where the sum over j only includes nuclei. In the above expression, we did not specify the choice of P, which could be either adiabatic (Eq. 26 or diabatic states (Eq 31). Fig. shows numerical results for this Hamiltonian for a simple 1-D proton-transfer ) molecular model. The left graph shows a characterization of this model with its adiabatic and diabatic states, diabatic coupling, and dipole matrix elements as a function of the proton's 1-D coordinate, R. Additionally, the small insets pictorially depict the ions, proton, and electron positions for different R. The middle figure then plots the Born-Oppenheimer surfaces as a function of R for different Hamiltonians, compared to the zero coupling case. For R values where the polariton states differ from the uncoupled case, the naively truncated Coulomb gauge Hamiltonian results differ from the gauge invariant results. The right figure, similarly, shows how the naively truncated Coulomb gauge Hamiltonian behaves very poorly as the coupling strength is increased for a given R value. For this model, the dipole gauge results converge to the accuracy of the graph with two levels, so for the results in these graphs, the dipole gauge can be considered "exact" for this model. This numerically demonstrates the necessity of maintaining gauge invariance.