  Using a state-of-the art combination of density functional theory and
impurity solver techniques we present a complete and parameter-free picture of
the Kondo effect in the high-spin ($S=3/2$) coordination complex known as
Manganese Phthalocyanine adsorbed on the Pb(111) surface. We calculate the
correlated electronic structure and corresponding tunnel spectrum and find an
asymmetric Kondo resonance, as recently observed in experiments. Contrary to
previous claims, the Kondo resonance stems from only one of three possible
Kondo channels with origin in the Mn 3d-orbitals, its peculiar asymmetric shape
arising from the modulation of the hybridization due to strong coupling to the
organic ligand. The spectral signature of the second Kondo channel is strongly
suppressed as the screening occurs via the formation of a many-body singlet
with the organic part of the molecule. Finally, a spin-1/2 in the 3d-shell
remains completely unscreened due to the lack of hybridization of the
corresponding orbital with the substrate, hence leading to a spin-3/2
underscreened Kondo effect.
