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 H2 and LiH potential energy surfaces, keto-enol tautomerization energy barrier, van der Waals interactions between two H2 molecules, and the torsional potential energy surface of the ethylene molecule, C2H4. 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.