Appendix
297
Thus, the formal solution of the linear set of equations takes on the form
ph
(∞) = (1 − A
ph
)
−1 b
ph
(A.5.13)
Once the p-h and h- p components of (∞) have been obtained by solving the
Eqs. (A.5.9) or (A.5.10), the h-h and p- p matrix elements can be evaluated according
to
i j (∞) =
b,l
1
l − b
(V il[ jb] bl (∞) + V ib[ jl] lb (∞)) + b i j
(A.5.14)
ac (∞) =
b,l
1
l − b
(V al[cb] bl (∞) + V ab[cl] lb (∞)) + b ac
(A.5.15)
Treatment of the Inhomogenities
The inhomogenities b pq , as given by Eqs. (A.5.6), (A.5.7), establish the connection
with the dynamic part M(ω) of the self-energy via the contour integrals (A.5.7). If
M(ω) is available in the explicit form of spectral representation (8.19), the contour
integrals Q rs can readily be evaluated. Here, it is useful to distinguish between p- p,
h-h, and p-h elements of Q. In the p- p case, the calculation is as follows:
Q ab =
1
2πi
2 G
0
a (ω)M ab (ω)G
0
b (ω) dω
=
ν∈{N −1}
1
2πi
2
1
ω − a + iη
m
(ν)
a m
(ν)∗
b
ω − ω ν − iη
1
ω − b + iη
dω
(A.5.16)
=
ν∈{N −1}
m
(ν)
a m
(ν)∗
b
( a −ω ν )( b −ω ν )
(A.5.17)
Note that here only the M
−
(ω) part contributes. In a similar way, one obtains the
h-h and p-h integrals
Q kl = −
μ∈{N +1}
m
(μ)
k m
(μ)∗
l
( k −ω μ )( l −ω μ )
(A.5.18)
Q ak = −
μ∈{N +1}
m
(μ)
a m
(μ)∗
k
( a − k )( k −ω μ )
−
ν∈{N −1}
m
(ν)
a m
(ν)∗
k
( a − k )( a −ω ν )
(A.5.19)
In the latter case, both parts M
±
(ω) of the self-energy contribute, whereas only
M
+
(ω) is involved in the h-h results. Note that Q ka = Q
∗
ak .
The computation of the Q rs integrals according to the explicit expressions above
presupposes that the full spectral information of M(ω) is available. Usually, however,
the acquisition of that information within a given approximation is computationally
expensive and hardly expedient. In the ADC approximation for M(ω) presented