In this lecture we present many-body perturbation theory as a method to determine quasiparticle excitations in solids, especially electronic band structures, accurately from first principles.The main ingredient is the electronic self-energy that, in principle, contains all many-body exchange and correlation effects beyond the Hartree potential.As its exact mathematical expression is unknown, approximations must be used in practical calculations.The approximation is obtained using a systematic algebraic approach on the basis of Green function techniques.It constitutes an expansion of the self-energy up to linear order in the screened Coulomb potential, which describes the interaction between the quasiparticles and includes dynamic screening through the creation of exchange-correlation holes around the bare particles.The implementation of the approximation relies on a perturbative treatment starting from density functional theory.Besides a detailed mathematical discussion we focus on the underlying physical concepts and show some illustrative applications.
Calculations of the electronic energy of the Be-atom ground state are performed using a configuration-interaction technique and a nondegenerate formulation of diagrammatic many-body perturbation theory. Both methods are applied within the algebraic approximation defined by three finite-basis sets. One of the basis sets contains only
s functions, and it defines an algebraic problem without degeneracies where agreement between the two methods is excellent. The other two basis sets, although more complete, both contain
p functions, and they define algebraic problems involving a near degeneracy where agreement between the two methods is found to be less satisfactory. It is concluded that either a degenerate formulation of perturbation theory is required, or certain diagrams in the model perturbative scheme need to be summed to high order through denominator shifts in order to properly handle the nondynamic-correlation effect found in the Be atom.