Tensor product expansions for correlation in quantum many-body systems

We explore a 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 that 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 Hartree-Fock in describing the homogeneous electron gas for the range of densities typically found in the valence regions of solids.