FIG. 1: Schematic figure showing the simplest likely scenario for the Klein total energy functional. What is shown are level curves of the total energy functional F [ G ] as a function of the self- energy Σ$_{xc}$( ω ) that determines the Green’s function G via the Dyson Eq. (4) (not necessarily self- consistently). The horizontal axis represents self-energies that are static and Hermitian operators U$_{0}$ that will generate non-interacting Green’s functions G$_{0}$ via Eq. (3). The vertical axis represents the deviation of the self-energy from a static and Hermitian operator. The black circle represents the extremum of the total energy functional corresponding to the true self-consistent self-energy which must occur for a dynamic and non-Hermitian Σ$_{xc}$( ω ) since F has no extremum along the horizontal axis. minimization over allowed densities, or equivalently over any set of occupied orthonormal non-interacting wave functions, leads to a single minimum with the correct ground-state energy [1, 2]. For the Klein functional, the analogous approach fails completely because an unconstrained optimization over non-interacting Green’s functions drives the system to an unphysical minimum with negative infinite energy. The analogous situation in DFT would be the (fictitious) situation where the correct ground-state density locates a minimum but the total energy functional has no minimum or lower bound when evaluated on a subset of allowed densities. In our mind, there are two basic ways to overcome this hurdle. One could decide to solve the Dyson equation for the true self-consistent G and avoid the instabilities. However, this requires storing the entire Green’s function G ( x, x $^{′}$, ω ) — a full matrix as a function of continuous frequency ω — which is highly prohibitive in terms of storage and computation and unlikely to lead to a method that will deal with realistic systems with many atoms in the near future. The other approach is to stick with the simple and appealing picture of a noninteracting G$_{0}$ generated by a static and non-local U$_{0}$ but to constrain the extremization so as to avoid unphysical behavior. In other words, one constrains the allowed forms of U$_{0}$ in some way. In this light, the use of strictly local V$_{xc}$ , the QS GW , and the scCOHSEX 19