Numerical time integration for variational approximations in quantum molecular dynamics is the topic of this talk. The proposed method is based on a splitting of the quantum Hamiltonian, though not on the level of the working equations but of the underlying Dirac-Frenkel time-dependent variational principle. The abstract integrator is made concrete for the cases of MCTDH and Gaussian wavepacket dynamics:

* In contrast to direct time discretizations of the MCTDH equations, the integrator suffers no step size restriction caused by the unbounded separable part of the Hamiltonian.

* For Gaussian wavepacket dynamics, one obtains an explicit integrator that preserves exactly norm, linear and angular momentum, and the Poisson structure. The total energy is preserved approximately over "exponentially long" times. In the semi-classical limit the approximations to position and momentum tend to those obtained by applying the Stoermer-Verlet method to the classical limit system.

Christian Lubich

A variational splitting integrator for quantum molecular dynamics

Appl. Numer. Math. 48 (2004), 355-368.

Christian Lubich

On variational approximations in quantum molecular dynamics

Report (2003), to appear in Math. Comp.

E. Faou and Ch. Lubich

A Poisson integrator for Gaussian wavepacket dynamics

Report (2004).