Adsorption of a water molecule on a Si (001) surface and its dissociation were studied using density functional theory to study the distribution of -OH fragments on the Si surface. The Si (001) surface was composed of Si dimers, which buckle in a zigzag pattern below the order-disorder transition temperature to reduce the surface energy. When a water molecule approached the Si surface, the O atom of the water molecule favored the down-buckled Si atom, and the H atom of the water molecule favored the up-buckled Si atom. This is explained by the attractions between the negatively charged O of the water and the positively charged down-buckled Si atom and between the positively charged H of the water and the negatively charged up-buckled Si atom. Following the adsorption of the first water molecule on the surface, a second water molecule adsorbed on either the inter-dimer or intra-dimer site of the Si dimer. The dipole-dipole interaction of the two adsorbed water molecules led to the formation of the water dimer, and the dissociation of the water molecules occurred easily below the order-disorder transition temperature. Therefore, the 1/2 monolayer of -OH on the water-terminated Si (001) surface shows a regular distribution. The results shed light on the atomic layer deposition process of alternate gate dielectric materials, such as $HfO_2$.