Connections between Richardson-Gaudin states, perfect-pairing, and pair coupled-cluster theory

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 multi-reference 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 states. We show that PP arises as an eigenvector of a simplified, reduced Bardeen-Cooper-Schrieffer 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 ansätze and open avenues for hybrid approaches that combine the strengths of orbital- and geminal-based methods.