to (0, 0) becomes anomalously weak \[\]. The increased state density around the singularity has been argued to enhance $T_c$ of the $d$-wave superconductivity \[\]. The value of $t'$ was argued to correlate with the maximum $T_c$ values of cuprate subfamilies \[\]. Quantum M. C. studies indicated that $t'$ enables the bulk superconductivity in the 2D Hubbard model \[\].

We have examined the $t'$-dependence of the energy gain, taking account of $H_{nnn}$ in (\[eq:H2\]) in our model. In the case of $N_e$ = 32 on the $6 \times 6$ lattice, the energy minimum became slightly deeper with the change of $t'$ from zero to $-0.25$, but at $t' = 0.25$ the minimum became shallower by a factor of 2. In the case of $N_e = 84$ on the $10 \times 10$ lattice the energy gain was maximized around $t' \cong -0.1 \sim -0.15$ as shown in Fig. \[fig:DEg-t’\]. The error bar for $\Delta E_g/N_s$ in the figure is about $0.0003$. $\Delta$ is also displayed in the figure with the right-hand-side vertical scale. The error bar for $\Delta$ is about 10 percents of the magnitude. With a positive value of $t' = 0.10$, the energy gain quickly decreases, nearly vanishing when 