Alternative formulation of many-body perturbation theory for electron-proton correlation

We present a formulation of nuclear-electronic many-body perturbation theory for treating electron-proton correlation. Our analysis indicates that removal of the proton-proton Coulomb-exchange operator from the reference Hamiltonian can lead to a significantly lower second-order energy and potentially faster convergence of the perturbation series for many-electron systems with a single quantum nucleus and many classical nuclei. This alternative reference Hamiltonian gives negative virtual nuclear orbital energies that are related to proton ionization potentials. Our applications to the chemical systems [HeHHe]⁺, [FHF]^-, and [ClHCl] ^- illustrate the substantial decrease in the second-order energy. This formulation is applicable to numerous biologically important hydrogen transfer reactions in which the transferring hydrogen nucleus is treated quantum mechanically.