Appendix
295
A.5 Dyson Expansion Method for the Static Self-Energy
Part
In this Appendix, we briefly describe an approximation method for the static part of
the self-energy referred to as Dyson expansion method (DEM) [8, 9].
According to Eq. (8.18), the self-energy is the sum of a static and a dynamic part,
(ω) = (∞) + M(ω)
(A.5.1)
Here, the static part, (∞), can be expressed directly in terms of density matrix
elements (Eq. 8.34), or likewise, using Eq. (3.34), in terms of electron propagator
integrals,
pq (∞) =
u,v
V pu[qv]
ρ vu − ρ
(0)
vu
=
u,v
V pu[qv]
1
2πi
2 dω
G vu (ω) − G
0
vu (ω)
(A.5.2)
The static part is related to the dynamic part and can be determined consistently once
the dynamic part M(ω) or an approximation to it has been established. This is seen
by writing the Dyson equation (8.10) more explicitly as
G(ω) = G
0
(ω) + G
0
(ω)((∞) + M(ω))G(ω)
(A.5.3)
and inserting this form in Eq. (A.5.2). The result is an implicit equation for (∞).
For a given M(ω), the associated result for (∞) can be obtained, in principle, via an
obvious iteration scheme. However, such a self-consistent procedure may not be very
practical, and it is more advisable to relinquish the quest for a fully self-consistent
solution for (∞) and rather resort to an approximation such as the DEM considered
below.
Dyson Expansion
The starting point is the truncation of the Dyson expansion (8.12) after the linear
term in (ω):
G(ω) = G
0
(ω) + G
0
(ω)((∞) + M(ω))G
0
(ω) + . . .
(A.5.4)
Note that the truncation error here is of fourth order because the perturbation expansion of (ω) begins in second order. Inserting the truncated Dyson expansion in
Eq. (A.5.2) yields the following linear set of equations for the matrix elements of
(∞):
pq (∞) =
r,s
V pr[qs]
1
2πi
2 G
0
s (ω)) sr (∞)G
0
r (ω)dω + b pq + O(5) (A.5.5)
295
A.5 Dyson Expansion Method for the Static Self-Energy
Part
In this Appendix, we briefly describe an approximation method for the static part of
the self-energy referred to as Dyson expansion method (DEM) [8, 9].
According to Eq. (8.18), the self-energy is the sum of a static and a dynamic part,
(ω) = (∞) + M(ω)
(A.5.1)
Here, the static part, (∞), can be expressed directly in terms of density matrix
elements (Eq. 8.34), or likewise, using Eq. (3.34), in terms of electron propagator
integrals,
pq (∞) =
u,v
V pu[qv]
ρ vu − ρ
(0)
vu
=
u,v
V pu[qv]
1
2πi
2 dω
G vu (ω) − G
0
vu (ω)
(A.5.2)
The static part is related to the dynamic part and can be determined consistently once
the dynamic part M(ω) or an approximation to it has been established. This is seen
by writing the Dyson equation (8.10) more explicitly as
G(ω) = G
0
(ω) + G
0
(ω)((∞) + M(ω))G(ω)
(A.5.3)
and inserting this form in Eq. (A.5.2). The result is an implicit equation for (∞).
For a given M(ω), the associated result for (∞) can be obtained, in principle, via an
obvious iteration scheme. However, such a self-consistent procedure may not be very
practical, and it is more advisable to relinquish the quest for a fully self-consistent
solution for (∞) and rather resort to an approximation such as the DEM considered
below.
Dyson Expansion
The starting point is the truncation of the Dyson expansion (8.12) after the linear
term in (ω):
G(ω) = G
0
(ω) + G
0
(ω)((∞) + M(ω))G
0
(ω) + . . .
(A.5.4)
Note that the truncation error here is of fourth order because the perturbation expansion of (ω) begins in second order. Inserting the truncated Dyson expansion in
Eq. (A.5.2) yields the following linear set of equations for the matrix elements of
(∞):
pq (∞) =
r,s
V pr[qs]
1
2πi
2 G
0
s (ω)) sr (∞)G
0
r (ω)dω + b pq + O(5) (A.5.5)