I . Second, it is bound from above by zero, as the spectral expression is always negative. Third, it is not bound from below, as we can simply choose Ĥ(0) such that E I=1 is arbitrarily close to E (0) . This implies that there exists some Ĥ(0) which recreates any possible correlation energy, including the exact correlation energy and which, by the previous discussion, only depends on ρ.