\Sigma}] / \delta {\bf \Sigma} = 0 
  \; \Leftrightarrow \; {\bf G}[{\bf \Sigma}] 
  = ({\bf G}_0^{-1} - {\bf \Sigma})^{-1} \: .
\label{eq:stat}$$ Thus $\Omega_{\bf t}[{\bf \Sigma}]$ is stationary at the exact (physical) self-energy and its value is the exact grand potential of the system. Again, the problem is that the functional $\Omega_{\bf t}[{\bf \Sigma}]$ is in general not known explicitly.

As the domain of the self-energy functional $\Omega_{\bf t}[{\bf \Sigma}]$ we define the class of all ${\bf t}'$ representable self-energies. ${\bf \Sigma}$ is termed ${\bf t}'$ representable, if there is a set of hopping parameters ${\bf t}'$ such that ${\bf \Sigma}$ is the exact self-energy of the model $H_0({\bf t}') + H_1 ({\bf U})$. This implies that any self-energy in the domain of $\Omega_{\bf t}[{\bf \Sigma}]$ can be parameterized as ${\bf \Sigma} = {\bf \Sigma}({\bf t}')$. The interaction parameters ${\bf U}$ are taken to be fixed. Suppose we are interested in the model $H = H_0({\bf t}) + H_1 ({\bf U})$. Then the function $\Omega_{\bf t}({\bf t}') \equiv \Omega_{\bf t}[{\bf \Sigma}({\bf t}')]$ is stationary at ${\bf t}' = {\bf t}$. Thus $\partial \Omega_{\bf t}({\bf t}') / \partial {\bf t}' = 0$.

It is important to note that $F[{\bf \Sigma}]$ is 