot G ₀ W ₀ calculations. The method significantly improves one-shot G ₀ W ₀ and for large systems is very accurate. Δ ¯ G W 0 is similar in spirit to ev GW ₀ where the self-consistency is only applied on the eigenvalues entering Green's function, while both W and the eigenvectors of Green's function are held fixed. Δ ¯ G W 0 further assumes that the shift of the eigenvalues is rigid scissors-like so that all occupied states are shifted by the same amount and analogously for all the unoccupied states. We show that this results in a trivial modification of the time-dependent G ₀ W ₀ self-energy, enabling an a posteriori self-consistency cycle. The method is applicable for our recent stochastic- GW approach, thereby enabling self-consistent calculations for giant systems with thousands of electrons. The accuracy of Δ ¯ G W 0 increases with the system size. For molecules, it is up to 0.4-0.5 eV away from coupled-cluster single double triple (CCSD(T)), but for tetracene and hexacene, it matches the ionization energies from both CCSD(T) and ev GW ₀ to better than 0.05 eV. For solids, as exemplified here by periodic supercells of semiconductors and insulators with 6192 valence electrons, the method matches ev GW ₀ quite well and both methods are in good agreement with the experiment.