The behavior of an electronic system when subjected to external observation or perturbation can be understood in terms of appropriate time-dependent response functions (many-body Green's functions) describing the dynamics of the interacting many-body system. The two most widely useful functions are (1) the one-particle Green's function, describing the dynamics when particles are added or removed from the system (quasielectron and quasihole properties), and (2) the two particle Green's function, which describes the coupled dynamics of pairs of particles, and in particular can answer questions abou the excitation spectrum of the system and couplied electron-hole behavior (excitons).
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We rely on the GW approximation to the self-energy to explore quasiparticle properties of our systems: these can include quasiparticle energies, lifetimes, as well as their spatial wave functions. If quasielectron and quasihole interactions are not very strong, then the quasiparticle description can also provide us with the excitation energies and optical properties of the system.
Starting with quasiparticles from the GW description, we can allow the quasielectrons and quasiholes to interact. The solution to this two-particle problem, the Bethe-Salpeter equation, then provides us, in principle, with the exact two-particle Green's function. We solve the problem approximately by using a simplified but physically motivated interaction kernel between quasiparticles. Solution of the problem provides us with the energies of the excited states as well as the requisite matrix elements between ground and excited states so that the absorption spectrum can be computed. In addition, solving for the eigenstates of the Bethe-Salpeter equation provides us with the exciton wave functions, we can be further analyzed to shed light on the physical properties of specific excited states.