as a stationary point of $\Omega_{\bf t}[{\bf \Sigma}]$ when restricting the search to local self-energies representable by a SIAM.

Within the DMFT the computation of the self-energy requires an iterative procedure: ${\bf \Sigma}_{\rm old} \mapsto {\bf \Sigma}_{\rm new}$. Here it turns out that this corresponds to a certain (discrete) path on the hypersurface of SIAM trial self-energies. Convergence of the iteration (${\bf \Sigma}_{\rm old} = {\bf \Sigma}_{\rm new}$), however, is by no means guaranteed physically but depends on the contracting properties of the map ${\bf \Sigma}_{\rm old} \mapsto {\bf \Sigma}_{\rm new}$. The self-energy-functional approach offers an alternative as instead of solving Eq. (\[eq:dmft\]) one may calculate $\Omega_{\bf t}[{\bf \Sigma}({\bf t}')]$ by Eq. (\[eq:vvv\]) and determine the stationary point. Hence, the DMFT can also be obtained by a more [*direct*]{} computation avoiding any iterations – similar (in this respect) to the random-dispersion approximation. [@NG99] Note that in case of more than a single stationary point there is also an equally direct access to metastable phases.

For any inhomogeneous situation, Eq. (\[eq:dmft\]) represents a system of self-consistency equations to fix the parameters of non-equivalent impurity models labeled by the site index $i$. The models can be solved independently but are coupled indirectly 