The energy ratio, α (Eq. 11), the maximum eigenvalues (γ) of 2 Γ αα and 2 Γ αβ , the maximum eigenvalues (δ) of 2 ∆ αα and 2 ∆ αβ , the trace of 2 ∆, and the square norm of 2 ∆ with respect to the exact ground-state wavefunction of the 6-site half-filled Hubbard model are shown in Fig. as a function of U with t=1. As the ratio U/t increases, we expect the electronic structure to change from the non-interacting limit with freely delocalized electrons to a strongly interacting regime in which electrons are localized on each site, exhibiting antiferromagnetic order. All of the 2 ∆ quantities are seen to monotonically increase in magnitude, as expected. The ratio α is always negative due to the denominator, and (almost by construction) increases in magnitude as the U/t ratio increases. The largest eigenvalues of 2 Γ are found to decrease, rather surprisingly, though this will be discussed in detail in the final section.