  The binding energies (BE) of molecules on the interstellar grains are crucial
in the chemical evolution of the interstellar medium (ISM). Both temperature
programmed desorption (TPD) laboratory experiments and quantum chemistry
computations have often provided, so far, only single values of the BE for each
molecule. This is a severe limitation, as the ices enveloping the grain mantles
are structurally amorphous, giving rise to a manifold of possible adsorption
sites, each with different BEs. However, the ice amorphous nature prevents the
knowledge of structural details, hindering the development of a common accepted
atomistic icy model. In this work, we propose a computational framework that
closely mimics the formation of the interstellar grain mantle through a water
by water accretion. On that grain, an unbiased random (but well reproducible)
positioning of the studied molecule is then carried out. Here we present the
test case of NH$_3$, an ubiquitous species in the molecular ISM. We provide the
BE distribution computed by a hierarchy approach, using the semiempirical
xTB-GFN2 as low-level method to describe the whole icy cluster combined with
the B97D3 DFT functional as high-level method on the local zone of the NH$_3$
interaction. The final ZPE corrected BE is computed at
ONIOM(DLPNO-CCSD(T)//B97D3:xTB-GFN2) level, ensuring the best cost/accuracy
ratio. The main peak of the predicted NH$_3$ BE distribution is in agreement
with experimental TPD and literature computed data. A second broad peak at very
low BE values is also present, never detected before. It may provide the
solution to a long-standing puzzle about the presence of gaseous NH$_3$
observed also in cold ISM objects.
