  The electronic structure and magnetic moments of free Mn$_2^+$ and Mn$_3^+$
are characterized by $2p$ x-ray absorption and x-ray magnetic circular
dichroism spectroscopy in a cryogenic ion trap that is coupled to a synchrotron
radiation beamline. Our results show directly that localized magnetic moments
of 5 $\mu_B$ are created by $3d^5 (^6\mathrm{S})$ states at each ionic core,
which are coupled in parallel to form molecular high-spin states via indirect
exchange that is mediated in both cases by a delocalized valence electron in a
singly-occupied $4s$ derived orbital with an unpaired spin. This leads to total
magnetic moments of 11 $\mu_B$ for Mn$_2^+$ and 16 $\mu_B$ for Mn$_3^+$, with
no contribution of orbital angular momentum.
