Abstract

In this chapter, we discuss the theoretical and physical aspects governing two methods for the computational study of electronically excited states: time-dependent density-functional theory and many-body Green׳s functions theory, drawing a close link between the two towards the end. To ensure a self-contained presentation in the chapter, we begin with a recapitulation of ground-state density-functional theory (DFT) and then build towards the time-dependent DFT. Finally, we will treat many-body perturbation theory, which uses concepts from DFT: both the ground-state and the time-dependent version. Keeping the variety of readers in mind, we present a mix of the physical interpretation of the theory, the mathematical details to arrive at important equations, and the numerical formulation for computational purposes as seen in modern-day codes.