  The probability that the primordial black hole (PBH) binaries formed in the
early Universe can be affected by the Hubble expansion of background, which is
non-negligible when the number density of PBHs is very low (it is actually this
case for supermassive PBHs). In this paper, taking into account the effect of
cosmic expansion on the comoving distance of PBH pairs, we worked out the
merger rate of PBHs with any extended mass function. The torques by all PBHs
and linear density perturbations are also considered. It is found that the
merger rate of PBH, $M\gtrsim 10^6M_\odot$, binaries is significantly lower for
$f_\text{pbh}\lesssim 0.01$ than expected.
