Numerical integrators for quantum dynamics close to the adiabatic limit

Tobias Jahnke, Christian Lubich

Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle 10, D-72076 Tübingen, Germany

jahnke@na.uni-tuebingen.de

http://na.uni-tuebingen.de/~jahnke/

Contributed talk

Schrödinger equations of the form

$$\dot{\psi}(t) = - \frac{i}{\epsilon} H(t) \psi(t)$$

with a small parameter $0 < \epsilon \ll 1$ appear as a computationally critical problem in mixed quantum-classical models for molecular dynamics. If evaluations of $H(t)$ are expensive, it is desirable to use large time steps to integrate this equation, but since the solution $\psi$ oscillates rapidly on a time scale $\sim \epsilon$, this can not be done in a straightforward way.

In the talk, several new time-reversible numerical integrators will be presented which can be used efficiently with step sizes larger than one period of oscillation $(h>\epsilon)$. The performance of the methods will be demonstrated both in an almost adiabatic setup and in case of an avoided energy level crossing, where non-adiabatic state transitions occur, posing additional problems for any numerical treatment.