An improved density matrix functional (DMF) combining the properties of the "corrected Hartree" (CH) and "corrected Hartree-Fock" (CHF) approximations is proposed. Functionals of the CH/CHF type and the closely related natural orbital functional of Goedecker and Umrigar (GU) are tested in fully variational finite basis set calculations of light atoms, the lowest energy singlet methylene, and, for the first time, potential energy curves of diatomic molecules. Although CH/CHF-style DMFs may give reasonable energies for atoms and molecules near equilibrium geometries, they predict unrealistically shallow minima in the potential energy curves for diatomic molecules with more than two electrons. The calculated CH and CHF molecular dissociation curves exhibit the same patterns of over- and under-correlations as the corresponding correlation energy plots for the homogeneous electron gas undergoing a transition from high to low densities. In contrast, the GU functional yields not only accurate atomic and molecular energies but also plausible dissociation curves. The reasons behind the observed performance are analyzed.