Algebraic–Diagrammatic Construction (ADC)

Abstract

As was discussed in the beginning of Chap. 8 , a finite perturbation expansion of the electron propagator does not reproduce its correct analytical form. While this applies also to the self-energy part beyond second order, here finite perturbation expansions together with the Dyson equation may provide viable approximations to the electron propagatorPropagatorelectron, as shown by the third-order ovgf Outer-Valence Green’s function (OVGF) method approach discussed in Sect. 8.3 . The applicability of the ovgf approximation, however, is restricted to the energy region above the highest pole of $$\varvec{M}^-(\omega )$$ and below the first pole of $$\varvec{M}^+(\omega )$$ . For the treatment of ionization energies and electron affinities outside that outer valence regime, the behavior of the self-energy part near its poles matters. This means that one has to recover the proper analytical form ( 8.19 ) of the dynamical self-energy part. Approximations of that type are obtained as a result of performing infinite summations of a certain class of diagrams. An example of such an infinite partial summation of diagrams in the case of the self-energy part is Hedin’s gw approximation [1] GW approximation , which, in turn, is based on the famous random-phase approximation (rpa) Random-Phase Approximation (RPA) diagrams , being itself an infinite partial summation of diagrams of the polarization propagatorPropagatorpolarization (see Chap. 15 ). An alternative way of generating infinite partial summationsInfinite partial summations (of diagrams) in a diagrammatic perturbation expansion is the algebraic–diagrammatic construction (adc) [2, 3]. In the following, we will discuss how the adc procedure can be performed in the case of the dynamic self-energy part. A direct adc approach to the $$\varvec{G}^-(\omega )$$ and $$\varvec{G}^+(\omega )$$ parts of the electron propagator is presented in Chap. 10 .