DIA, $n_{\rm s}=4$: dynamical-impurity approximation with $n_{\rm s}=4$ sites. DIA, $n_{\rm s}=\infty$: DMFT limit (circles: NRG, [@Bul99] line: ED using 8 sites). []{data-label="fig:zofu"}](fig5.eps){width="80mm"}

So far we discussed the case $U=4=W$ only. As a function of $U$ the half-filled paramagnetic Hubbard model at $T=0$ is expected to undergo a transition from a metal to a Mott-Hubbard insulator. [@Geb97] This is marked by a divergence of the effective mass or, equivalently, by a vanishing quasi-particle weight $z=1/(1-d \Sigma(\omega)/d\omega|_{\omega=0})$ as $U$ approaches a critical value $U_{\rm c}$ from below. [@GKKR96; @BR70] The result for $z(U)$ as obtained by the use of the two-site reference system (“dynamical impurity approximation”, DIA with $n_{\rm s}=2$) is shown in Fig. \[fig:zofu\]. As there are less degrees of freedom contained in $H'$, the approximation should be considered to be [*inferior*]{} as compared to the full DMFT the results of which are in Fig. \[fig:zofu\], too. It is remarkable that the simple $n_{\rm s}=2$-DIA (which requires the diagonalization of a dimer model only) yields $z(U)$ in an almost quantitative agreement with the full DMFT.

The results of the $n_{\rm s}=2$-DIA may also be compared with those of the recently developed “linearized” or “two-site” DMFT [@BP00; @Pot01] where the Hubbard 