Validity of the Adiabatic Born-Oppenheimer Approximation in the Tight-Binding Model of Graphene

The adiabatic Born-Oppenheimer (ABO) approximation is widely used in molecular and atomic physics to simplify quantum mechanical calculations. For a molecular system, an ABO electronic state can be computed by solving the time-independent Schrödinger equation (TISE) at several time points for a Hamiltonian that is dependent on the nuclear configuration. We show that this approach breaks down when calculating the time evolution of the single-electron wavefunction of a finite graphene sheet in the presence of thermal nuclear vibrations. By classically solving for the time-dependence of the nuclear coordinates, we construct the electronic Hamiltonian explicitly as a matrix in the nearest-neighbor tight-binding approximation. We then calculate an allowed ABO state by solving the TISE at multiple time points. Comparing the ABO state with the true state determined from the time-dependent Schrödinger equation, we find that the ABO approximation breaks down in the timescale of the nuclear vibrational period when the ABO state’s time-dependent energy approaches avoided crossings.