9.2 Dyson-ADC Secular Equations
145
The exact (N −1)-electron states |
N−1
n can be classified according to their descent
from CI configuration classes, mh-(m − 1) p, m = 1, . . . . We will denote the excitation class of |
N−1
n by [n], that is, [n] = μ if |
N−1
n derives from a μh-(μ − 1) p CI
configuration. The amplitude
N−1
n |c p | 0 fulfills the following remarkable order
relation [3, 7, 8]:
N−1
n |c p | 0 ∼ O([n] − 1)
(9.38)
This means that the lowest non-vanishing term in the perturbation expansion of the
amplitude is of the order [n] − 1. Now, we may ask at which order states of class [n]
will appear in the (diagrammatic) perturbation expansion of the electron propagator.
Using the order relation (9.38), and the fact that the energy denominators on the righthand side of (9.37) always begin at zeroth order, that is, in the form ω + a + · · · −
k . . . , the answer is 2[n] − 2, which is in agreement with the diagrammatic argument
outlined above. A completely analogous reasoning applies to the (N + 1)-electron
states.
The size-consistency of the ADC(n) approximations is a consequence of the diagrammatic perturbation theory, more specifically, the linked-cluster theorem. The
Feynman diagrams constituting the electron propagator or the self-energy part are
locally correct. Let us again inspect the case of separate fragments. If a diagram
begins with a local free fermion line, associated, for example, with a one-particle
state of fragment A, then the entire diagram pertains to fragment A, because any part
of the diagram is ultimately linked to the initial free fermion line and the interaction dots (or interaction lines) can have only fragment- A entries due to the separate
fragment model (which implies that interaction points with both fragment-A and
fragment-B entries vanish). The separation of the diagrams into subsets of A- and
B-type is reflected in a corresponding separation of the ADC secular matrices. We
will come back to the size-consistency of the ADC approximations in Chap. 12.
Exercises
9.1 Apply the systematic construction of Abrikosov diagrams discussed in Sect. 6.3
to the self-energy diagrams and verify that Fig. 9.1 shows all fourth-order diagrams for M(ω).
9.2 Use the matrix representation of self-energy diagrams (Exercise 8.4) to generate
the fourth-order diagrams for M(ω).
9.3 Redraw the diagrams 3–5 in Fig. 9.1 with the order of the two inner vertices
inverted. Apply the same procedure to diagram 8. Discuss the finding for diagram
8 with regard to the rule (A5) for Abrikosov diagrams (Sect. 6.3).
9.4 Use RSPT for the (N −1)-electron state | abjkl deriving from c
†
a c
†
b c j c k c l | 0
and verify that the matrix element abjkl |c p | 0 vanishes in first order.
145
The exact (N −1)-electron states |
N−1
n can be classified according to their descent
from CI configuration classes, mh-(m − 1) p, m = 1, . . . . We will denote the excitation class of |
N−1
n by [n], that is, [n] = μ if |
N−1
n derives from a μh-(μ − 1) p CI
configuration. The amplitude
N−1
n |c p | 0 fulfills the following remarkable order
relation [3, 7, 8]:
N−1
n |c p | 0 ∼ O([n] − 1)
(9.38)
This means that the lowest non-vanishing term in the perturbation expansion of the
amplitude is of the order [n] − 1. Now, we may ask at which order states of class [n]
will appear in the (diagrammatic) perturbation expansion of the electron propagator.
Using the order relation (9.38), and the fact that the energy denominators on the righthand side of (9.37) always begin at zeroth order, that is, in the form ω + a + · · · −
k . . . , the answer is 2[n] − 2, which is in agreement with the diagrammatic argument
outlined above. A completely analogous reasoning applies to the (N + 1)-electron
states.
The size-consistency of the ADC(n) approximations is a consequence of the diagrammatic perturbation theory, more specifically, the linked-cluster theorem. The
Feynman diagrams constituting the electron propagator or the self-energy part are
locally correct. Let us again inspect the case of separate fragments. If a diagram
begins with a local free fermion line, associated, for example, with a one-particle
state of fragment A, then the entire diagram pertains to fragment A, because any part
of the diagram is ultimately linked to the initial free fermion line and the interaction dots (or interaction lines) can have only fragment- A entries due to the separate
fragment model (which implies that interaction points with both fragment-A and
fragment-B entries vanish). The separation of the diagrams into subsets of A- and
B-type is reflected in a corresponding separation of the ADC secular matrices. We
will come back to the size-consistency of the ADC approximations in Chap. 12.
Exercises
9.1 Apply the systematic construction of Abrikosov diagrams discussed in Sect. 6.3
to the self-energy diagrams and verify that Fig. 9.1 shows all fourth-order diagrams for M(ω).
9.2 Use the matrix representation of self-energy diagrams (Exercise 8.4) to generate
the fourth-order diagrams for M(ω).
9.3 Redraw the diagrams 3–5 in Fig. 9.1 with the order of the two inner vertices
inverted. Apply the same procedure to diagram 8. Discuss the finding for diagram
8 with regard to the rule (A5) for Abrikosov diagrams (Sect. 6.3).
9.4 Use RSPT for the (N −1)-electron state | abjkl deriving from c
†
a c
†
b c j c k c l | 0
and verify that the matrix element abjkl |c p | 0 vanishes in first order.