  Molecular hydrogen is the most abundant molecular species in the Universe.
While no doubts exist that it is mainly formed on the interstellar dust grain
surfaces, many details of this process remain poorly known. In this work, we
focus on the fate of the energy released by the H$_2$ formation on the dust icy
mantles, how it is partitioned between the substrate and the newly formed
H$_2$, a process that has a profound impact on the interstellar medium. We
carried out state-of-art \textit{ab-initio} molecular dynamics simulations of
H$_2$ formation on periodic crystalline and amorphous ice surface models. Our
calculations show that up to two thirds of the energy liberated in the reaction
($\sim$300 kJ/mol $\sim$3.1 eV) is absorbed by the ice in less than 1 ps. The
remaining energy ($\sim$140 kJ/mol $\sim$1.5 eV) is kept by the newly born
H$_2$. Since it is ten times larger than the H$_2$ binding energy on the ice,
the new H$_2$ molecule will eventually be released into the gas-phase. The ice
water molecules within $\sim$4 \AA ~from the reaction site acquire enough
energy, between 3 and 14 kJ/mol (360--1560 K), to potentially liberate other
frozen H$_2$ and, perhaps, frozen CO molecules. If confirmed, the latter
process would solve the long standing conundrum of the presence of gaseous CO
in molecular clouds. Finally, the vibrational state of the newly formed H$_2$
drops from highly excited states ($\nu = 6$) to low ($\nu \leq 2$) vibrational
levels in a timescale of the order of ps.
