  We show that using complex, spin-restricted orbitals (cR) in Kohn-Sham
density functional theory (KS-DFT) allows one to access a new class of
densities that is not accessible by either spin-restricted (RKS) or
spin-unrestricted (UKS) orbitals. We further show that the real part of a cRKS
density matrix can be non-idempotent when the imaginary part of the density
matrix is not zero. Using cRKS orbitals shows significant improvements in the
triplet-singlet gaps of the TS12 benchmark set for the SPW92, PBE, BLYP, and
TPSS functionals. Moreover, it was shown that RKS and UKS yield qualitatively
wrong charge densities and spin densities, respectively, leading to worse
energetics. We demonstrate that the modern functionals SCAN, MN15-L, and B97M-V
show surprisingly no improvement even with a qualitatively more accurate
density from cRKS orbitals. To this end, our work not only paves the way to
escape the symmetry dilemma whenever there exists a cRKS solution, but also
suggests a new route to design better approximate XC functionals.
