304
Appendix
As opposed to these unlinked diagrams, there are strictly connected (SC) Hubbard
diagrams, D
h|SC
I
, forming a subset of the Hubbard diagrams of the form (A.6.10).
Summing up all SC diagrams for a given configuration (or edge structure) I ,
D
h|SC
I
= t I ˆ
C I
defines a new amplitude, t I , which differs from x I . As should be noted, the t I can
likewise be obtained from the diagrams for the ground-state amplitude (A.6.8) by
discarding all diagrams that are not strictly connected:
t I = lim
→0
0 | ˆ
C
†
I
ˆ
U (0, −∞)| 0 SC
(A.6.13)
In analogy to Eq. (A.6.11), the sum of all strictly connected Hubbard diagrams
establishes a particular operator expansion,
lim
→0
ˆ
U (0, −∞)
h
SC
=
I
t I ˆ
C I
(A.6.14)
and this operator expansion allows for a representation of the ground state in the
form
| 0 = e
I t I ˆ
C I | 0
(A.6.15)
This is the central result of Hubbard’s extended linked-cluster theorem. The latter
two equations can be seen as a foundation of the CC ansatz (17.1), and, moreover,
establish a diagrammatic PT approach to the CC amplitudes.
The validity of the order relations (A.6.1) for the t-amplitudes can be seen by
inspecting the construction principle underlying the strictly connected diagrams as
shown in Fig. A.6. Beginning with a double excitation, [I ] = 2, O(t I ) = 1, it takes
each one interaction point (order 1) to increase the excitation class by 1, that is, by
a 1p-1h excitation. This means it takes (at least) ν − 2 interaction points to arrive at
an excitation of class ν > 2. The ν − 2 interaction points have a cumulated order of
ν − 2, to which the order 1 of the initial double excitation vertex has to be added.
A.6.2 Proof of Canonical Order Relations
Originally, a proof of the canonical order relations obeyed by the ISR-ADC secular
matrix elements was given in Ref. [11]. However, a more elegant proof [12] is enabled
by recourse to the closed-form expressions of the biorthogonal coupled cluster (BCC)
representation discussed in Chap. 17. Following the latter concept, we shall first
derive the BCC order relations and then use these results to establish the canonical
ISR-ADC order relations.
Appendix
As opposed to these unlinked diagrams, there are strictly connected (SC) Hubbard
diagrams, D
h|SC
I
, forming a subset of the Hubbard diagrams of the form (A.6.10).
Summing up all SC diagrams for a given configuration (or edge structure) I ,
D
h|SC
I
= t I ˆ
C I
defines a new amplitude, t I , which differs from x I . As should be noted, the t I can
likewise be obtained from the diagrams for the ground-state amplitude (A.6.8) by
discarding all diagrams that are not strictly connected:
t I = lim
→0
0 | ˆ
C
†
I
ˆ
U (0, −∞)| 0 SC
(A.6.13)
In analogy to Eq. (A.6.11), the sum of all strictly connected Hubbard diagrams
establishes a particular operator expansion,
lim
→0
ˆ
U (0, −∞)
h
SC
=
I
t I ˆ
C I
(A.6.14)
and this operator expansion allows for a representation of the ground state in the
form
| 0 = e
I t I ˆ
C I | 0
(A.6.15)
This is the central result of Hubbard’s extended linked-cluster theorem. The latter
two equations can be seen as a foundation of the CC ansatz (17.1), and, moreover,
establish a diagrammatic PT approach to the CC amplitudes.
The validity of the order relations (A.6.1) for the t-amplitudes can be seen by
inspecting the construction principle underlying the strictly connected diagrams as
shown in Fig. A.6. Beginning with a double excitation, [I ] = 2, O(t I ) = 1, it takes
each one interaction point (order 1) to increase the excitation class by 1, that is, by
a 1p-1h excitation. This means it takes (at least) ν − 2 interaction points to arrive at
an excitation of class ν > 2. The ν − 2 interaction points have a cumulated order of
ν − 2, to which the order 1 of the initial double excitation vertex has to be added.
A.6.2 Proof of Canonical Order Relations
Originally, a proof of the canonical order relations obeyed by the ISR-ADC secular
matrix elements was given in Ref. [11]. However, a more elegant proof [12] is enabled
by recourse to the closed-form expressions of the biorthogonal coupled cluster (BCC)
representation discussed in Chap. 17. Following the latter concept, we shall first
derive the BCC order relations and then use these results to establish the canonical
ISR-ADC order relations.