The short time dynamics of tagged pairs of molecules that, at the initial time, are in the first coordination shell, is investigated in supercooled liquid water at 245 K by using the molecular dynamics technique with the four-points transferable intermolecular potential of Jorgensen [J. Chem. Phys. 79, 926 (1983)]. The instantaneous normal mode approach and the results of the local structure investigations are exploited to build up a correlation function of the relative displacements that represents the projection of normal modes along the initial center of mass separation vector. By imposing simple constraints to the initial dynamical conditions, localized damped oscillations of the centers of mass are detected along the hydrogen bond directions. The corresponding density of states shows a maximum around the frequency of 230 cm(-1) and its shape agrees with the frequency contributions expected from the translational phonon branches of ice. Total and radial correlation functions of the relative velocity are also computed. The radial component is dominated by oscillations at the frequency of 230 cm(-1); it compares fairly well with that derived from the translational density of states, thus supporting the reliability of the employed method. A slower component of motion, absent along the hydrogen bond direction, is derived; in the density of states, it produces a contribution peaked around 50 cm(-1).