${\bf \Sigma}({\bf t}') = {{\bf G}'_0}^{-1} - {{\bf G}'}^{-1}$. Since the self-energy is local, as in the DMFT, the lattice structure enters via the free ($U=0$) density of states only. Therefore, the ${\bf k}$-sum which appears in the first trace in Eq. (\[eq:vvv\]) can be performed conveniently by a one-dimensional density-of-states integration. A semi-elliptical free density of states with the band width $W=4$ is used for the calculations. This sets the energy scale for the results discussed below.

Fig. \[fig:e1\] shows the grand potential $\Omega$ and the three different contributions as functions of $V$. The interaction is kept fixed at $U=W=4$. The remaining variational parameters are set to $\epsilon_c=0$ and $\epsilon_a=U/2=2$, as required by particle-hole symmetry. Each of the three contributions strongly depends on $V$ and none of them has a stationary point at a finite $V\ne 0$. Two of them show a singular behavior at $V=0$. Contrary, the resulting $\Omega$ is regular for any $V$ and shows a much weaker $V$ dependence. There are three stationary points which are indicated by the arrows. The maximum at $V=0$ corresponds to an insulating phase since ${\Sigma}(\omega)$ for $n_{\rm s}=2$ and $V=0$ is the Hubbard-I self-energy which implies a vanishing spectral density 