152
10 Direct ADC Procedure for the Electron Propagator
f
(1)
μ = 0, C
(1)
1μ = 0 for μ > 2
(10.25)
and need no longer be considered in second and higher order.
The non-vanishing ADC terms, to be determined by the comparison with the
second-order diagrams, are f
(2)
1 , f
(1)
2 , C
(1)
12 , and C
(2)
11 . As discussed in Chap. 7, there
are each 12 second-order Goldstone diagrams contributing to G
+
(ω) and G
−
(ω),
respectively. The diagrams for G
+
(ω) are shown in Fig. 7.3; the diagrams for G
−
(ω)
are obtained by simply turning those of Fig. (7.3) upside down; corresponding to the
numbering (1)–(12) in Fig. 7.3, we use the denotations (1), . . . , (12) for the upsidedown diagrams. Obviously, the diagrams (1)−(6), (11), and (12) can directly be
assigned to individual terms in the ADC expansion (10.24), namely (1) to (d); (2),
(3) to (e); (4) to (c); (5), (6), (11), (12) to (a). Here, the derivation of the ADC terms
from the diagrammatic expressions is straightforward. As an example, let us consider
diagram (3). This diagram can directly be identified with the ADC contribution (e),
while the hermitian conjugate term relates to diagram (2). Exchanging the indices
p and q to be consistent with ˜
G pq the diagrammatic expression for (3) reads
˜
A
(2,3)
pq (ω) = −
1
ω − p
a,i< j
V i j[ pa]
ω − i − j + a
V qa[i j]
q + a − i − j
n p n q
(10.26)
Here, again, the infinitesimal −iη in the ω-denominators has been omitted. The
comparison of Eq. (10.26) and the ADC expression (e) of Eq. (10.24) allows us to
determine the first-order contributions to the ADC matrix blocks C 12 and f 2 :
C
(1)
p,akl = −V kl[ pa] n p
(10.27)
f
(1)
akl,q =
V qa[kl]
q + a − k − l
n q
(10.28)
It should be noted that only the relative sign of the C
(1)
12 and f
(1)
2 matrix elements
can be extracted from diagram (3); the absolute signs can be chosen consistently,
herewith fixing the absolute sign of the entire C 12 and f 2 matrix blocks. Obviously,
C
(1)
12 and f
(1)
2 could have been derived as well by equating the diagrams (1) and (4)
with the terms (d) and (c), respectively. This redundancy can, of course, be used for
checking the results in various ways.
Note that the f
(2)
1 contributions deriving from comparing term (a) with the diagrams (11), (12) (or the h.c. part with (5), (6)) are of particle type, n q = 1.
The remaining four diagrams (7)−(10) need special consideration since they do
not individually match any of the ADC terms. Moreover, each diagram introduces
ω 5 -denominators, that is, denominators of the type (ω + a + b − i − j − k )
−1
as there are cuts between successive vertices crossed by two particle and three hole
lines (plus the ω-line). This seems to bring into play the 3h-2 p configurations of
class μ = 3. However, when all four diagrams are combined, the ω 5 -denominators
10 Direct ADC Procedure for the Electron Propagator
f
(1)
μ = 0, C
(1)
1μ = 0 for μ > 2
(10.25)
and need no longer be considered in second and higher order.
The non-vanishing ADC terms, to be determined by the comparison with the
second-order diagrams, are f
(2)
1 , f
(1)
2 , C
(1)
12 , and C
(2)
11 . As discussed in Chap. 7, there
are each 12 second-order Goldstone diagrams contributing to G
+
(ω) and G
−
(ω),
respectively. The diagrams for G
+
(ω) are shown in Fig. 7.3; the diagrams for G
−
(ω)
are obtained by simply turning those of Fig. (7.3) upside down; corresponding to the
numbering (1)–(12) in Fig. 7.3, we use the denotations (1), . . . , (12) for the upsidedown diagrams. Obviously, the diagrams (1)−(6), (11), and (12) can directly be
assigned to individual terms in the ADC expansion (10.24), namely (1) to (d); (2),
(3) to (e); (4) to (c); (5), (6), (11), (12) to (a). Here, the derivation of the ADC terms
from the diagrammatic expressions is straightforward. As an example, let us consider
diagram (3). This diagram can directly be identified with the ADC contribution (e),
while the hermitian conjugate term relates to diagram (2). Exchanging the indices
p and q to be consistent with ˜
G pq the diagrammatic expression for (3) reads
˜
A
(2,3)
pq (ω) = −
1
ω − p
a,i< j
V i j[ pa]
ω − i − j + a
V qa[i j]
q + a − i − j
n p n q
(10.26)
Here, again, the infinitesimal −iη in the ω-denominators has been omitted. The
comparison of Eq. (10.26) and the ADC expression (e) of Eq. (10.24) allows us to
determine the first-order contributions to the ADC matrix blocks C 12 and f 2 :
C
(1)
p,akl = −V kl[ pa] n p
(10.27)
f
(1)
akl,q =
V qa[kl]
q + a − k − l
n q
(10.28)
It should be noted that only the relative sign of the C
(1)
12 and f
(1)
2 matrix elements
can be extracted from diagram (3); the absolute signs can be chosen consistently,
herewith fixing the absolute sign of the entire C 12 and f 2 matrix blocks. Obviously,
C
(1)
12 and f
(1)
2 could have been derived as well by equating the diagrams (1) and (4)
with the terms (d) and (c), respectively. This redundancy can, of course, be used for
checking the results in various ways.
Note that the f
(2)
1 contributions deriving from comparing term (a) with the diagrams (11), (12) (or the h.c. part with (5), (6)) are of particle type, n q = 1.
The remaining four diagrams (7)−(10) need special consideration since they do
not individually match any of the ADC terms. Moreover, each diagram introduces
ω 5 -denominators, that is, denominators of the type (ω + a + b − i − j − k )
−1
as there are cuts between successive vertices crossed by two particle and three hole
lines (plus the ω-line). This seems to bring into play the 3h-2 p configurations of
class μ = 3. However, when all four diagrams are combined, the ω 5 -denominators