The random phase approximation (RPA) and the $GW$ approximation share the same total energy functional but RPA is defined on a restricted domain of Green's functions determined by a local Kohn-Sham (KS) potential. In this work, we perform self-consistent RPA calculations by optimizing the local KS potential through the optimized effective potential equation. We study a number of solids (C, Si, BN, LiF, MgO, ${\mathrm{TiO}}_2$) and find in all cases a lowering of the total energy with respect to non-self-consistent RPA. We then propose a variational approach to optimize parameter-dependent hybrid functionals based on the minimization of the RPA total energy with respect to the fraction of exact exchange used to generate the input KS orbitals. We show that this scheme leads to hybrid functionals with a KS band structure in close agreement with RPA and with lattice constants of similar accuracy as within RPA. Finally, we evaluate $G_0{W}_0$ gaps using RPA and hybrid KS potentials as starting points. Special attention is given to ${\mathrm{TiO}}_2$, which exhibits a strong starting-point dependence.

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• Electronic structure
• First-principles calculations
• Density functional theory
• GW method
• Quantum chemistry methods
• Random phase approximation