. This then links directly to the excited-state energies of the corresponding (N + 1) and (N -1) systems. Extensive numerical evidence supports this claim. Thus, the excited-state energies of N electron systems can be obtained from ground-state calculations on the (N -1) or (N +1) electron systems. that is equivalent to making a wave-function description of the combined system. When the electron number in the open quantum subsystem is varied between two adjacent integers, its exact total energy and electron density vary linearly with the electron number (piecewise linearity), so the exact energy minimizes at an integer electron number. This has important practical consequences. In particular, local and semilocal approximations predict incorrect energies and densities for a diatomic molecule AB in the dissociation limit. In fact, these approximations are much more accurate for integer than for fractional electron number. This problem still plagues density functional approximations. A non-self-consistent cure is to evaluate the approximate functionals on Hartree-Fock densities, which localize an integer charge around each separated nucleus. 118 Doing that also cures some related problems, such as spurious charge transfers at smaller internuclear separations.