The polaritonic non-adiabatic coupling can be analytically evaluated as follows ing a shifted harmonic oscillator basis instead of the traditional Fock basis of the pure light Hamiltonian. This is because the light-matter coupling itself causes a shift in the photonic Hamiltonian (as shown in Eq. 30). Thus, a shifted harmonic oscillator basis is a more natural basis in which to represent this system. This leads to a significant improvement in Fock state convergence over the d•E Hamiltonian, as shown numerically in Fig. . Additionally, Fig. shows in further detail the advantageous convergence properties of the PFS representation compared to Ĥd•E and Ĥ′ p•A . In fact, this PFS representation is equivalent to the properly truncation Coulomb gauge Hamiltonian in Eq. 27. To see this, we first look at how Ûθ rotates qc :