total energy methods via the Luttinger-Ward [22] approach that should, in principle, allow one to obtain accurate total energies and band structures. Separate from self-consistent band structure methods, there has been ongoing work on using GW -RPA type correlation functionals for computing total energies. Much of the activity was sparked by initial work on the uniform electron gas showing that Luttinger- Ward functionals with the GW -RPA correlation energy provided very accurate ground- state total energies [23–26]. Other model calculations cast doubt on whether such high accuracy was generic [27]. However, actual calculations on atoms, small molecules, and simple solids find ground-state energies that improve over the standard DFT functionals, especially when short-ranged corrections are added [28–33, 35–39]. The vast majority of such calculations are post-DFT calculations. Namely, they use Green’s functions based on LDA or GGA wave functions and eigenvalues to evaluate the correlation energy instead of using a self-consistent one-particle description coming from a more elaborate, and presumably more accurate, theory like QS GW or scCOHSEX. In addition, the correlation energies are evaluated using the standard formula relying on frequency integration of the trace of a matrix logarithm (see Section 5). In this work, we report on three main points of progress in this general area for the GW -RPA correlation energy functional. The first is an exact rewriting of the correlation energy in terms of plasma and interband energies, a result that was found recently using very different methods [21]. This exact form is amenable to computations of identical com- plexity to present-day linear response time-dependent DFT (TDDFT) or Bethe-Salpeter (BSE) calculations [11, 43–46, 49]. In addition, this new form has much better convergence properties when compared to the standard frequency integration method as evinced by our atomic calculations. Second, we prove that the GW -RPA correlation energy functional is not bounded from below, has a minimum with negative infinite value, and when evaluated using non-interacting Green’s functions, it has no extrema. This means blind optimization of total energy functionals that are based on the GW -RPA correlation energy is highly prob- lematic, and physical constraints are required. Third, we rewrite the GW -RPA correlation energy approximately as a sum over screened interband transition contributions. This allows us to create a ladder of approximations of which the COHSEX [10] is the lowest wrung. In addition, this allows us to put schemes such as scCOHSEX on a firm footing by showing how they can originate from a variational principle. The ladder provides a series of more accurate functionals and associated self-energy operators that may deliver improved band structure and Green’s functions. We hope that these findings pave the way towards efficient and accurate computation of total energies within GW -RPA as well as the creation of effi- cient self-consistent schemes for computing total energies and band structures that improve over DFT. This paper is organized as follows. In Section 2, we describe our notation and provide basic definitions. Section 3 provides a brief description and the necessary ingredients of the Luttinger-Ward approach. Section 4 describes how the total energy Klein functional we focus on in this work actually has no dependence on the choice of non-interacting auxiliary Green’s function that enters the theory. This greatly simplifies the form of the functional if we restrict ourselves to evaluating the total energy on non-interacting Green’s functions. We also make a connection to the Sham-Schl¨ uter equation [47, 48] appropriate for the variational problem associated with the functional. In Section 5, we rewrite the GW -RPA correlation energy 3