Electron - Nuclear coupling in the classical limit for the electronic degrees of freedom

The use of a classical limit for the electronic degrees of freedom avoids the need to keep the nuclei clamped while solving for the dynamics of the electrons. The Hamiltonian for the electrons will then depend on the nuclear coordinates as dynamical variables. The resulting (classical) electron-nuclear coupled equations of motion exhibit dynamical symmetry and are shown to depend only on the ratio, κ⁴, of the electron to nuclear mass. We explore the coupled electron-nuclear dynamics as a function of κ for the special case of a single electron moving between two centers. In the dynamical regime where the nuclei are heavy and the Born-Oppenheimer separation should work, the full dynamical procedure is in excellent agreement with the nuclear dynamics as computed using the Born-Oppenheimer separation. In the opposite regime where the period of the electronic motion is long, a case that can be physically realized for very high Rydberg states, one reaches an 'inverse' behavior where the nuclei adiabatically adjust to the slow electronic motion. The failure of the Born-Oppenheimer separation, as judged by the electronic coupling not being governed solely by the instantaneous position of the nuclei, is more severe when the initial electronic state is not stationary.