Electronic Schrödinger equation with nonclassical nuclei

We present a rigorous reformulation of the quantum mechanical equations of motion for the coupled system of electrons and nuclei that focuses on the dynamics of the electronic subsystem. Usually the description of electron dynamics involves an electronic Schrödinger equation where the nuclear degrees of freedom appear as parameters or as classical trajectories. Here we derive the exact Schrödinger equation for the subsystem of electrons, staying within a full quantum treatment of the nuclei. This exact Schrödinger equation features a time-dependent potential energy surface for electrons (e-TDPES). We demonstrate that this exact e-TDPES differs significantly from the electrostatic potential produced by classical or quantum nuclei.

Figure 1

Electron localization probabilities along the negative (solid line) and the positive (dashed line) z axes as a function of time, obtained from exact dynamics (black), dynamics on the traditional potential ${\epsilon }_e^{\mathrm{trad}}$ evaluated at the exact mean nuclear position (red or dark gray), and dynamics on the approximate potential ${\epsilon }_e^{\mathrm{approx}}$ (green or light gray). The field is shown in the top panel.

Figure 2

Top panel: Electronic potentials at the times indicated: exact ${\epsilon }_e$ (black), and traditional ${\epsilon }_e^{\mathrm{trad}}$ evaluated at the exact mean nuclear position (red or gray). Lower panel: Electron densities obtained from dynamics on the electronic potentials shown at the indicated times in the top panel.