Generalized many-body perturbation theory for the electron correlation energy: multi-reference random phase approximation via diagrammatic resummation

Many-body perturbation theory (MBPT) based on Green's functions and Feynman diagrams provides a fundamental theoretical framework for various \emph{ab initio} computational approaches in molecular and materials science, including the random phase approximation (RPA) and $GW$ approximation. Unfortunately, this perturbation expansion often fails in systems with strong multi-reference characters. Extending diagrammatic MBPT to the multi-reference case is highly nontrivial and remains largely unexplored, primarily due to the breakdown of Wick's theorem. In this work, we develop a diagrammatic multi-reference generalization of MBPT for computing correlation energies of strongly correlated systems, by using the cumulant expansion of many-body Green's function in place of Wick's theorem. This theoretical framework bridges the gap between MBPT in condensed matter physics and multi-reference perturbation theories (MRPT) in quantum chemistry, which had been almost exclusively formulated within time-independent wavefunction frameworks prior to this work. Our formulation enables the explicit incorporation of strong correlation effects from the outset as in MRPT, while treating residual weak interactions through a generalized diagrammatic perturbation expansion as in MBPT. As a concrete demonstration, we formulate a multi-reference (MR) extension of the standard single-reference (SR) RPA by systematically resumming generalized ring diagrams, which naturally leads to a unified set of equations applicable to both SR and MR cases. Benchmark calculations on prototypical molecular systems reveal that MR-RPA successfully resolves the well-known failure of SR-RPA in strongly correlated systems. This theoretical advancement paves the way for advancing \emph{ab initio} computational methods through diagrammatic resummation techniques in future.