due to the matrix inversion in (\[eq:dmft\]). This exactly recovers the DMFT generalized to systems with reduced translational symmetry. [@PN99c; @PN99d]

A consistent ED method {#sec:ed}
======================

A brief discussion of two limiting cases of the Hubbard model may be instructive. Consider the band limit with $U=0$ first. Here $H = \sum_{ij\sigma} t_{ij} c_{i\sigma}^\dagger c_{j\sigma}$ describes a system of non-interacting electrons. This case is exceptional as obviously the functional $F[{\bf \Sigma}] 
\equiv 0$ and therefore $\Omega_{\bf t}[{\bf \Sigma}] = {\rm Tr} \ln 
(- ({\bf G}_0^{-1} - {\bf \Sigma})^{-1})$. Any valid reference system $H'$ must have the same (i.e. a vanishing) interaction part as $H$, and thus ${\bf \Sigma}({\bf t}') \equiv 0$ and $\Omega_{\bf t}[{\bf \Sigma}({\bf t}')] = {\rm Tr} \ln (-{\bf G}_0^{-1})$, the exact grand potential for non-interacting electrons.

The atomic limit, $H = \sum_{i\sigma} ( t_{0} c_{i\sigma}^\dagger c_{i\sigma}
+ (U/2) n_{i\sigma} n_{i-\sigma})$ is more interesting as $\Phi[{\bf G}]$ and $F[{\bf \Sigma}]$ cannot be constructed explicitly. Within the self-energy-functional approach one has to compute ${\bf \Sigma({\bf t}')}$, ${\bf G}'$, and $\Omega'$ for a suitable reference system $H'$ and to insert into Eq. (\[eq:vvv\]) for optimization. The only meaningful choice for the reference system is $H'=H$ in this case. Obviously, this 