We present a zero temperature ensemble spin density functional theory. We discuss the ensemble quantities that arise from derivative discontinuities, including the nonvanishing asymptotic potential and band gap shift, in the context of the Kohn-Sham formalism, and hybrid exact exchange theories, such as the Hartree-Fock-Kohn-Sham formalism. We describe and implement a general method of calculating these quantities in atomic and molecular systems. Finally we discuss how our results explain the deficiencies of existing functionals, and how new functionals should be constructed, illustrating our conclusions by examining the dissociation of H-2(+).