Chapter 9
Algebraic–Diagrammatic Construction
(ADC)
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
propagator, as shown by the third-order OVGF approach discussed in Sect. 8.3. The
applicability of the OVGF approximation, however, is restricted to the energy region
above the highest pole of M
−
(ω) and below the first pole of M
+
(ω). 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], which,
in turn, is based on the famous random-phase approximation (RPA), being itself an
infinite partial summation of diagrams of the polarization propagator (see Chap. 15).
An alternative way of generating infinite partial summations 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 G
−
(ω) and G
+
(ω) parts
of the electron propagator is presented in Chap. 10.
9.1 ADC Formulation of the Dynamic Self-Energy Part
According to Eq. (8.19), the dynamical self-energy consists of two parts, M
+
(ω) and
M
−
(ω), referred to as affinity and ionization part, respectively. The ADC formulation
applies independently to either of them. To be specific, we will consider the affinity
part M
+
(ω) in the following and drop the superscripts + for notational ease. The
treatment of M
−
(ω) is completely analogous.
© Springer Nature Switzerland AG 2018
J. Schirmer, Many-Body Methods for Atoms, Molecules and Clusters, Lecture
Notes in Chemistry 94, https://doi.org/10.1007/978-3-319-93602-4_9
135
Algebraic–Diagrammatic Construction
(ADC)
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
propagator, as shown by the third-order OVGF approach discussed in Sect. 8.3. The
applicability of the OVGF approximation, however, is restricted to the energy region
above the highest pole of M
−
(ω) and below the first pole of M
+
(ω). 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], which,
in turn, is based on the famous random-phase approximation (RPA), being itself an
infinite partial summation of diagrams of the polarization propagator (see Chap. 15).
An alternative way of generating infinite partial summations 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 G
−
(ω) and G
+
(ω) parts
of the electron propagator is presented in Chap. 10.
9.1 ADC Formulation of the Dynamic Self-Energy Part
According to Eq. (8.19), the dynamical self-energy consists of two parts, M
+
(ω) and
M
−
(ω), referred to as affinity and ionization part, respectively. The ADC formulation
applies independently to either of them. To be specific, we will consider the affinity
part M
+
(ω) in the following and drop the superscripts + for notational ease. The
treatment of M
−
(ω) is completely analogous.
© Springer Nature Switzerland AG 2018
J. Schirmer, Many-Body Methods for Atoms, Molecules and Clusters, Lecture
Notes in Chemistry 94, https://doi.org/10.1007/978-3-319-93602-4_9
135