On intermolecular interactions in the Hamiltonian used in polaritonic chemistry

Experiments have shown that strong coupling between molecular excitations and a mode of a Fabry--P\'erot cavity can significantly alter molecular properties, such as reaction rates and equilibrium constants. However, in spite of the large body of theoretical work, the mechanism behind this change is still not well understood. In order to make progress, we first take a step back and investigate the appropriateness of the Hamiltonian that most recent studies are based on. In particular, we investigate the dipole self-energy, which can be divided into in self terms and cross terms. While the self terms are an indispensable part of the Hamiltonian, the cross terms -- which have received attention as they seem to mediate distance-independent interactions between all molecules in the cavity -- are known to, under certain conditions, cancel exactly with the usually neglected intermolecular Coulombic interactions. In this work, we revisit how this cancellation comes about in free space and in a perfect cavity, clarifying that it can only be found when looking beyond the single-mode approximation and taking the full continuum of light modes into account. We also provide numerical evidence suggesting that this cancellation may extend to the case of an imperfect cavity, and show how the situation changes for a more realistic cavity in the framework of macroscopic QED. Finally, we discuss the implications of this cancellation for the single-mode Hamiltonian.