H$_4$ system have been calculated in the literature. Some of the earlier studies treated the diatomic molecules as a rigid rotor (RR) as in the work of Zarur and Rabitz (ZR) [@Zarur74], Schaefer and Köhler (SK) [@Schaefer89], Diep and Johnson (DJ) [@Diep00], and the recent study of Patkowski et al. [@Patkowski08]. Some of these PESs have been employed in several studies of rotational energy transfer in H$_2-$H$_2$ collisions within the rigid-rotor approximation at thermal energies [@Zarur74; @Forrey01; @Forrey02; @Mate05; @Lee06; @Sultanov06a; @Sultanov06b] and also at ultralow energies [@Forrey01; @Forrey02; @Lee06]. Several full-dimensional potential surfaces have also been reported for the H$_4$ system in recent years. They include the PESs of Schwenke [@Schwenke88], Aguado, Su[á]{}rez and Paniagua (ASP) [@Aguado94], Boothroyd, Martin, Keogh and Peterson (BMKP) [@Boothroyd02], and the more recent work of Hinde [@Hinde08]. These PESs have been adopted in a number of quantum scattering calculations of H$_2-$H$_2$ that go beyond the rigid rotor approximation [@Flower98a; @Flower98b; @Flower99; @Flower00a; @Flower00b]. The BMKP and Schwenke PESs were recently employed in full-dimensional quantum calculations by Pogrebyna et al. [@Pogrebnya02; @Pogrebnya03]. They reported the first quantum calculation of vibrational relaxation in collisions between H$_2(v=1,j=0)$ and H$_2(v=0,j=0)$ molecules using a time-independent quantum mechanical (TIQM) method 