324
Appendix
A.9.2 Direct ADC Expressions for the Ionization Part of the
Electron Propagator
In the following, direct ADC expressions (Chap. 10) are compiled for the G
−
(ω) part
of the electron propagator, covering specifically the secular matrix K + C at the thirdorder (ADC(3)) level, and the effective transition moments f at the ADC(2) level.
Here, the ADC configuration manifold comprises the 1h and 2h-1 p configurations.
ADC(3) Secular Matrix
1h diagonal block:
K kk = k δ kk
(A.9.9)
C kk =
a
a + b − j −
1
2
k −
1
2
k
(A.9.10)
+ C
(A)
kk + C
(B)
kk + C
(C)
kk + C
(D)
kk +
(3)
kk (∞)
(A.9.11)
where
C
(A)
kk =
1
4
a,b,c,d
l
v abkl v
∗
cdk l V cd[ab]
(A.9.12)
C
(B)
kk =
a,b,c
l,m
v abkl v
∗
ack m V lc[bm]
(A.9.13)
C
(C)
kk =
1
4
a,b
l,m, j
v ablm v
∗
abjk V lm[ jk] + h.c.
(A.9.14)
C
(D)
kk =
a,b,c
l,m
v ablm v
∗
bck m V lc[ka] + h.c.
(A.9.15)
The last term in Eq. (A.9.11) is the third-order contribution to the constant self-energy
part, kk (∞) (see Sect. 8.2). In actual ADC(3) computations, it is recommendable to
go beyond the strict third-order level here, which is afforded by the Dyson expansion
method (DEM) presented in Sect. A.5. The actual third-order expressions can readily
be derived from diagram T 3 in Fig. 8.9 (see Exercise 8.2). Alternatively, one may
resort to Eq. (8.34),
(3)
pq (∞) =
r,s
V pr[qs] ρ
(2)
sr
(A.9.16)
and relate the second-order density matrix elements via Eq. (10.37) to the effective
transition amplitudes,
Appendix
A.9.2 Direct ADC Expressions for the Ionization Part of the
Electron Propagator
In the following, direct ADC expressions (Chap. 10) are compiled for the G
−
(ω) part
of the electron propagator, covering specifically the secular matrix K + C at the thirdorder (ADC(3)) level, and the effective transition moments f at the ADC(2) level.
Here, the ADC configuration manifold comprises the 1h and 2h-1 p configurations.
ADC(3) Secular Matrix
1h diagonal block:
K kk = k δ kk
(A.9.9)
C kk =
a
a + b − j −
1
2
k −
1
2
k
(A.9.10)
+ C
(A)
kk + C
(B)
kk + C
(C)
kk + C
(D)
kk +
(3)
kk (∞)
(A.9.11)
where
C
(A)
kk =
1
4
a,b,c,d
l
v abkl v
∗
cdk l V cd[ab]
(A.9.12)
C
(B)
kk =
a,b,c
l,m
v abkl v
∗
ack m V lc[bm]
(A.9.13)
C
(C)
kk =
1
4
a,b
l,m, j
v ablm v
∗
abjk V lm[ jk] + h.c.
(A.9.14)
C
(D)
kk =
a,b,c
l,m
v ablm v
∗
bck m V lc[ka] + h.c.
(A.9.15)
The last term in Eq. (A.9.11) is the third-order contribution to the constant self-energy
part, kk (∞) (see Sect. 8.2). In actual ADC(3) computations, it is recommendable to
go beyond the strict third-order level here, which is afforded by the Dyson expansion
method (DEM) presented in Sect. A.5. The actual third-order expressions can readily
be derived from diagram T 3 in Fig. 8.9 (see Exercise 8.2). Alternatively, one may
resort to Eq. (8.34),
(3)
pq (∞) =
r,s
V pr[qs] ρ
(2)
sr
(A.9.16)
and relate the second-order density matrix elements via Eq. (10.37) to the effective
transition amplitudes,