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The main focus of this thesis was to use quantum dynamics techniques to probe the process of a charge transfer in a dye-semiconductor complex. The use of photovoltaic cells in solar electricity relies strongly on these types of charge transfer systems, and therefore an increased knowledge on this process can help to increase efficiency of these cells, and lead to better design of photovoltaic cells. In order to achieve this goal, firstly, the charge transfer along the radically cationic state of allene was investigated, as a precursor to the more complicated dye-semiconductor system. The populations of the charge donor and acceptor states were analysed, and the photoelectron spectrum was calculated and compared to experimental data to verify the results. These computations were calculated using a vibronic coupling Hamiltonian coupled with the multi-configuration time-dependent Hartree (MCTDH) method, as well as with multilayer form (ML-MCTDH). Following on from allene, a dye-semiconductor system was investigated, using a Coumarin-343-TiO2 complex. The model used for this process was akin to a donor-acceptor system, comprising of the S1 state of the dye molecule as the donor state, and the conduction band of the semiconductor as a continuum of acceptor states. In order to represent the conduction band of the semiconductor, the band was discretised and coupled to the donor state. The couplings between the donor and acceptor states were approached from two different angles, with varying results of success. Again, employing a vibronic Hamiltonian, the main vibrational modes of the dye were included in the dynamics. Using the multilayer multi-configurational time-dependent Hartree (ML-MCTDH) method, the wavepacket dynamics were analysed and the population of the donor state was investigated. Whilst the calculations performed so far has been done at 0 Kelvin, this is not an accurate model of the charge transfer that occurs inside a solar cell. Solar cells often have normal working temperatures of over 300 K. Therefore, the next step was to see if a new model can be employed which can study this quantum behaviour at temperatures >0 K. Using the molecule Salicylaldimine as a smaller test model, a ground state proton transfer was probed at various temperatures. This was done using density matrices. Using the ML-MCTDH formalism of a density matrix is a previously unexplored method, the results of which are presented in this thesis. This new approach to studying the quantum behaviour of larger systems at temperatures above 0 K offers a promising avenue to investigating the dye-semiconductor system further.