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We present a novel theory and implementation for computing coupled electronicand quantal nuclear subsystems on a single potential energy surface, movingbeyond the standard Born-Oppenheimer (BO) separation of nuclei and electrons.We formulate an exact self-consistent nucleus-electron embedding potential fromthe single product molecular wavefunction, and demonstrate that the fundamentalbehavior of correlated nucleus-electron can be computed for mean-fieldelectrons that are responsive to a quantal anharmonic vibration of selectednuclei in a discrete variable representation. Geometric gauge choices arediscussed and necessary for formulating energy invariant biorthogonalelectronic equations. Our method is further applied to characterizevibrationally averaged molecular bonding properties of molecular energetics,bond length, protonic and electron density. Moreover, post-Hartree-Fockelectron correlation can be conveniently computed on the basis ofnucleus-electron coupled molecular orbitals, as demonstrated to correlatedmodels of second-order M{\o}llet-Plesset perturbation and full configurationinteraction theories. Our approach not only accurately quantifies non-classicalnucleus-electron couplings for revising molecular bonding properties, but alsoprovides an alternative time-independent approach for deploying non-BOmolecular quantum chemistry.

Nucleus-electron correlation revising molecular bonding fingerprints from the exact wavefunction factorization