Recovering Exact Vibrational Energies within a Phase Space Electronic Structure Framework

In recent years, there has been a push to go beyond the Born-Oppenheimer theory and build electronic states from a phase space perspective, i.e., parametrize electronic states by both nuclear position (<b><i>R</i></b>) and nuclear momentum (<b><i>P</i></b>). Previous empirical studies have demonstrated that such approaches can yield improved single-surface observables, including vibrational energies, electronic momenta, and vibrational circular dichroism spectra. That being said, unlike the case of the BO theory, there is no unique phase space electronic Hamiltonian nor any theory for using phase space eigenvectors (as opposed to BO eigenvectors) to recover exact quantum vibrational eigenvalues. As such, one might consider such phase space approaches <i>ad hoc</i>. To that end, here we show how to formally extract exact quantum energies from a coupled nuclear-electronic Hamiltonian using perturbation theory on top of a phase space electronic framework. Thus, while we cannot isolate an "optimal" phase space electronic Hamiltonian, this work does justify a phase space electronic structure approach by offering a rigorous framework for correcting the zeroth-order phase space electronic states.