where the ĤM is the bare molecular Hamiltonian, hω c â † â+ 1 2 is the cavity photon field Hamiltonian (under the single mode assumption), ω c A 0 • μ(â † + â) is the molecular-cavity coupling term under the dipole (length) gauge, and ωc h (A 0 • μ) 2 is the dipole self-energy, which is an essential for the correct description of a bounded ground state as well as for including other effects for energy and off-diagonal coupling corrections at large light-matter coupling strengths.