In recent years, time-dependent density-functional theory (TDDFT) has been the method of choice for calculating optical excitations in medium sized to large systems, due to its good balance between computational cost and achievable accuracy. In this thesis, TDDFT is reformulated to fit the framework of the linear-scaling density-functional theory (DFT) code ONETEP. The implementation relies on representing the optical response of the system using two sets of localised, atom centered, in situ optimised orbitals in order to ideally describe both the electron and the hole wavefunctions of the excitation. This dual representation approach requires only a minimal number of localised functions, leading to a very efficient algorithm. It is demonstrated that the method has the capability of computing low energy excitations of systems containing thousands of atoms in a computational effort that scales linearly with system size.

The localised representation of the response to a perturbation allows for the selective convergence of excitations localised in certain regions of a larger system. The excitations of the whole system can then be obtained by treating the coupling between different subsystems perturbatively. It is shown that in the limit of weakly coupled excitons, the results obtained with the coupled subsystem approach agree with a full treatment of the entire system, with a large reduction in computational cost.

The strengths of the methodology developed in this work are demonstrated on a number of realistic test systems, such as doped p-terphenyl molecular crystals and the exciton coupling in the Fenna-Matthews-Olson complex of bacteriochlorophyll. It is shown that the coupled subsystem TDDFT approach allows for the treatment of system sizes inaccessible by previous methods.