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

The Ground State Correlation Energy of the Random Phase Approximation from a Ring Coupled Cluster Doubles Approach