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The time-dependent Schrödinger equation models the quantum nature of molecular processes. Numerical simulations of these models help in understanding and predicting the outcome of chemical reactions.

In this thesis, several numerical algorithms for evolving the Schrödinger equation with an explicitly time-dependent Hamiltonian are studied and their performance is compared for the example of a pump-probe and an interference experiment for the rubidium diatom. For the important application of interaction dynamics between a molecule and a time-dependent field, an efficient fourth order Magnus-Lanczos propagator is derived. Error growth in the equation is analyzed by means of a posteriori error estimation theory and the self-adjointness of the Hamiltonian is exploited to yield a low-cost global error estimate for numerical time evolution. Based on this theory, an h,p-adaptive Magnus-Lanczos propagator is developed that is capable to control the global error. Numerical experiments for various model systems (including a three dimensional model and a dissociative configuration) show that the error estimate is effective and the number of time steps needed to meet a certain accuracy is reduced due to adaptivity.

Moreover, the thesis proposes an efficient numerical optimization framework for the design of femtosecond laser pulses with the aim of manipulating chemical reactions. This task can be formulated as an optimal control problem with the electric field of the laser being the control variable. In the algorithm described here, the electric field is Fourier transformed and it is optimized over the Fourier coefficients. Then, the frequency band is narrowed which facilitates the application of a quasi-Newton method. Furthermore, the restrictions on the frequency band make sure that the optimized pulse can be realized by the experimental equipment. A numerical comparison shows that the new method can outperform the Krotov method, which is a standard scheme in this field.