Finally, one can reconstruct the full Hamiltonians using the forms from Eqs. 49 and 50 and the truncated operators, P ĤM P, P Ĥph P, and Û (from Eq. 73). The properly truncated Coulomb gauge Hamiltonian takes the form Ĥp•A = Û † P ĤM P Û + P Ĥph P. (74) By ensuring proper confinement of all operators, this method strictly ignores any information from the Q subspace. The Û operator is also strictly unitary in its own Hilbert subspace, so the gauge invariance between the dipole and Coulomb gauges is ensured. This transformed Hamiltonian can then be explicitly written for molecular systems as