Appendix
323
A.9 Compilation of ADC Expressions
In the explicit ADC expressions given below, the indices i, j, k, l, . . . and a, b, c, . . .
refer to occupied and unoccupied orbitals, respectively, with respect to the HF ground
state. The subscripts p, q, r . . . label both occupied and unoccupied orbitals. The
short notation
v rsuv =
V rs[uv]
r + s − u − v
(A.9.1)
is used throughout. Note that v uvrs = −v
∗
rsuv .
A.9.1 ADC(3) Expressions for the Dynamical Self-energy
According to Eq. (9.2), the affinity and ionization part M
+
(ω) and M
−
(ω), respectively, of the dynamical self-energy can be written in the ADC form
M
±
pq (ω) = U
±†
p (ω1 − K
±
− C
±
)
−1 U
±
q
(A.9.2)
At the ADC(3) level, the matrices K
± , C
± , and U
±
q are defined with respect to 2 p1h configurations, ( jab), a < b, in the affinity part (+); and 2h-1 p configurations,
(akl), k < l, in the ionization part (−).
K
+
jab, j a b = (− j + a + b ) δ j j δ aa δ bb
(A.9.3)
K
−
akl,a k l = (− a + k + l ) δ aa δ kk δ ll
(A.9.4)
C
+
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
)
(A.9.5)
C
−
akl,a k l = −δ aa V kl[k l ] + ( δ kk V a l[al ] + δ ll V a k[ak ] ) − (k
↔ l
)
(A.9.6)
U
+
jab,q = V ab[q j] −
1
2
k,l
v abkl V kl[q j] + (
k,c
v ack j V kb[qc] ) − (a ↔ b)
(A.9.7)
U
−
akl,q = V kl[qa] +
1
2
b,c
v klbc V bc[qa] − (
i,b
v kiba V bl[qi] ) + (k ↔ l)
(A.9.8)
Here, (r ↔ s) means the preceding term (in brackets) upon interchanging the indices
r and s.