Beyond the Born-Oppenheimer Approximation: A Treatment of Electronic Flux Density in Electronically Adiabatic Molecular Processes

Intuition suggests that a molecular system in the electronic ground state Phi(0) should exhibit an electronic flux density (EFD) in response to the motion of its nuclei. If that state is described by the Born-Oppenheimer approximation (BOA), however, a straightforward calculation of the EFD yields zero, since the electrons are in a stationary state, regardless. of the state of the nuclear motion. Here an alternative. pathway to a nonzero EFD from a knowledge of only the BOA ground-state wave function is proposed. Via perturbation theory a complete set of approximate vibronic eigenfunctions of the whole Hamiltonian is generated. If the complete non-BOA wave function is expressed in the basis of these vibronic eigenfunctions, the ground-state contribution to the EFD is found to involve a summation over excited states. Evaluation of this sum through the so-called average excitation energy approximation produces a nonzero EFD. An explicit formula for the EFD for the prototypical system, namely, oriented H-2(+) vibrating in the electronic ground state, is derived.