Chapter 16
Algebraic Propagator Methods
Algebraic propagator methods, as opposed to methods based on diagrammatic perturbation theory, have been derived within the so-called superoperator formulation
[1–3], in which the respective propagator is written and evaluated in the form of
a superoperator resolvent [4]. The resulting secular equations are fully equivalent
with those obtained in the context of the EOM method [5, 6], originally devised by
Rowe [7] in the field of nuclear physics. To some extent, the EOM formulation is
more general than the algebraic propagator approach since the former provides a
genuine wave-function representation of the (generalized) excited states in terms of
an extended set of CE states. In this chapter, we briefly review the EOM method,
emphasizing here in particular its ISR characteristics. For a more comprehensive
presentation as well as references to the original literature, we refer to the cited
review articles and to Ref. [8]. A direct connection to the algebraic propagator methods is established in Sect. 16.2, while the superoperator formulation is reviewed in
Appendix A.8.
16.1 Equations-of-Motion (EOM) Method for N ± 1
Electrons
Depending on the choice of the excitation operators, EOM schemes can be derived
for neutral (N -electron) excitations, as well as for generalized excitations of N ±
1, N ±2, . . . electrons. Exemplarily, we will consider the (N ± 1)-electron EOM
scheme in the following. The N -electron case, featuring some particularities, will be
addressed in Sect. 16.3.
In the EOM formulation, a general cationic state (not necessarily an energy eigenstate) |
N −1
n
is written in the form
© 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_16
241
Algebraic Propagator Methods
Algebraic propagator methods, as opposed to methods based on diagrammatic perturbation theory, have been derived within the so-called superoperator formulation
[1–3], in which the respective propagator is written and evaluated in the form of
a superoperator resolvent [4]. The resulting secular equations are fully equivalent
with those obtained in the context of the EOM method [5, 6], originally devised by
Rowe [7] in the field of nuclear physics. To some extent, the EOM formulation is
more general than the algebraic propagator approach since the former provides a
genuine wave-function representation of the (generalized) excited states in terms of
an extended set of CE states. In this chapter, we briefly review the EOM method,
emphasizing here in particular its ISR characteristics. For a more comprehensive
presentation as well as references to the original literature, we refer to the cited
review articles and to Ref. [8]. A direct connection to the algebraic propagator methods is established in Sect. 16.2, while the superoperator formulation is reviewed in
Appendix A.8.
16.1 Equations-of-Motion (EOM) Method for N ± 1
Electrons
Depending on the choice of the excitation operators, EOM schemes can be derived
for neutral (N -electron) excitations, as well as for generalized excitations of N ±
1, N ±2, . . . electrons. Exemplarily, we will consider the (N ± 1)-electron EOM
scheme in the following. The N -electron case, featuring some particularities, will be
addressed in Sect. 16.3.
In the EOM formulation, a general cationic state (not necessarily an energy eigenstate) |
N −1
n
is written in the form
© 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_16
241