Chapter 6
Feynman Diagrams
Having established the three pillars of the formalism in the two foregoing chapters,
we now can fully implement the diagrammatic approach to the PT expansion of
the electron propagator. Specifically, we derive and formulate a list of rules for
drawing and evaluating Feynman diagrams (Sect. 6.1). Adopting a HF one-particle
representation leads to a considerable reduction of the number of diagrams to be
considered, which will be discussed in Sect. 6.2. The systematic construction of
higher-order diagrams is greatly facilitated by using the compact Abrikosov notation,
in which sets of related Feynman diagrams can be incorporated within individual
Abrikosov diagrams (Sect. 6.3). Abrikosov diagrams can be represented by specific
matrices, which allows for the algorithmic construction of higher-order diagrams in
a systematical way.
6.1 Second-Order Diagrams
To further establish the diagrammatic formulation of the perturbation expansion of
the electron propagator, we will now take a look at the second-order contribution,
i G
(2)
pq (t, t
) =
(−i)
2
2!
∞
−∞
dt 1
∞
−∞
dt 2 (
1
2
)
2
uvrs
i jkl
V uvrs V i jkl
Φ 0 | ˆ
T T T [c
†
u (t 1 )c
†
v (t 1 )c s (t 1 )c r (t 1 )c
†
i (t 2 )c
†
j (t 2 )c l (t 2 )c k (t 2 )c p (t)c
†
q (t
)]|Φ 0 C
(6.1)
as deriving from the general expression (5.43). The subscript C on the righthand side indicates that only linked diagrams are taken into account. The restriction to linked diagrams discards from the outset many of the 5! = 120 individual
contraction schemes arising at second order. According to Remark 3 in Chap. 5,
© Springer Nature Switzerland AG 2018
J. Schirmer, Many-Body Methods for Atoms, Molecules and Clusters, Lecture
Notes in Chemistry 94, https://doi.org/10.1007/978-3-319-93602-4_6
75
Feynman Diagrams
Having established the three pillars of the formalism in the two foregoing chapters,
we now can fully implement the diagrammatic approach to the PT expansion of
the electron propagator. Specifically, we derive and formulate a list of rules for
drawing and evaluating Feynman diagrams (Sect. 6.1). Adopting a HF one-particle
representation leads to a considerable reduction of the number of diagrams to be
considered, which will be discussed in Sect. 6.2. The systematic construction of
higher-order diagrams is greatly facilitated by using the compact Abrikosov notation,
in which sets of related Feynman diagrams can be incorporated within individual
Abrikosov diagrams (Sect. 6.3). Abrikosov diagrams can be represented by specific
matrices, which allows for the algorithmic construction of higher-order diagrams in
a systematical way.
6.1 Second-Order Diagrams
To further establish the diagrammatic formulation of the perturbation expansion of
the electron propagator, we will now take a look at the second-order contribution,
i G
(2)
pq (t, t
) =
(−i)
2
2!
∞
−∞
dt 1
∞
−∞
dt 2 (
1
2
)
2
uvrs
i jkl
V uvrs V i jkl
Φ 0 | ˆ
T T T [c
†
u (t 1 )c
†
v (t 1 )c s (t 1 )c r (t 1 )c
†
i (t 2 )c
†
j (t 2 )c l (t 2 )c k (t 2 )c p (t)c
†
q (t
)]|Φ 0 C
(6.1)
as deriving from the general expression (5.43). The subscript C on the righthand side indicates that only linked diagrams are taken into account. The restriction to linked diagrams discards from the outset many of the 5! = 120 individual
contraction schemes arising at second order. According to Remark 3 in Chap. 5,
© Springer Nature Switzerland AG 2018
J. Schirmer, Many-Body Methods for Atoms, Molecules and Clusters, Lecture
Notes in Chemistry 94, https://doi.org/10.1007/978-3-319-93602-4_6
75