138
9 Algebraic–Diagrammatic Construction (ADC)
M pq (ω) = U
(1)†
p (ω1 − K )
−1 U
(1)
q
+ U
(2)†
p (ω1 − K )
−1 U
(1)
q + U
(1)†
p (ω1 − K )
−1 U
(2)
q
+ U
(1)†
p (ω1 − K )
−1 C
(1)
(ω1 − K )
−1 U
(1)
q + O(4)
(9.12)
The new quantities arising here, namely U
(2)
p and C
(1) , are to be determined by comparing the second and third line with the diagrams for M
(3)+
pq (ω). Before inspecting
the diagrams, let us note that, as in second order, the K denominators are restricted
to those of 2 p-1h type. Accordingly, only the 2 p-1h matrix elements of U
(2)
p and
C
(1) come into play at third order.
The two third-order diagrams T 1 and T 2, constituting M
(3)
(ω), are shown in
Fig. 8.9. Each three Goldstone diagrams (with t > t
), shown in Fig. 8.10 for T 1,
contribute to the affinity part, M
(3)+
(ω). The diagrams T 1(1) and T 2(1) can directly
be compared to the third line in the ADC expansion (9.12). Let us consider the T 1(1)
contribution,
T 1(1) pq =
j,a
ω + j − a − b
δ j j V ab[a b ]
V q j [a b ]
ω +
j −
a −
b
(9.13)
where a superfluous
j δ j j summation has been inserted for clarity. The comparison
with the ADC expression (third line of Eq. 9.12) reproduces the U
(1) expressions
already determined and yields the following contribution to C
(1) :
C
(1)
jab, j a b ← δ j j V ab[a b ]
(9.14)
In a similar way, the contribution of the T 2(1) diagram can be taken into account.
The final result can be written in the form
C
(1)
jab, j a b = δ j j V ab[a b ] −
δ aa V j b[ jb ] + δ bb V j a[ ja ]
+
a
↔ b
(9.15)
Here, the 2 p-1h configurations ( jab) and ( j
a
b
) are restricted by requiring a < b
and a
< b
; together with these restrictions, the form (9.15), being anti-symmetrized
with respect to the index pairs (ab) and (a
b
), is consistent with the diagrammatic
expressions.
To determine U
(2)
jab, p , we may inspect the diagrams T 1(2) and T 2(2) conforming
to the third term in Eq. (9.12); diagrams T 1(3) and T 2(3) simply reproduce the
hermitian conjugate expression (second term of Eq. 9.12).
The analytic expression for T 1(2) reads
T 1(2) pq =
j,a
1
ω + j − a − b
1
2
kl
V ab[kl] V kl[q j]
k + l − a − b
(9.16)
from which the contribution to U
(2)
jab,q is readily derived:
9 Algebraic–Diagrammatic Construction (ADC)
M pq (ω) = U
(1)†
p (ω1 − K )
−1 U
(1)
q
+ U
(2)†
p (ω1 − K )
−1 U
(1)
q + U
(1)†
p (ω1 − K )
−1 U
(2)
q
+ U
(1)†
p (ω1 − K )
−1 C
(1)
(ω1 − K )
−1 U
(1)
q + O(4)
(9.12)
The new quantities arising here, namely U
(2)
p and C
(1) , are to be determined by comparing the second and third line with the diagrams for M
(3)+
pq (ω). Before inspecting
the diagrams, let us note that, as in second order, the K denominators are restricted
to those of 2 p-1h type. Accordingly, only the 2 p-1h matrix elements of U
(2)
p and
C
(1) come into play at third order.
The two third-order diagrams T 1 and T 2, constituting M
(3)
(ω), are shown in
Fig. 8.9. Each three Goldstone diagrams (with t > t
), shown in Fig. 8.10 for T 1,
contribute to the affinity part, M
(3)+
(ω). The diagrams T 1(1) and T 2(1) can directly
be compared to the third line in the ADC expansion (9.12). Let us consider the T 1(1)
contribution,
T 1(1) pq =
j,a
ω + j − a − b
δ j j V ab[a b ]
V q j [a b ]
ω +
j −
a −
b
(9.13)
where a superfluous
j δ j j summation has been inserted for clarity. The comparison
with the ADC expression (third line of Eq. 9.12) reproduces the U
(1) expressions
already determined and yields the following contribution to C
(1) :
C
(1)
jab, j a b ← δ j j V ab[a b ]
(9.14)
In a similar way, the contribution of the T 2(1) diagram can be taken into account.
The final result can be written in the form
C
(1)
jab, j a b = δ j j V ab[a b ] −
δ aa V j b[ jb ] + δ bb V j a[ ja ]
+
a
↔ b
(9.15)
Here, the 2 p-1h configurations ( jab) and ( j
a
b
) are restricted by requiring a < b
and a
< b
; together with these restrictions, the form (9.15), being anti-symmetrized
with respect to the index pairs (ab) and (a
b
), is consistent with the diagrammatic
expressions.
To determine U
(2)
jab, p , we may inspect the diagrams T 1(2) and T 2(2) conforming
to the third term in Eq. (9.12); diagrams T 1(3) and T 2(3) simply reproduce the
hermitian conjugate expression (second term of Eq. 9.12).
The analytic expression for T 1(2) reads
T 1(2) pq =
j,a
1
ω + j − a − b
1
2
kl
V ab[kl] V kl[q j]
k + l − a − b
(9.16)
from which the contribution to U
(2)
jab,q is readily derived: