viewpoint, working with an arbitrary function G ( x, x $^{′}$, ω ) of three variables is computation- ally very demanding so that any simplifying assumptions on the form of G are enormously helpful in practice. In what follows, we will restrict G ( ω ) to be of non-interacting form, and the next section explains why this is a sensible choice. Clearly some approximations are required to make progress. We will first replace the true interacting G ( ω ) by something simpler in order to reduce the search space for the extremum. Restricting to noninteracting Green’s functions that are generated by Hermitian Hamiltonians is sensible from a physical viewpoint: we will try to search for the “best” noninteracting picture of the electronic system. As an added benefit, we can employ known algorithms for Hermitian and orthonormal eigensystems. In fact, this restriction greatly simplifies the structure of F . This is because F does not in fact depend on the choice of U$_{0}$ or equivalently G$_{0}$ . Namely, δF/δU$_{0}$ = 0 for fixed G . Physically, this is sensible since the final result for G should not depend on the arbitrary initial guess G$_{0}$ . For example, in the Dyson Eq. (4), we remove U$_{0}$ and replace it by the self-energy. To demonstrate this lack of dependence, we use Eq. (9) to find the variation of F versus G$_{0}$ at fixed G : Using G$_{0}$ ( ω ) = ( ωI − H$_{0}$ ) − 1 and the variation of an inverse δ [ A − $^{1}$] = − A − $^{1}$[ δA ] A − $^{1}$, we have Plugging these in gives Using the cyclicity of the trace, this simplifies to where we have traced over the orthonormal basis of eigenstates | n 〉 and used the diagonal nature of H$_{0}$ and G$_{0}$ in this basis. The e iω 0 + factor has us close the integral over the upper ω complex half-plane, 4. LACK OF DEPENDENCE OF F ON G$_{0}$ δF ∣ ∣ G = ∫ ∞ −∞ dω 2 πi e iω 0 $^{+}$tr $^{{}$[ δH$_{0}$ ] G$_{0}$ ( ω ) + H$_{0}$ [ δG$_{0}$ ( ω )] − [ δG$_{0}$ ( ω ) − $^{1}$] G ( ω ) + G ( ω ) − $^{1}$G$_{0}$ ( ω )[ δG$_{0}$ ( ω ) − $^{1}$] G ( ω ) − [ δU$_{0}$ ] G ( ω ) . } δH$_{0}$ = δU$_{0}$ , δG$_{0}$ ( ω ) − 1 = − δU$_{0}$ , δG$_{0}$ ( ω ) = G$_{0}$ ( ω )[ δU$_{0}$ ] G$_{0}$ ( ω ) . (11) δF ∣ ∣ G = ∫ ∞ −∞ dω 2 πi e iω 0 $^{+}$tr $^{{}$[ δU$_{0}$ ] G$_{0}$ ( ω ) + H$_{0}$G$_{0}$ ( ω )[ δU$_{0}$ ] G$_{0}$ ( ω ) + [ δU$_{0}$ ] G ( ω ) − G ( ω ) − $^{1}$G$_{0}$ ( ω )[ δU$_{0}$ ] G ( ω ) − [ δU$_{0}$ ] G ( ω ) $^{}}$. δF ∣ ∣ G = ∫ ∞ −∞ dω 2 πi e iω 0 $^{+}$tr $^{{}$G$_{0}$ ( ω ) H$_{0}$G$_{0}$ ( ω ) δU$_{0}$ } = ∫ ∞ −∞ dω 2 πi e iω 0 + ∑ n ϵ$_{n}$ 〈 n | δU$_{0}$ | n 〉 ( ω − ϵ$_{n}$ ) 2 δF ∣ ∣ G = ∮ dω 2 πi ∑ n ϵ$_{n}$ 〈 n | δU$_{0}$ | n 〉 ( ω − ϵ$_{n}$ ) 2 . 8