Spurious self-interaction is shown to be responsible for essentially exact H-2 potential energy curves calculated using simple one-electron density matrix functionals. For molecules with more than two electrons, bond-stretching potentials are unrealistically shallow due to overcorrelation that is most severe in the separated-atom limit. In addition, too much population is shifted into orbitals beyond the formal valence shell. Both problems are remedied by a facile self-interaction correction. At large internuclear distance, the corrected potentials are superior to those obtained from Hartree-Fock and density functional theories. (C) 2003 Elsevier B.V. All rights reserved.