Charge Migration in Eyring, Walter and Kimball's 1944 Model of the Electronically Excited Hydrogen-Molecule Ion

In an elementary variational treatment of the electronic structure of H₂⁺, Eyring, Walter, and Kimball (EWK) serendipitously discovered charge migration (CM) in 1944. Using an analytic expression for the electronic probability density (EPD), they found that if the electron is initially localized on one of the protons (by taking the initial state to be a superposition of the ground and first excited electronic energy eigenstates), then it oscillates adiabatically between fixed protons with a period T inversely proportional to the energy gap between the eigenstates. At the equilibrium internuclear separation, T = 550.9 as. As shown here, the EWK model also yields an analytic formula for the electronic flux density (EFD). While the EPD indicates where the electron is at any instant, the EFD reveals the pathways the electron follows during its migration. Thus, the EFD complements the EPD, providing valuable new insight into the mechanism of CM. The formula for the EFD is a simple product of a time factor and a spatial factor. This factoring exposes a plethora of spatial-temporal symmetry relations which imply novel and surprising properties. An especially significant finding is that, in contrast to multielectron systems, where electron correlation may play a role in CM, in the EWK model of H₂⁺, CM is due strictly to quantum interference between the ground and first excited electronic states.