1. INTRODUCTION Density Functional Theory (DFT) [1, 2] with the local density (LDA) or generalized gra- dient approximations (GGA) [2–4] is the most widely used framework for first principles calculations of materials. In practice, it is often found to provide a good description of ground-state total energies, atomic geometries, vibrational modes, etc. of a variety of ma- terials. A major shortcoming is its inability to predict accurate electronic band structure energies [5]. For band insulators with weak correlation the failure is mainly quantitative [6]. However, for the classic case of transition metal oxides, the failures can be qualitative and serious such as predicting a metallic instead of an insulating ground state [7]. Ideally, a computationally efficient ab initio method with accurate total energies and band structures would yield a major advance in predictive power. A number of approaches aim to improve electronic band structures. For systems with transition metals, one fundamental problem is that the LDA or GGA does not capture the proper electronic correlations for the spatially localized d - and f -state derived bands. Two current solutions to this deficiency are LDA+U [7] and dynamical mean-field theory (DMFT) [8, 9]. The LDA+U approach is popular and easy-to-use: one “manually” adds static and localized correlation effects within a Hubbard-like model to the DFT energy functional. More sophisticated is DMFT where localized but dynamic and high-level correlations are included using exact solutions of interacting quantum impurity models. Both approaches are physically motivated in that they create frameworks that include to physics deemed important for the problem at hand. A shared drawback is their requirement of an unspecified localized basis set in order to define key quantities such as the Hubbard parameters or the impurity site. This raises questions of transferability, i.e. , the dependence of their predictions on the chosen orbitals or parameter values. A different approach is to use many-body perturbation theory of Green’s functions. The most successful is Hedin’s GW approximation to the electron self-energy [10]. This approx- imation delivers high quality band structures of many band insulators and simple metals [6, 11]. The GW approximation includes a great deal of physics including exact exchange, localized Coulomb repulsion, dynamic screening, and dispersion forces: e.g. , the LDA+U is a static approximation to GW [7]. However, most GW calculations are perturbative in that they compute corrections to a mean-field DFT input, and the quality of the final result depends on the DFT description. In certain transition metal oxides where the LDA pro- vides a decent starting point, GW corrections on this DFT starting point can yield a good description of the electronic bands [12–17]. However, in other situations, the inadequacy of the DFT description can lead to quantitative errors [12, 18–20]. Clearly, it is advantageous to apply GW beyond the perturbation-off-DFT prescription. Such an approach would not assume a localized basis or any set of parameters. Recent methods such as the Quasiparticle Self-Consistent GW (QS GW ) [18] or the self-consistent COHSEX (scCOHSEX) [20] have successfully moved away from using DFT as the starting point. These two methods find the noninteracting initial guess for the band structure ap- proximately but self-consistently within GW . Such approaches allow for inclusion of both static and dynamic screening effects in addition to some localized (Hubbard U) Coulombic physics in a single, general, and parameter-free framework. QS GW and scCOHSEX are self-consistent band structure methods, but it would be highly desirable to turn them into 2