  We present an analytic proof demonstrating the equivalence between the Random
Phase Approximation (RPA) to the ground state correlation energy and a
ring-diagram simplification of the Coupled Cluster Doubles (CCD) equations. In
the CCD framework, the RPA equations can be solved in $\mathcal{O}(N^4)$
computational effort, where $N$ is proportional to the number of basis
functions.
