Mathematical and Numerical Methods for Complex Quantum Systems

The Time-Dependent Born-Oppenheimer Approximation and Non-Adiabatic Transitions

George Hagedorn

Virginia Tech


We describe the mathematics of the time-dependent Born-Oppenheimer approximation of molecular quantum mechanics. We then consider non-adiabatic transitions for the electrons, which are typically exponentially small effects (in terms of the Born-Oppenheimer parameter). There are interesting, important open problems in this area, although a few results have been proved rigorously.