in terms of integrals of the standard, time-ordered RPA polarizability. Section 6 provides expressions for the derivatives of the total energy functional versus the wave functions and eigenenergies or equivalently the static and Hermitian non-local potential determining the non-interacting Green’s function. In Section 7 we perform an exact rewriting of the GW - RPA correlation energy functional in terms of plasma and interband transition contributions. Section 8 provides a proof of the unboundedness and lack of extremum of the Klein total energy functional when evaluated on non-interacting Green’s functions. We discuss what this result means in more physical terms, how it relates to other results in the literature, and the requirement of constrained optimization that stems from this result. In Section 9, we switch gears and derive approximate forms of the GW -RPA correlation energy written in terms of contributions from self-interaction-free screened interband transitions. Section 10 derives a ladder of approximations that allows us to connect to the COHSEX approximation for the self-energy and its associated correlation energy expression. We provide a number of approximate correlation energies from which appropriate self-energies can be derived. In Section 11, we report numerical results for atoms demonstrating the lack of a lower bound to the correlation energy, the superior convergence properties of the exact plasmon form, and a tabulation of the quality of the various approximate forms derived in Sections 9 and 10. We summarize and provide an outlook in Section 12. In this work, we restrict ourselves to systems with time-independent, non-relativistic, many-body Hamiltonians so that all response or Green’s functions are functions only of time differences. Furthermore, we will consider systems with time reversal symmetry so that quantities such as Hamiltonians, density matrices, or one-particle states are real-valued. We set ℏ = 1 so energies and frequencies are interchangeable. Wherever possible, we use matrix notation. For example, the one-particle electron Green’s function for a time-independent system in the frequency domain, G ( x, x $^{′}$, ω ), is a function of three arguments. The x and x ′ arguments include both spatial coordinates and spin: x = ( ⃗ r, σ ) where ⃗ r is a three-vector and σ = ± 1 labels the two spin projections. In matrix notation, we write the matrix G ( ω ) whose matrix elements are 〈 x | G ( ω ) | x $^{′}$〉 = G ( x, x $^{′}$, ω ). The time-ordered, non-interacting, one-particle Green’s function G$_{0}$ ( ω ) is given by The eigenenergies ϵ$_{n}$ have imaginary parts Im ϵ$_{n}$ = sgn( µ − ϵ$_{n}$ )0 + where 0 + is a positive infinitesimal and µ is the Fermi energy. Thus the poles of G$_{0}$ are above the real ω axis for occupied or valence states, labeled by v so ϵ$_{v}$ < µ , and below the real axis for unoccupied or conduction states, labeled by c so ϵ$_{c}$ > µ . The non-interacting one-particle Hamiltonian H$_{0}$ generates the orthonormal eigenstates | n 〉 and real eigenvalues ϵ$_{n}$ , 2. DEFINITIONS & NOTATION G$_{0}$ ( ω ) = ( ωI − H$_{0}$ ) − 1 = ∑ n | n 〉〈 n | ω − ϵ$_{n}$ . (1) H$_{0}$ | n 〉 = ϵ$_{n}$ | n 〉 . The wave functions in coordinate space are denoted as ψ$_{n}$ ( x ) = 〈 x | n 〉 . In the time domain, 4