Many-Body Second Order Green's Function Theory for Ab Initio Molecular Quantum Electrodynamics

In this work, we develop two many-body quantum electrodynamic methods to calculate the ground-state energies of strongly coupled light-matter molecular systems. Specifically, we extend the second-order many-body Green's function theory (GF2) for electronic systems to incorporate electron-boson couplings. We employ two ansätze to treat the bosonic part of the system, namely the coherent-state (CS) and Lang-Firsov (LF) transformed vacuum state. These are combined with the GF2 method to construct two new approaches, which we refer to as CS-GF2 and LF-GF2. We benchmark CS- and LF-GF2 by studying various molecular systems inside an optical cavity. We investigate H 2 and LiH potential energy surfaces, keto-enol tautomerization energy barrier, van der Waals interactions between two H 2 molecules and the torsional potential energy surface of the ethylene molecule, C 2H 4. Both methods provide similar energy profiles, with relatively small differences between CS-GF2 and LF-GF2. Comparisons with the reference calculations also reveal remaining limitations at stretched bond lengths and near the twisted ethylene configuration. We additionally benchmark CS-GF2 against exact diagonalization for a two-site Hubbard-Holstein model and observe that it closely reproduces the coupling-induced energy change in the weakly interacting regime while improving upon CS-MP2 in the strongly interacting regime. Overall, the results demonstrate that CS-GF2 and LF-GF2 provide self-consistent descriptions of correlated molecular electron-boson systems, offering a many-body Green's function alternative for studying ground-state cavity-modified systems.