Tr} \ln (-{\bf G}') \: .
\label{eq:vvv}$$ Here it has been used that $\Omega_{{\bf t}'}[{\bf \Sigma}({\bf t}')] = \Omega'$, the exact grand potential of the reference system $H'$, and $({{\bf G}'_0}^{-1} - {\bf \Sigma}({\bf t}'))^{-1} = {\bf G}'$, the exact Green’s function of $H'$. Suppose that the reference system $H'$ is much simpler than the original system $H$ so that it can be solved exactly for any ${\bf t}'$ belonging to a certain subspace of the entire space of hopping parameters. The resulting Eq. (\[eq:vvv\]) is remarkable, as it shows that the functional $\Omega_{{\bf t}}[{\bf \Sigma}]$ can be evaluated rigorously for trial self-energies ${\bf \Sigma}={\bf \Sigma}({\bf t}')$ taken from the reference system $H'$.

This is the main result. Contrary to previous approaches (e.g. conserving theories, LDA), there is no need to approximate the functional dependence in a fundamental variational principle. Approximations are constructed by searching for a stationary point of $\Omega_{{\bf t}}[{\bf \Sigma}]$ on a [*restricted*]{} set of trial self-energies ${\bf \Sigma}({\bf t}')$.

The stationary point is determined by the Euler equation: $\partial \Omega_{\bf t}[{\bf \Sigma}({\bf t}')] / \partial {\bf t}'=0$. Calculating the derivative, $$T \sum_{\omega} \sum_{\alpha\beta}
   \left( 
   \frac{1}{{\bf G}_0^{-1} - {\bf \Sigma}({\bf t}')} 
   -  {\bf G}' \right)_{\beta \alpha} 
   \frac{\partial 