The interaction of -$Al(CH_3)_2$ with -OH on a fully OH-terminated Si (001) surface was studied using density functional theory. Two sites for $Al(CH_3)_3$ to react with the -OH on the surface were identified. The $-Al(CH_3)_2$ product energetically favored the dimer-row site rather than the inter-row site because the Al atom of $-Al(CH_3)_2$ at the dimer-row site was attracted by the lone pair electrons of the O atom in the neighboring -OH. The energy barrier for the transfer of the $-Al(CH_3)_2$ between the two sites was 0.11 eV, and therefore, the $-Al(CH_3)_2$ at the inter-row site can easily transfer to the dimer-row site at room temperature.