Time-Dependent Density Functional Theory and Green’s Functions Methods with the Bethe–Salpeter Equation

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.