Chapter 3
One-Particle Green’s Function
The one-particle Green’s function or electron propagator, which we shall introduce
in this chapter, is the first and simplest member in the hierarchy of many-body
Green’s functions [1–3]. While the formal definition looks rather abstract and even
forbidding, the benefits afforded by an approach based on the electron propagator
should become clear after the theory has been more fully described. Before working
through the various derivations, the reader might take a first look at Eqs. (3.24), (3.25)
in which the essence of the electron propagator is apparent: Its elements are matrix
elements of the many-body resolvent operator taken with respect to states of N +1 or
N −1 electrons. This indicates that the physics conveyed by the electron propagator
relates to excitations of the system following the addition of one electron (electron attachment) or the removal of one electron (ionization). We shall refer exclusively to the electron propagator in much of the book, that is, when we develop the
formalism of diagrammatic perturbation theory in Chaps. 4–7 and establish practical
approximation methods in Chaps. 8–12. The polarization propagator and the physics
of N -electron excitations will be considered in the Chaps. 13–15 of Part IV.
3.1 Definition and Relation to Physical Quantities
In the following, we suppose a basis set of one-particle states | p and the associated
creation and destruction operators c
†
p , c p as introduced in Chap. 2. We consider an
N -electron system with the hamiltonian
ˆ
H = ˆ
T + ˆ
V =
t pq c
†
p c q +
1
2
V pqrs c
†
p c
†
q c s c r
(3.1)
and a non-degenerate (normalized) ground state | 0 of energy E 0 . Moreover, we
define time-dependent or Heisenberg operators according to
c
†
p [t] = e
i ˆ
Ht c
†
p e
−i ˆ
Ht
, c p [t] = e
i ˆ
Ht c p e
−i ˆ
Ht
(3.2)
© 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_3
31
One-Particle Green’s Function
The one-particle Green’s function or electron propagator, which we shall introduce
in this chapter, is the first and simplest member in the hierarchy of many-body
Green’s functions [1–3]. While the formal definition looks rather abstract and even
forbidding, the benefits afforded by an approach based on the electron propagator
should become clear after the theory has been more fully described. Before working
through the various derivations, the reader might take a first look at Eqs. (3.24), (3.25)
in which the essence of the electron propagator is apparent: Its elements are matrix
elements of the many-body resolvent operator taken with respect to states of N +1 or
N −1 electrons. This indicates that the physics conveyed by the electron propagator
relates to excitations of the system following the addition of one electron (electron attachment) or the removal of one electron (ionization). We shall refer exclusively to the electron propagator in much of the book, that is, when we develop the
formalism of diagrammatic perturbation theory in Chaps. 4–7 and establish practical
approximation methods in Chaps. 8–12. The polarization propagator and the physics
of N -electron excitations will be considered in the Chaps. 13–15 of Part IV.
3.1 Definition and Relation to Physical Quantities
In the following, we suppose a basis set of one-particle states | p and the associated
creation and destruction operators c
†
p , c p as introduced in Chap. 2. We consider an
N -electron system with the hamiltonian
ˆ
H = ˆ
T + ˆ
V =
t pq c
†
p c q +
1
2
V pqrs c
†
p c
†
q c s c r
(3.1)
and a non-degenerate (normalized) ground state | 0 of energy E 0 . Moreover, we
define time-dependent or Heisenberg operators according to
c
†
p [t] = e
i ˆ
Ht c
†
p e
−i ˆ
Ht
, c p [t] = e
i ˆ
Ht c p e
−i ˆ
Ht
(3.2)
© 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_3
31