  Kohn-Sham density functional theory (DFT) has long struggled with the
accurate description of strongly correlated and open shell systems and
improvements have been minor even in the newest hybrid functionals. In this
Letter we treat the static correlation in DFT when frontier orbitals are
degenerate by the means of using a semidefinite programming (SDP) approach to
minimize the system energy as a function of the $N$-representable,
non-idempotent 1-electron reduced density matrix. While showing greatly
improved singlet-triplet gaps for linear density approximation and generalized
gradient approximation (GGA) functionals, the SDP procedure reveals flaws in
modern meta and hybrid GGA functionals, which show no major improvements when
provided with an accurate electron density.
