Therefore, global minimization of F [ G$_{0}$, G$_{0}$ ] over all allowed G$_{0}$ — more precisely over all non-local potentials U$_{0}$ that generate the G$_{0}$ — is guaranteed to give unphysical results. However, one may still hope to find a local minimum or an extremum: after all, the Luttinger- Ward approach only guarantees that the true G is an extremum of the energy functional. However, this is also not possible because ∂F [ G$_{0}$, G$_{0}$ ] /∂ϵ$_{n}$ is never zero. Referring back to Eq. (28) in Section 6, we see that the derivative versus ϵ$_{n}$ is always positive for unoccupied states and always negative for occupied states. Hence, a search for an extremum also drives us to the unphysical global minimum. No other extremum exists because the derivatives are never zero. This proof confirms the suggestion [54] that direct minimization of the GW -RPA total energy will give meaningless results. 

We now discuss the meaning and relation of this result to what is known in the literature. A number of studies have used local (in coordinate x ) exchange-correlation potentials V$_{xc}$ = U$_{0}$ − φ$_{H}$ when generating G$_{0}$ for atoms and molecules. The choices have included LDA, GGA, or optimized effective potentials [33–35]. In addition, a more recent work [37] has solved the linearized Sham-Sch¨ ulter (LSS) equation for spherical atoms to find the optimal local V$_{xc}$ . No sign of any instability has been found in any of these works and the numerical evidence shows that the LSS-optimal local V$_{xc}$ locates an energy minimum [37]. In addition, there are strong theoretical arguments for why the total energy functional will have a minimum when varied over local potentials [35]. 

Unfortunately, the optimal local potential is not necessarily the best potential in terms of physical predictions. In practice, using a non-local potential to generate G$_{0}$ can greatly improve results: e.g. , for atoms and diatomic molecules, the Hartree-Fock non-local potential yields total energies that are essentially indistinguishable from those given by using the exact, self-consistent Green’s function G [35]. Therefore, including some nonlocality in the potential U$_{0}$ is important physically: for atoms and small molecules, the Hartree-Fock description is significantly closer to the true Green’s function than any description based on a local potential [35]. This result is sensible since the true Green’s function G obeying Dyson’s Eq. (4) is generated by a non-local self-energy operator Σ$_{xc}$( ω ), and we would expect a non-local U$_{0}$ to be a better approximation than a local operator. 

Alas, our proof above shows that choosing the proper form of nonlocality is not straight- forward. The na¨ıve idea of extremizing the total energy functional over all non-local U$_{0}$ to find the “best” G$_{0}$ leads to unphysical results. For example, the simple and appealing idea of holding the wave functions ψ$_{n}$ ( x ) fixed and varying the eigenenergies ϵ$_{n}$ to find an improved band structure via optimizing the total energy leads to the unphysical minimum. Of course, we know that there is an exact G that extremizes the total energy functional and solves the Dyson equation self-consistently: there are examples of such calculations in the literature for model systems such as atoms, small molecules, or the electron gas [23–26, 35]. The problem is that the energy functional has no extremum in the subspace of Green’s functions that correspond to non-interacting Green’s functions. To put this in pictures, the simplest likely scenario is illustrated schematically in Figure 1 where one assumes that the total energy functional has a single extremum corresponding to the true Green’s function. Obviously, that extremum must occur for a dynamic and non-local self-energy Σ$_{xc}$( x, x $^{′}$, ω ). 

The situation here is quite different from DFT where one can represent the electron density in terms of occupied non-interacting wave functions and where an unconstrained 

