Dissociative adsorption of water molecules on Si(001) clean surface becomes a common understanding now. The mechanism, however, has not been clarified yet. This is because it is always tacitly assumed that a water molecule is adsorbed to a surface and dissociates as a monomer. Therefore, we investigated the adsorption of water molecules on Si(001) clean surface based on the first-principles density-functional-theory (DFT) calculations and found that the adsorption energy as a water cluster dramatically increases at a silicon down-dimer site. This adsorption state accompanies a change in quality of hydrogen bond, and the water molecule can be easily dissociated via a proton-relay mechanism. Since this mechanism allows the dissociation between adjacent silicon dimers, the previous experimental results can be explained in more natural ways. We also obtained the result that implies the generality of this mechanism by analyzing of the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO) of the water monomer, dimer and Si(001) clean surface based on molecular orbital theory. These theoretical results propose the necessity of taking account of the multi-molecular process as well as the single molecular process.