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 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 ab initio computational methods through diagrammatic resummation techniques in future.