The dynamics of many chemical systems can be influenced significantly by interactions with their environment (bath). An exact treatment of this is often infeasible due to the exponential scaling of Hilbert space with the number of degrees of freedom. In this thesis, dynamical methods, which overcome this difficulty, are considered for spin systems coupled to spin baths as well as systems coupled to harmonic oscillator baths.

The thesis begins by considering recombination reactions of radical pairs, in which two electron spins interact with baths of nuclear spins. Motivated by inaccuracies observed in semiclassical treatments of the dynamics of a series of cryptochrome-based radical pairs, a new approach capable of treating the quantum dynamics of spin systems coupled to large baths of spins is presented. This method is validated for a series of central spin models, and is found to correctly describe the quantum dynamics of spin systems containing up to 1000 spins.

Following this, the validity of the conventional treatment of radical pair recombination reactions is explored by considering the coupled electronic, nuclear and spin dynamics of a model radical ion pair. The dynamics of this model is treated using the hierarchical equations of motion (HEOM) formalism, and the results obtained are used to validate a recently proposed series of master equations. The HEOM formalism exactly captures the quantum dynamics of systems that are coupled to harmonic baths. However, due to the scaling of its computational cost with the strength of the coupling, it often becomes impractical. To overcome this difficulty, an efficient tree tensor network-based approach is presented and valided by treating a series of simple models. Following this, it is applied to the calculation of electron transfer rates for a series of models spanning a wide range of physical regimes in order to validate a recently proposed interpolation formula for the calculation of electron transfer rates.