Connections between Richardson-Gaudin States, Perfect-Pairing, and Pair Coupled-Cluster Theory

Abstract: Slater determinants underpin most electronic structure methods, but orbital-based approaches often struggle to describe strong correlation efficiently. Geminal-based theories, by contrast, naturally capture static correlation in bond-breaking and multireference problems, though at the expense of implementation complexity and limited treatment of dynamic effects. In this work, we examine the interplay between orbital and geminal frameworks, focusing on perfect-pairing (PP) wavefunctions and their relation to pair coupled-cluster doubles (pCCD) and Richardson-Gaudin (RG) states. We show that PP arises as an eigenvector of a simplified reduced Bardeen-Cooper-Schrieffer (BCS) Hamiltonian expressed in bonding/antibonding orbital pairs, with the complementary eigenvectors enabling a systematic treatment of weak correlation. Second-order Epstein-Nesbet perturbation theory on top of PP is found to yield energies nearly equivalent to pCCD. These results clarify the role of pair-based ans\"atze and open avenues for hybrid approaches that combine the strengths of orbital- and geminal-based methods.