Several "reconstructive" proposals for density matrix functional theory are investigated, each of which expresses the two-electron density matrix, and therefore the electronic energy, as a functional of the natural orbitals and their occupation numbers. It is shown that for each of these functionals, half of the parallel-spin eigenvalues of the reconstructed two-electron density matrix are necessarily negative. Illustrative all-electron calculations for Be and LiH, in a variety of Gaussian basis sets, demonstrate that these spurious negative eigenvalues lower the electronic energy substantially. In spite of this, there is no indication that the variationally optimized energy diverges as the basis set approaches completeness, as has been suggested based on calculations with a small number of active orbitals. The apparent variational instability reported previously is attributed to qualitative differences between the minimal-basis and extended-basis potential curves, for certain functionals. However, we identify one functional that yields accurate LiH potential curves-comparable to full configuration interaction results-in both minimal and extended basis sets. Explicitly antisymmetric reconstructions are recommended as a remedy for the positivity problem. (C) 2003 American Institute of Physics.