Chapter 8
Self-Energy and the Dyson Equation
In the four preceding chapters, we have established the formalism of diagrammatic
perturbation theory for the electron propagator, which allows one to derive successively higher-order contributions G
(n)
(ω). However, a finite perturbation expansion,
e.g., through third order,
G(ω) = G
0
(ω) + G
(2)
(ω) + G
(3)
(ω) + O(4)
(8.1)
does not result in a useful approximation scheme to determine the physical quantities
of interest, that is, ionization energies, electron affinities, and the corresponding
spectral factors. The reason is that the components G pq (ω) are analytical functions,
and a finite perturbation expansion does not recover the proper analytical structure
(3.17), being a sum over simple poles, from which the desired information could be
extracted. So the question is how to translate the diagrammatic perturbation expansion
into a viable computational scheme. What is needed here is to sum the perturbation
expansion, even if only partially, through infinite order. A possible path toward such
infinite partial summations, recovering the proper analytical structure of the electron
propagator, is provided by the Dyson equation, which we will address in this chapter.
8.1 Diagrammatic Approach to the Self-Energy
According to the diagram rules, the Feynman or Abrikosov diagrams of order n ≥ 1
begin and end with a free fermion line. This suggests to write the perturbation expansion of the electron propagator in a form depicted graphically in Fig. 8.1. Here, the
hatched symbol represents the quantity
pq (t, t
), referred to as improper selfenergy part.
pq (t, t
) is defined as the sum (n > 0) of all diagrammatic contributions obtained by stripping off the respective two outer free fermion lines. The
analytical expression corresponding to Fig. 8.1 reads
© 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_8
111
Self-Energy and the Dyson Equation
In the four preceding chapters, we have established the formalism of diagrammatic
perturbation theory for the electron propagator, which allows one to derive successively higher-order contributions G
(n)
(ω). However, a finite perturbation expansion,
e.g., through third order,
G(ω) = G
0
(ω) + G
(2)
(ω) + G
(3)
(ω) + O(4)
(8.1)
does not result in a useful approximation scheme to determine the physical quantities
of interest, that is, ionization energies, electron affinities, and the corresponding
spectral factors. The reason is that the components G pq (ω) are analytical functions,
and a finite perturbation expansion does not recover the proper analytical structure
(3.17), being a sum over simple poles, from which the desired information could be
extracted. So the question is how to translate the diagrammatic perturbation expansion
into a viable computational scheme. What is needed here is to sum the perturbation
expansion, even if only partially, through infinite order. A possible path toward such
infinite partial summations, recovering the proper analytical structure of the electron
propagator, is provided by the Dyson equation, which we will address in this chapter.
8.1 Diagrammatic Approach to the Self-Energy
According to the diagram rules, the Feynman or Abrikosov diagrams of order n ≥ 1
begin and end with a free fermion line. This suggests to write the perturbation expansion of the electron propagator in a form depicted graphically in Fig. 8.1. Here, the
hatched symbol represents the quantity
pq (t, t
), referred to as improper selfenergy part.
pq (t, t
) is defined as the sum (n > 0) of all diagrammatic contributions obtained by stripping off the respective two outer free fermion lines. The
analytical expression corresponding to Fig. 8.1 reads
© 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_8
111