Chapter 15
Random-Phase Approximation (RPA)
In this chapter, we will take a look at the famous random-phase approximation (RPA)
to the polarization propagator. The computational benefit afforded by the RPA is
rather modest, at least for atoms and molecules, as the resulting excitation energies
and transition moments are only consistent through first order of perturbation theory.
From a theoretical point of view, though, the RPA represents a highly interesting
concept, to be seen as an integral part of general knowledge in many-body physics.
Let us note some essential features:
– The RPA was originally devised in solid-state physics [1, 2] as a means to study
collective excitations (plasmons) of the interacting electron gas.
– The RPA can be obtained by summing a distinct class of diagrams (RPA diagrams)
through infinite order (see for example Thouless [3]), representing the paradigm
of an infinite partial summation of diagrams in propagator perturbation theory.
– There are other independent derivations of the RPA, such as via the time-dependent
Hartree–Fock (TDHF) approach (going back to Dirac [4]), or the equation-ofmotion (EOM) method (see Sect. 16.3).
– The RPA introduces interesting mathematics, featuring a specific pseudoeigenvalue secular problem.
– The RPA solutions fulfill the dipole sum rule and the equivalence of the length
and velocity forms of the transition moments.
As a point of interest in the present context, the ADC procedure can be specifically
applied to the series of RPA diagrams. The corresponding reformulation of the RPA
and the derivation of ADC(n) schemes, approximating the full RPA solution, are
discussed in Sect. 15.2.
© 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_15
223
Random-Phase Approximation (RPA)
In this chapter, we will take a look at the famous random-phase approximation (RPA)
to the polarization propagator. The computational benefit afforded by the RPA is
rather modest, at least for atoms and molecules, as the resulting excitation energies
and transition moments are only consistent through first order of perturbation theory.
From a theoretical point of view, though, the RPA represents a highly interesting
concept, to be seen as an integral part of general knowledge in many-body physics.
Let us note some essential features:
– The RPA was originally devised in solid-state physics [1, 2] as a means to study
collective excitations (plasmons) of the interacting electron gas.
– The RPA can be obtained by summing a distinct class of diagrams (RPA diagrams)
through infinite order (see for example Thouless [3]), representing the paradigm
of an infinite partial summation of diagrams in propagator perturbation theory.
– There are other independent derivations of the RPA, such as via the time-dependent
Hartree–Fock (TDHF) approach (going back to Dirac [4]), or the equation-ofmotion (EOM) method (see Sect. 16.3).
– The RPA introduces interesting mathematics, featuring a specific pseudoeigenvalue secular problem.
– The RPA solutions fulfill the dipole sum rule and the equivalence of the length
and velocity forms of the transition moments.
As a point of interest in the present context, the ADC procedure can be specifically
applied to the series of RPA diagrams. The corresponding reformulation of the RPA
and the derivation of ADC(n) schemes, approximating the full RPA solution, are
discussed in Sect. 15.2.
© 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_15
223