Combination of Hedin's $GW$ and dynamical mean-field theory tested on H$_2$ molecule

We compare various flavors of ``$GW$+DMFT" approach with LDA+DMFT for the simplest strongly correlated system, the H$_2$ molecule. The following $GW$+DMFT methodologies are compared: (i) the fully self-consistent GW+DMFT, (ii) the quasi-particle self-consistent QS-GW+DMFT with dynamic double-counting, (iii) QS-GW+DMFT with static double-counting schemes. We found that fully self-consistent $GW$+DMFT with exact double-counting yields very precise spectra around equilibrium H-H distance, as well as reasonable total energy (comparable to LDA+DMFT). However, this scheme breaks down in the correlated regime due to causality violation. The QS-GW+DMFT approaches, which are not derivable from a functional, yield similar spectra as full GW+DMFT near equilibrium distance, and in static double-counting schemes, can also be extended into correlated regime. However, the total energy of these approaches is much worse than the total energy of LDA+DMFT. In summary, this toy model of correlated physics suggests that QS-GW+DMFT with constant double-counting should give accurate predictions of spectra, but not total energy, while LDA+DMFT gives very precise total energy, but somewhat less precise spectra.