$t' = 0.25$. It also nearly vanishes for $t' \sim -0.40$. Figure \[fig:cor2\] shows the SC pair correlation functions as functions of the distance between pair operators in the optimal state for the case of $t' = - 0.10$. At the farthest distance their values are twice larger that those for $t' = 0$ (Fig \[fig:cor1\]), as expected. Thus, an appropriate negative value of $t'$ enhances the $d$-wave SC in the hole-doped case in the Hubbard model, which is in qualitative agreement with the above-mentioned references. The $t'$-dependences of the energy gain and $\Delta$ in the case of $N_e = 80$ exhibited in Fig. \[fig:DEg-t’2\] show similar features as in Fig. \[fig:DEg-t’\].

When one plots one-electron levels (Fig. \[fig:levels\]), $\varepsilon_{\mib k} = \xi_{\mib k} + \mu$, for ${\mib k}$-points in the non-interacting limit as a function of $t'$, a few occupied levels and a few empty levels are found to be bunched into a close neighborhood of the Fermi energy $\varepsilon_{\mbox{\scriptsize F}}$ within $\pm 0.07$ as $t'$ takes $-0.1 \sim -0.15$. These levels are located in the ${\mib k}$-space mainly in the neighborhood of the level of the van Hove singularity. Since each one-electron level is orbitally four- (or rarely 