  Standard approximations for the exchange-correlation functional have been
found to give big errors for the linearity condition of fractional charges,
leading to delocalization error, and the constancy condition of fractional
spins, leading to static correlation error. These two conditions are now
unified for states with both fractional charge and fractional spin: the exact
energy functional is a plane, linear along the fractional charge coordinate and
constant along the fractional spin coordinate with a line of discontinuity at
the integer. This sheds light on the nature of the derivative discontinuity and
calls for explicitly discontinuous functionals of the density or orbitals that
go beyond currently used smooth approximations. This is key for the application
of DFT to strongly correlated systems.
