The paper discusses the construction of a practical phase-space electronic Hamiltonian that depends on both nuclear position and momentum, aiming to better capture both nuclear and electronic properties. The authors use electron translation and rotational factors to couple electronic transitions to nuclear motion, leading to a Hamiltonian that conserves total linear and angular momentum. They derive necessary conditions for momentum conservation and propose a one-electron Hamiltonian operator \(\Gamma_{\mu\nu}\) that satisfies these conditions. The proposed Hamiltonian is tested numerically using Hartree-Fock theory, showing good agreement with finite-difference calculations for linear and angular momentum. The results demonstrate that the phase-space Hamiltonian can effectively capture electronic motion beyond the Born-Oppenheimer approximation, particularly in systems with nonadiabatic interactions.The paper discusses the construction of a practical phase-space electronic Hamiltonian that depends on both nuclear position and momentum, aiming to better capture both nuclear and electronic properties. The authors use electron translation and rotational factors to couple electronic transitions to nuclear motion, leading to a Hamiltonian that conserves total linear and angular momentum. They derive necessary conditions for momentum conservation and propose a one-electron Hamiltonian operator \(\Gamma_{\mu\nu}\) that satisfies these conditions. The proposed Hamiltonian is tested numerically using Hartree-Fock theory, showing good agreement with finite-difference calculations for linear and angular momentum. The results demonstrate that the phase-space Hamiltonian can effectively capture electronic motion beyond the Born-Oppenheimer approximation, particularly in systems with nonadiabatic interactions.