Hedin Equations, GW, GW+DMFT, and All That
  Historically, the GW approach was put forward by Hedin as the simplest approximation to the so-called Hedin equations. In Section 2, we will derive these Hedin equations from a Feynman-diagrammatical point of view. Section 3.1 shows how GW arises as an approximation to the Hedin equations. In Section 3.2, we briefly present some typical GW results for materials, including quasiparticle renormalizations, lifetimes, and band gap enhancements. In Section 4, the combination of GW and DMFT is summarized. Finally, as a prospective outlook, ab initio dynamical vertex approximation D$\Gamma$A is introduced in Section 5 as a unifying scheme for all that: GW, DMFT and non-local vertex correlations beyond. 
K. Held, C. Taranto, G. Rohringer, and A. Toschi
cond-mat.mtrl-sci