As expected, this Properly Truncated Coulomb Gauge Hamiltonian is now gauge invariant. This leads to equivalent results to the d•E Hamiltonian under the same level of truncation. Additionally, it preserves the favorable Fock state convergence behavior of the Coulomb Gauge Hamiltonian. Consequently, it requires fewer matter and Fock states than either of the Hamiltonians discussed above. This convergence behavior is demonstrated numerically in Fig. (e) for the Shin-Metiu molecular model.