10.2 Explicit ADC Procedure Through Second Order
155
C 11 (or parts thereof) could be transferred into f 1h , resulting ultimately in a diagonal
C 11 matrix. While such a transfer from C contributions to f is formally possible, it
does not lead to viable computational schemes, as is indicated by the “dangerous”
denominator ( p − q )
−1 in ˜
f pq . Inversely, any anti-hermitian contributions in f pq
can be transferred into the off-diagonal matrix elements C pq and C qp of C 11 . A unique
definition of both C 11 and f 1h can be obtained by requiring that f 1h is hermitian.
At the third-order ADC level, the three diagrams T 1, T 2, and T 3 shown in
Fig. 6.8 have to be considered. Each diagram gives rise to 120 time-ordered (Goldstone) diagrams of which 60 contribute to G
−
(ω). The full ADC analysis has been
given in Ref. [1], and we may confine ourselves to a few remarks. Due to their topological similarity, the treatment of the T 1 and T 2 Goldstone manifolds is largely
analogous. Many of the 60 time-orderings are redundant for the ADC procedure as
they repeat only terms already established at second and first order. Accordingly,
it suffices to focus on certain key diagrams. The T 3 diagrams can be treated in a
closed analytical way, without the need to spell out the 60 time-orderings. The ADC
contributions coming into play at the third-order level are C
(3)
11 , C
(2)
12 , C
(1)
22 in the secular matrix, and f
(3)
1 , f
(2)
2 in the spectroscopic amplitudes. The explicit ADC(3)
expressions are listed in Appendix A.9.
10.3 Properties of the non-Dyson ADC Schemes
The ADC procedure for the ionization part G
−
(ω) of the electron propagator as
outlined in the previous section leads to direct (or non-Dyson) ADC schemes for the
computation of ionization energies and spectroscopic amplitudes. In Fig. 10.1, the
structure of the second- and third-order ADC matrices is displayed. In both cases,
the configuration space is spanned by the 1h and 2h-1 p configurations. The secular
matrix elements read
(K + C) kl = k δ kl +
a
1
2
k −
1
2
l ) + [ C
(3)
kl ]
C k,ak l = −V k l [ka] + [ C
(2)
k,ak l ]
(K + C) akl,a k l = (− a + k + l )δ aa δ kk δ ll + [ C
(1)
akl,a k l ]
(10.33)
where the ADC(2) scheme is constituted by the explicit expressions, while the terms
in brackets are surrogates for the additions needed at the ADC(3) level. In a similar
way, the ADC expressions for the transition amplitudes can be written as
155
C 11 (or parts thereof) could be transferred into f 1h , resulting ultimately in a diagonal
C 11 matrix. While such a transfer from C contributions to f is formally possible, it
does not lead to viable computational schemes, as is indicated by the “dangerous”
denominator ( p − q )
−1 in ˜
f pq . Inversely, any anti-hermitian contributions in f pq
can be transferred into the off-diagonal matrix elements C pq and C qp of C 11 . A unique
definition of both C 11 and f 1h can be obtained by requiring that f 1h is hermitian.
At the third-order ADC level, the three diagrams T 1, T 2, and T 3 shown in
Fig. 6.8 have to be considered. Each diagram gives rise to 120 time-ordered (Goldstone) diagrams of which 60 contribute to G
−
(ω). The full ADC analysis has been
given in Ref. [1], and we may confine ourselves to a few remarks. Due to their topological similarity, the treatment of the T 1 and T 2 Goldstone manifolds is largely
analogous. Many of the 60 time-orderings are redundant for the ADC procedure as
they repeat only terms already established at second and first order. Accordingly,
it suffices to focus on certain key diagrams. The T 3 diagrams can be treated in a
closed analytical way, without the need to spell out the 60 time-orderings. The ADC
contributions coming into play at the third-order level are C
(3)
11 , C
(2)
12 , C
(1)
22 in the secular matrix, and f
(3)
1 , f
(2)
2 in the spectroscopic amplitudes. The explicit ADC(3)
expressions are listed in Appendix A.9.
10.3 Properties of the non-Dyson ADC Schemes
The ADC procedure for the ionization part G
−
(ω) of the electron propagator as
outlined in the previous section leads to direct (or non-Dyson) ADC schemes for the
computation of ionization energies and spectroscopic amplitudes. In Fig. 10.1, the
structure of the second- and third-order ADC matrices is displayed. In both cases,
the configuration space is spanned by the 1h and 2h-1 p configurations. The secular
matrix elements read
(K + C) kl = k δ kl +
a
1
2
k −
1
2
l ) + [ C
(3)
kl ]
C k,ak l = −V k l [ka] + [ C
(2)
k,ak l ]
(K + C) akl,a k l = (− a + k + l )δ aa δ kk δ ll + [ C
(1)
akl,a k l ]
(10.33)
where the ADC(2) scheme is constituted by the explicit expressions, while the terms
in brackets are surrogates for the additions needed at the ADC(3) level. In a similar
way, the ADC expressions for the transition amplitudes can be written as