Generalized random phase approximations

The approximation where we ignore 0 " and keep 0 can be solved exactly. This approximation corresponds to a generalized random-phase approximation or a linearized Vlasov approximation where the bare potential V(r) is replaced by — Co(r). Since 0 does not take into account any damping... 

Let us consider the Sawada model [58, 59] which corresponds to generalized random phase approximation and allows for clear definitions and interpretation. Space 11 represents the complete one-electron basis of spin-orbitals which is divided into two subspaces and it is assumed that only transitions between them are allowed ... 

Examples of this last type of functionals are the generalized random phase approximation (RPA) [67-72] and the interaction strength interpolation (ISI) [73]. 

In this contribution, we have shown that the Bethe sum rule, like the Thomas-Reiche-Kuhn sum rule, is satisfied exactly in the random phase approximation for a complete basis. Thus, in calculations that are related to the generalized oscillator strengths of a system, the Bethe sum rule may be used as an indicator of completeness of the basis set, much as the Thomas-Reiche-Kuhn... 

We presented fully self-consistent separable random-phase-approximation (SRPA) method for description of linear dynamics of different finite Fermi-systems. The method is very general, physically transparent, convenient for the analysis and treatment of the results. SRPA drastically simplifies the calculations. It allows to get a high numerical accuracy with a minimal computational effort. The method is especially effective for systems with a number of particles 10 — 10, where quantum-shell effects in the spectra and responses are significant. In such systems, the familiar macroscopic methods are too rough while the full-scale microscopic methods are too expensive. SRPA seems to be here the best compromise between quality of the results and the computational effort. As the most involved methods, SRPA describes the Landau damping, one of the most important characteristics of the collective motion. SRPA results can be obtained in terms of both separate RPA states and the strength function (linear response to external fields). 

This is a rigorous result generalizing the random phase approximation, as will be shown in Sect. 5.4. 

The relativistic or non-relativistic random-phase approximation (RRPA or RPA)t is a generalized self-consistent field procedure which may be derived making the Dirac/Hartree-Fock equations time-dependent. Therefore, the approach is often called time-dependent Dirac/Hartree-Fock. The name random phase comes from the original application of this method to very large systems where it was argued that terms due to interactions between many alternative pairs of excited particles, so-called two-particle-two-hole interactions ((2p-2h) see below) tend to... 

This general theory can be made more specific by introducing the explicit form of the wavefunction in such a way, by using an HF description, we obtain the random phase approximation (RPA) (or TDHF). Within this formalism, the free energy Hessian terms yield... 

In describing polarization propagator methods it is instructive to start out with the simplest consistent method of the kind, namely the random-phase approximation (RPA). Within the framework we use here, RPA is described as the approximation to the general equation of motion (Eq. (58)) in which we set h = hj and assume 0> = HF>, that is, use the simplest truncation in both Eqs (64) and (89). It is convenient to split hj up into p-h and h-p excitation operators... 

As in general all the y-coefficients do not vanish one has to assume a more general reference state than the single determinant SCF state. This is the rather well-known problem of finding the consistent reference state for the Random Phase Approximation (RPA). It also means that the field operator basis can be enlarged and can for instance include the iV-electron occupation number operators (in this discussion, electron field operators and their adjoints are used referring to a basis of spin orbitals that are the natural spin orbitals of the reference state, as will be discussed below, i.e., the spin orbitals that diagonalize the one-matrix)... 

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