where Ĥel is the bare electronic Hamiltonian (diagonal in the adiabatic representation) and Ĵ is the dipole-dipole interaction term. 50 R A is the location of molecule A, and êA is the unit vector of molecule A (not to be confused with the unit vector of the cavity polarization e). Adding the interaction term Ĵ to the many-molecule Hamiltonian in Eq. 17 will lead to additional physics and direct competition between the intermolecular DSE terms and the direct Coulomb interactions. These two interactions were shown to cancel each other when there is no electron density overlaps (i.e. in the absence of exchange couplings) . However, this textbook result 193 has, to our knowledge, never been explicitly investigated in molecular cavity QED. Thus, in this modified light-matter Hamiltonian, the molecules now communicate between one another through both the light-matter and Coulomb interactions, and each molecule's electronic structure can be evaluated individually with massively parallelized electronic structure calculations. .