Chapter 14
ADC and ISR Approaches to the
Polarization Propagator
The direct ADC procedure presented in Chap. 10 for the G
±
(ω) parts of the electron
propagator can essentially be transferred to the polarization propagator, where it
suffices to deal with
+
(ω) due to the redundancy of the two parts. This will be
demonstrated in the following. Moreover, in analogy to Chap. 11, the intermediatestate representation (ISR), here based on the correlated excited N -electron states,
will be established and discussed.
14.1 General Framework
The spectral representation for the
+
(ω) part of the polarization propagator, given
by the first term on the right-hand side of Eq. (13.1), can be written in matrix
notation as
+
(ω) = x
†
(ω − )
−1 x
(14.1)
where is the diagonal matrix of excitation energies, ω m = E m − E 0 , and x denotes
the matrix of transition amplitudes x m,rs specified by Eq. (13.5). The ADC formulation, by contrast, sets out from a non-diagonal representation
+
(ω) = f
†
(ω − M)
−1 f
(14.2)
deriving from a set of intermediate states | ˜
J to be further specified below. Here,
the secular matrix M is defined according to
M I J = = ˜
I | ˆ
H − E 0 | ˜
J
(14.3)
as the intermediate-state representation (ISR) of the (shifted) hamiltonian ˆ
H − E 0 .
The matrix f denotes a matrix of “effective” transition amplitudes
© 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_14
205
ADC and ISR Approaches to the
Polarization Propagator
The direct ADC procedure presented in Chap. 10 for the G
±
(ω) parts of the electron
propagator can essentially be transferred to the polarization propagator, where it
suffices to deal with
+
(ω) due to the redundancy of the two parts. This will be
demonstrated in the following. Moreover, in analogy to Chap. 11, the intermediatestate representation (ISR), here based on the correlated excited N -electron states,
will be established and discussed.
14.1 General Framework
The spectral representation for the
+
(ω) part of the polarization propagator, given
by the first term on the right-hand side of Eq. (13.1), can be written in matrix
notation as
+
(ω) = x
†
(ω − )
−1 x
(14.1)
where is the diagonal matrix of excitation energies, ω m = E m − E 0 , and x denotes
the matrix of transition amplitudes x m,rs specified by Eq. (13.5). The ADC formulation, by contrast, sets out from a non-diagonal representation
+
(ω) = f
†
(ω − M)
−1 f
(14.2)
deriving from a set of intermediate states | ˜
J to be further specified below. Here,
the secular matrix M is defined according to
M I J = = ˜
I | ˆ
H − E 0 | ˜
J
(14.3)
as the intermediate-state representation (ISR) of the (shifted) hamiltonian ˆ
H − E 0 .
The matrix f denotes a matrix of “effective” transition amplitudes
© 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_14
205