  Electronic states with fractional spins arise in systems with large static
correlation (strongly correlated systems). Such fractional-spin states are
shown to be ensembles of degenerate ground states with normal spins. It is
proven here that the energy of the exact functional for fractional-spin states
is a constant, equal to the energy of the comprising degenerate pure spin
states. Dramatic deviations from this exact constancy condition exist with all
approximate functionals, leading to large static correlation errors for
strongly correlated systems, such as chemical bond dissociation and band
structure of Mott insulators. This is demonstrated with numerical calculations
for several molecular systems. Approximating the constancy behavior for
fractional spins should be a major aim in functional constructions and should
open the frontier for DFT to describe strongly correlated systems. The key
results are also shown to apply in reduced density-matrix functional theory.
