$U$.

$t'$-Dependence
---------------

In the 2D Hubbard and the 2D $d$-$p$ models, the one-electron state density increases peak-wise around the energy of the van Hove singularities located in the ${\mib k}$-space region around $(0, \pi)$ and $(\pi, 0)$. When $t'$ is negative in the 2D Hubbard model with $H_{nnn}$ the energy level of van Hove singularity moves toward the Fermi energy in the hole-doped systems, as is seen from eq. (\[eq:ek\]), which realizes a situation favorable to get superconductivity for most theories. In fact Shimahara and Takada showed that in the RPA framework $T_c$ increases with the introduction of $t'$ \[\]. Importance of $t'$ in the physics of high-$T_c$ cuprates was pointed out in \[\]. From the viewpoint of the two-band mechanism of superconductivity in which the pair-wise transfer of electrons from a certain ${\mib k}$-region to another ${\mib k}$-region promotes SC, the location of the ${\mib k}$-space region highly contributing to the state density in the neighborhood of the Fermi energy should lead to a larger energy gain in the SC state \[\]. The state density peak is known to be further enhanced with the increase of electron correlation, as the single-particle dispersion along the lines from these points 