The adsorption and dissociation behavior of water molecule below and above the critical dissociation temperatures were studied by first principles calculations. We found that water-molecule adsorption (surface coverage, θ=0.25) on the down Si atom of a Si dimer in two dimers surface model was 0.26 eV more favorable than that on the up Si atom. The activation energies of water molecule on the down Si atom for interdimer and intradimer dissociations were 0.17 eV and 0.20 eV, respectively. Due to the lower activation energy, the water molecule splits into H and OH immediately once it adsorbs on down Si atom of the Si (001) surface at room temperature. There were three different adsorption sites among four sites of the two dimers for the second water molecule (θ=0.5): one was preoccupied by OH of the first water molecule; up Si atom of the same-dimer with 76.3 % probability, up Si atom of the adjacent-dimer with 23.6 % probability, and down Si atom of the adjacent-dimer with 0.1 % probability. Thus, ½ monolayer of OH sites on the Si (001) surface are irregularly distributed when water molecules are adsorbed and dissociated at room temperature.