In Eq. 1, H ˆe is the standard electronic Hamiltonian within the Born-Oppenheimer approximation , ωb ˆ †b ˆis the bare Hamiltonian for the photon mode where ω represents the frequency and b ˆ † and b ˆare raising and lowering operators for the photon mode. The final two terms capture interactions between the photonic and electronic degrees of freedom. In these interaction terms, known as the bilinear coupling and the quadratic dipole self energy, d ˆ= λ • µ ˆcouples the field associated with the photon mode to the molecular dipole operator .