Abstract: The time-dependent Schroedinger equation governs molecular quantums dynamics and, in particular, the interaction of light with molecules. The resulting dynamics are truely out-of-equilibrium and ask for very flexible numerical methods. We discuss the mathematics of the time dependent Dirac--Frenkel principle for this application, analyse its geometric properties and address the notorious ill-conditioning for variational gaussians, that occasionally become linearly dependent. The presented results are joint work with C. Lubich, T. Pedersen, S. Kvaal, and L. Adamowicz. "