By introducing the concept of self-energy the structure in Fig. 3.5 takes the form shown Fig. 3.6.
The corresponding analytic expression is given by
Another important concept is the proper self-energy insertion which is a self-energy insertion that can not be separated into two pieces by cutting a single-particle line. By definition, the proper self-energy is the sum of all proper self-energy insertions, and will be denoted by . Using the perturbation expansion, one can define the proper self-energy as an irreducible part of the GREEN's function. Based on this definition first-order proper self-energies, which are resulted from the first-order expansion of the GREEN's function (see Section 3.4.2), are shown in Fig. 3.7. These diagrams are irreducible parts of Fig. 3.4-b and Fig. 3.4-c and are referred to as the HARTREE ( ) and the FOCK ( ) self-energies.
The self-energy can also in principle be introduced variationally [203]. A variational derivation of the self-energies for the electron-electron and electron-phonon interactions is presented in Appendices F.1 and F.2, respectively. It follows from these definitions that the self-energy consists of a sum of all possible repetitions of the proper self-energy
Correspondingly, the GREEN's function in (3.41) can be rewritten as
The corresponding analytic expression is given by