New and efficient algorithms to solve the time-dependent Schroedinger equation for larger systems: Local propagating Gaussians and extending the viability of the Herman-Kluk propagator

The area of Quantum Molecular Dynamics studies quantum effects in molecules. Exact quantum calculations are only feasible for a very small number of dimensions; yet, quantum effects appear in many larger systems. There is a search for numerical algorithms that are both accurate and viable for larger molecules. Several possible time-dependent methods are presented in this work.; Chapters 1 lays the foundation for this work by giving the current background to the field of Quantum Molecular Dynamics. Challenges of the field and existing solutions to them are presented. In Chapters 2 and 3, the work on Local Propagating Gaussians (LPG) is presented. This method is an extension of mean-field techniques. It allows for coupling that is site specific, rather than an average of all sites. It uses the unique features of Gaussian wavepackets and their overlaps to model the coupling between different sites. In Chapter 2, the frozen Gaussian version of LPG is developed and then in Chapter 3, the flexible Gaussian formulation is presented. Initial results are given.; The emphasis is then switched to semi-classical methods. The semi-classical methods give accurate results for only a short time. In Chapters 4 and 5, we present ways of improving the results of the semi-classical Herman Kluk (HK) propagator. In Chapter 4, we focus on signal processing. When HK is combined with cross correlation Filter Diagonalization, and symmetrization, remarkably accurate results can be obtained. The Helium-Benzene system is used as an example.; In Chapter 5, another possible improvement to the Herman Kluk method is presented. We look at how to extend the signal. This is done using a trajectory dependent Filinov transformation. Preliminary results for this method are given for the Helium-Naphtalene system. The new method is able to resolve energy splitting with very few trajectories.; In the final chapter, a synopsis is given of this work. We conclude that these methods should be expanded to larger dimensional systems and embedded in other methods.