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-inter-action correction. At large internuclear distance, the corrected potentials are superior to those obtained from Hartree– Fock and density functional theories.