Motivated by the very low diffusivity recently found in ab initio simulations of liquid water, we have studied its dependence with temperature, system size, and duration of the simulations. We use ab initio molecular dynamics (AIMD), following the Born-Oppenheimer forces obtained from density-functional theory (DFT). The linear-scaling capability of our method allows the consideration of larger system sizes (up to 128 molecules in this study), even if the main emphasis of this work is in the time scale. We obtain diffusivities that are substantially lower than the experimental values, in agreement with recent findings using similar methods. A fairly good agreement with D(T) experiments is obtained if the simulation temperature is scaled down by approximate to20%. It is still an open question whether the deviation is due to the limited accuracy of present density functionals or to quantum fluctuations, but neither technical approximations (basis set, localization for linear scaling) nor the system size (down to 32 molecules) deteriorate the DFT description in an appreciable way. We find that the need for long equilibration times is consequence of the slow process of rearranging the H-bond network (at least 20 ps at AIMDs room temperature). The diffusivity is observed to be very directly linked to network imperfection. This link does not appear an artifact of the simulations, but a genuine property of liquid water.(C) 2004 American Institute of Physics.