A unified diagrammatic formulation of single-reference and multi-reference random phase approximations: The particle–hole and particle–particle channels

A diagrammatic multi-reference generalization of many-body perturbation theory was recently introduced [Wang et al., J. Phys. Chem. Lett. 16, 3047 (2025)]. This framework allows us to extend single-reference (SR) Green's function methods defined at the diagrammatic level naturally to the multi-reference case, as previously exemplified by the formulation of multi-reference direct random phase approximation (MR-dRPA) and the multi-reference second-order screened exchange approximation. In this work, we further elaborate on this framework and use it to develop MR generalizations of two other RPA variants, namely, particle-hole (ph) RPA with exchange (MR-RPAx) and particle-particle RPA (MR-ppRPA). We define these two MR generalizations by infinite-order resummations of the generalized "ring" and "ladder" diagrams with antisymmetrized interaction vertices, respectively, which incorporate the contributions from the active-space connected two-body Green's functions. As for MR-dRPA, we derive unified sets of equations that hold at both SR and MR levels for RPAx and ppRPA, respectively. We perform numerical studies of prototypical systems using the three MR-RPA methods and carry out a perturbative analysis to gain a deeper understanding of their behaviors. We find that error cancellation between the second and third orders is a key factor for both SR-RPA and MR-RPA. In addition, we observe that MR-phRPA (MR-dRPA and MR-RPAx) and MR-ppRPA tend to overestimate and underestimate correlation energies, respectively, suggesting that better accuracy can be achieved by further combining these two channels in the future.