  Qualitatively incorrect results are obtained for the Mn dimer in density
functional theory calculations using the generalized gradient approximation
(GGA) and similar results are obtained from local density and meta-GGA
functionals. The bond between the atoms is predicted to be an order of
magnitude too strong and about an {\AA}ngstr{\o}m too short, and the coupling
is predicted to be ferromagnetic rather than antiferromagnetic. Explicit,
self-interaction correction (SIC) applied to a commonly used GGA energy
functional, however, provides close agreement with both experimental data and
high-level, multi-reference wave function calculations. These results show that
the failure is not due to strong correlation but rather the single electron
self-interaction that is necessarily introduced in estimates of the classical
Coulomb and exchange-correlation interactions when only the total electron
density is used as input. The corrected functional depends explicitly on the
orbital densities and can, therefore, avoid the introduction of self-Coulomb
interaction. The error arises because of over-stabilization of bonding
$d$-states in the minority spin channel resulting from an overestimate of the
$d$-electron self-interaction in the GGA exchange-correlation functional. Since
the computational effort in the self-interaction corrected calculations scales
with system size in the same way as for regular semi-local functional
calculations, this approach provides a way to calculate properties of Mn
nanoclusters as well as biomolecules and extended solids where Mn dimers and
larger cluster are present, while multi-reference wave function calculations
can only be applied to small systems.
