326
Appendix
amplitudes constituting the second-order ADC(2) scheme are listed. The configuration manifold is spanned by the 1 p-1h and 2 p-2h configurations.
Secular Matrix
1 p-1h diagonal block:
K ak,a k =( a − k )δ aa δ kk ,
(A.9.26)
C ak,a k = − V ak [a k] + C
(A)
ak,a k + C
(B)
ak,a k + C
(C)
ak,a k
(A.9.27)
where
C
(A)
ak,a k =δ kk
1
2
c,i, j
v aci j v i ja c ( i + j − c −
1
2
a −
1
2
a ),
(A.9.28)
C
(B)
ak,a k =δ aa
1
2
c,d,i
v cdki v k icd (
1
2
k +
1
2
k + i − c − d ),
(A.9.29)
C
(C)
ak,a k =
c,i
v k ia c v acik (
1
2
k +
1
2
k −
1
2
a −
1
2
a + i − c ).
(A.9.30)
1 p-1h/2 p-2h block:
C
(1)
ak,a b k l = δ aa V k l [kb ] − δ ab V k l [ka ] − δ kk V al [a b ] + δ kl V ak [a b ] ,
(A.9.31)
2 p-2h diagonal block:
K abkl,a b k l = ( a + b − k − l )δ aa δ bb δ kk δ ll
(A.9.32)
The 2 p-2h diagonal block can be extended with the first-order secular matrix elements anticipated from the ADC(3) level:
C
(1)
abkl,a b k l =δ kk δ ll V ab[a b ] + δ aa δ bb V k l [kl]
−
δ bb δ ll V ak [a k] + δ bb δ kk V al [a l] + δ aa δ ll V bk [b k] + δ aa δ kk V bl [b l]
+(k
↔ l
) + (a
↔ b
) − (k
↔ l
, a
↔ b
).
(A.9.33)
While this extension (ADC(2)-x) improves the treatment of doubly excited states, it
does not afford consistently better results for the single excitations.
Effective Transition Amplitudes
1 p-1h part, p-h amplitudes:
f ak,a k = δ aa δ kk + f
(A)
ak,a k + f
(B)
ak,a k + f
(C)
ak,a k
(A.9.34)
where