  Gas-grain and gas-phase reactions dominate the formation of molecules in the
interstellar medium (ISM). Gas-grain reactions require a substrate (e.g. a dust
or ice grain) on which the reaction is able to occur. The formation of
molecular hydrogen (H$_2$) in the ISM is the prototypical example of a
gas-grain reaction. In these reactions, an atom of hydrogen will strike a
surface, stick to it, and diffuse across it. When it encounters another
adsorbed hydrogen atom, the two can react to form molecular hydrogen and then
be ejected from the surface by the energy released in the reaction. We perform
in-depth classical molecular dynamics (MD) simulations of hydrogen atoms
interacting with an amorphous water-ice surface. This study focuses on the
first step in the formation process; the sticking of the hydrogen atom to the
substrate. We find that careful attention must be paid in dealing with the
ambiguities in defining a sticking event. The technical definition of a
sticking event will affect the computed sticking probabilities and
coefficients. Here, using our new definition of a sticking event, we report
sticking probabilities and sticking coefficients for nine different incident
kinetic energies of hydrogen atoms [5 K - 400 K] across seven different
temperatures of dust grains [10 K - 70 K]. We find that probabilities and
coefficients vary both as a function of grain temperature and incident kinetic
energy over the range of 0.99 - 0.22.
