Chapter 10
Direct ADC Procedure for the Electron
Propagator
In the preceding Chap. 9, the algebraic–diagrammatic construction (ADC) was established as a procedure to derive systematically higher-order approximations to the
M
+
(ω) and M
−
(ω) parts of the dynamical self-energy M(ω). The respective ADC
matrices could then be incorporated within a common Dyson secular matrix allowing one to solve the Dyson equation in the form of a hermitian eigenvalue problem.
As a characteristic feature, the Dyson approach combines the (N −1)- and (N +1)particle problems as parts of a common computational scheme. The ADC procedure,
however, is quite general and can be applied to the electron propagator as well, more
precisely, to the G
−
(ω) or G
+
(ω) parts. Such a direct ADC approach, described
in this chapter, avoids the Dyson equation altogether and leads to separate ADC
schemes for the (N −1)- and (N +1)-particle problems. The obvious advantage of
these direct or non-Dyson schemes is the smaller size of the secular problem, being
roughly half the size of a comparable Dyson formulation. Moreover, the lowest ionization energies (or electron affinities) are at the edge of the eigenvalue spectrum
(and not in the middle of the joint (N ± 1)-particle energy spectrum). The price
to be paid here is a higher complexity in the PT expansions of the secular matrix
elements, which, however, is workable through the third-order ADC(3) level.
10.1 ADC Representation of G
−
(ω)
The direct ADC procedures for the (N ± 1)-parts of the electron propagator are completely analogous, and we shall confine ourselves to the G
−
(ω) part in the following.
As discussed in Sect. 3.1, the (N −1)-particle part of the electron propagator can be
written in the form of Eq. (3.25),
G
−
pq (ω) =
Ψ 0 |c
†
q (ω + ˆ
H − E 0 − iη)
−1 c p |Ψ 0
(10.1)
© 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_10
147
Direct ADC Procedure for the Electron
Propagator
In the preceding Chap. 9, the algebraic–diagrammatic construction (ADC) was established as a procedure to derive systematically higher-order approximations to the
M
+
(ω) and M
−
(ω) parts of the dynamical self-energy M(ω). The respective ADC
matrices could then be incorporated within a common Dyson secular matrix allowing one to solve the Dyson equation in the form of a hermitian eigenvalue problem.
As a characteristic feature, the Dyson approach combines the (N −1)- and (N +1)particle problems as parts of a common computational scheme. The ADC procedure,
however, is quite general and can be applied to the electron propagator as well, more
precisely, to the G
−
(ω) or G
+
(ω) parts. Such a direct ADC approach, described
in this chapter, avoids the Dyson equation altogether and leads to separate ADC
schemes for the (N −1)- and (N +1)-particle problems. The obvious advantage of
these direct or non-Dyson schemes is the smaller size of the secular problem, being
roughly half the size of a comparable Dyson formulation. Moreover, the lowest ionization energies (or electron affinities) are at the edge of the eigenvalue spectrum
(and not in the middle of the joint (N ± 1)-particle energy spectrum). The price
to be paid here is a higher complexity in the PT expansions of the secular matrix
elements, which, however, is workable through the third-order ADC(3) level.
10.1 ADC Representation of G
−
(ω)
The direct ADC procedures for the (N ± 1)-parts of the electron propagator are completely analogous, and we shall confine ourselves to the G
−
(ω) part in the following.
As discussed in Sect. 3.1, the (N −1)-particle part of the electron propagator can be
written in the form of Eq. (3.25),
G
−
pq (ω) =
Ψ 0 |c
†
q (ω + ˆ
H − E 0 − iη)
−1 c p |Ψ 0
(10.1)
© 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_10
147