  Tensor hypercontraction is a method that allows the representation of a
high-rank tensor as a product of lower-rank tensors. In this paper, we show how
tensor hypercontraction can be applied to both the electron repulsion integral
(ERI) tensor and the two-particle excitation amplitudes used in the parametric
reduced density matrix (pRDM) algorithm. Because only O(r) auxiliary functions
are needed in both of these approximations, our overall algorithm can be shown
to scale as O(r4), where r is the number of single-particle basis functions. We
apply our algorithm to several small molecules, hydrogen chains, and alkanes to
demonstrate its low formal scaling and practical utility. Provided we use
enough auxiliary functions, we obtain accuracy similar to that of the
traditional pRDM algorithm, somewhere between that of CCSD and CCSD(T).
