Chapter 13
Polarization Propagator
The electron propagator is the simplest many-body Green’s functions, and accordingly, the formalism of diagrammatic perturbation theory was specifically presented
for that paradigm in Chaps. 4–6. Also the ADC and ISR concepts in the design of
practical approximation schemes (Chaps. 10–12) were explicitly formulated for the
electron propagator and the underlying physics, being (N −1)- and (N +1)-electron
excitations. In this and the two following Chaps. 14 and 15 we consider the polarization propagator designed for the treatment of N -electron excitations. By and large,
the concepts developed for the electron propagator can be transferred to the case of
the polarization propagator, but there are also important differences that need to be
addressed. In this chapter we present the polarization propagator, discuss its physical
significance, and outline the pertaining diagrammatic perturbation expansion.
13.1 Definition and Physical Significance
Reversing the order adopted in Sect. 3.1 introducing the electron propagator, we here
begin with the spectral representation of the polarization propagator and from there
go back to the original definition.
The polarization propagator is constituted by a matrix (ω) of energy (or time)dependent functions rs,r s (ω) which can be written in an explicit spectral representation form as follows:
rs,r s (ω) =
m =0
0 |c
†
s c r | m m |c
†
r c s | 0
ω − E m + E 0 + iη
+
m =0
0 |c
†
r c s | m m |c
†
s c r | 0
−ω − E m + E 0 + iη
(13.1)
In addition to notations already used in the analogous representation (3.17) of
the electron propagator, E m and | m denote the energies and energy eigenstates,
© 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_13
195
Polarization Propagator
The electron propagator is the simplest many-body Green’s functions, and accordingly, the formalism of diagrammatic perturbation theory was specifically presented
for that paradigm in Chaps. 4–6. Also the ADC and ISR concepts in the design of
practical approximation schemes (Chaps. 10–12) were explicitly formulated for the
electron propagator and the underlying physics, being (N −1)- and (N +1)-electron
excitations. In this and the two following Chaps. 14 and 15 we consider the polarization propagator designed for the treatment of N -electron excitations. By and large,
the concepts developed for the electron propagator can be transferred to the case of
the polarization propagator, but there are also important differences that need to be
addressed. In this chapter we present the polarization propagator, discuss its physical
significance, and outline the pertaining diagrammatic perturbation expansion.
13.1 Definition and Physical Significance
Reversing the order adopted in Sect. 3.1 introducing the electron propagator, we here
begin with the spectral representation of the polarization propagator and from there
go back to the original definition.
The polarization propagator is constituted by a matrix (ω) of energy (or time)dependent functions rs,r s (ω) which can be written in an explicit spectral representation form as follows:
rs,r s (ω) =
m =0
0 |c
†
s c r | m m |c
†
r c s | 0
ω − E m + E 0 + iη
+
m =0
0 |c
†
r c s | m m |c
†
s c r | 0
−ω − E m + E 0 + iη
(13.1)
In addition to notations already used in the analogous representation (3.17) of
the electron propagator, E m and | m denote the energies and energy eigenstates,
© 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_13
195