Positrons are the anti-particles of electrons, and have the same mass but the opposite charge. As a consequence of the strong Coulomb repulsion with atomic nuclei, positrons are very sensitive to open volumes in materials, such as vacancies but also surfaces. The fact that electrons and positrons annihilate shortly after coming into contact, a process in which the resulting photons conserve the total momentum and energy of the original particles, makes that positrons are convenient to characterize these open volumes in materials.

A downside of positron annihilation experiments, is that one has little control of where positrons end up in the material. An additional complication is that different situations can result in qualitatively similar outcomes for the experiment. In many cases, first-principles calculations can help significantly in the interpretation of experiments. Indeed, from these calculations, it is possible to predict where in the material positrons will annihilate as well as the resulting spectrum measured in the experiment.

The existing theoretical description of positron states in bulk materials is quite accurate. Using local or semi-local approximations to describe electron-positron correlations, it is possible to obtain electron-positron annihilation properties, such as the positron's annihilation rate, which closely match the experiment. In the case of surfaces, however, the situation is quite different. Indeed, the aforementioned approximations are unable to describe long-range correlation effects that are critical to obtain a correct description of positron states at the surface of a material.

A non-local approximation that captures long-range correlation effects was developed in this thesis. Additionally, we critically investigated a phenomenological model from literature. Both approaches were applied to provide theoretical support for three recent positron annihilation experiments on surfaces. In all cases, we managed to obtain satisfactory agreement with the experiment, demonstrating the usefulness of our approach.