The Kohn-Sham equations are derived from the diagrammatic structure of many-body perturbation theory for the exact exchange-correlation energy,
Exc, without explicit recourse to a minimum or extremal principle, thus providing new support for this implementation of density-functional theory in a most common situation. Furthermore, it is shown that for a large class of approximate expressions for
Exc, there exist approximate Kohn-Sham equations which give the same density and energy as other means of calculation within the same diagrammatic approximation.
A diagrammatic expansion for the dynamic exchange-correlation kernel
fxc of time-dependent density functional theory is formulated. It is shown that
fxc has no singularities at Kohn-Sham transition energies in every order of the perturbation theory. However, it may diverge with the system size in extended systems. This signifies that any approximate perturbative substitute for
fxc requires a consistent perturbative treatment of the response function to avoid uncontrollable errors in the many-body corrections to excitations energies.
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.