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 Schr¨odinger equation where the nuclear degrees of freedom appear as parameters or as classical trajectories. Here we derive the exact Schr¨odinger 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.