  We explore a new class of computationally feasible approximations of the
two-body density matrix as a finite sum of tensor products of single-particle
operators. Physical symmetries then uniquely determine the two-body matrix in
terms of the one-body matrix. Representing dynamical correlation alone as a
single tensor product results in a theory which predicts near zero dynamical
correlation in the homogeneous electron gas at moderate to high densities. But,
representing both dynamical and statistical correlation effects together as a
tensor product leads to the recently proposed ``natural orbital functional.''
We find that this latter theory has some asymptotic properties consistent with
established many-body theory but is no more accurate than Hartee-Fock in
describing the homogeneous electron gas for the range of densities typically
found in the valence regions of solids.
  PACS 71.10.-w 71.15.Mb, Accepted for publication in Physical Review B
