Random-phase approximation matrix

The absorption spectrum is proportional to the imaginary part of the macroscopic dielectric function. Adopting the same level of approximation that we have introduced to obtain GW quasiparticle energies, i.e. neglecting the vertex correction by putting T = 55, we get the so called random phase approximation (RPA) for the dielectric matrix. Within this approximation, neglecting local field effects, the response to a longitudinal field, for q 0, is ... 

A comparison between experimental and theoretical values for the J (2p) parameter in neon is shown in Fig. 2.14. (The corresponding comparison between experimental and theoretical values for the partial cross section experimental data are given by the solid curve surrounded by a hatched area which takes into account the error bars. Theoretical results from advanced photoionization theories (many-body perturbation theory, R-matrix theory, and random-phase approximation) are represented by the other lines, and they are in close agreement with the experimental data (for details see [Sch86]). The theoretical / (2p) data of Fig. 2.13 are also close to the experimental values, except in the threshold region. 

The unknown density matrix coefficient vectors X and Y for each of the perturbing field-components can be determined from an equation system that resembles the random-phase approximation (RPA) equations [38] ... 

These equations result from the intimate mixing of electron-electron and electron-hole channels (the Parquet summation). This is of crucial importance in one dimension. The f-matrix or random-phase approximations are incapable of doing this and are fundamentally wrong in one dimension. Notice also that g4 is absent because it does not alone contribute any logarithmic term. It leads only to charge and spin velocity corrections. It is normally neglected in the RG treatments (see Refs. 15 and 39 for a discussion of this). It will only be taken into account for the uniform susceptibility in part d. 

We study the dielectric and energy loss properties of diamond via first-principles calculation of the (0,0)-element ( head element) of the frequency and wave-vector-dependent dielectric matrix eg.g CQ, The calculation uses all-electron Kohn-Sham states in the integral of the irreducihle polarizahility in the random phase approximation. We approximate the head element of the inverse matrix hy the inverse of the calculated head element, and integrate over frequencies and momenta to obtain the electronic energy loss of protons at low velocities. Numerical evaluation for diamond targets predicts that the band gap causes a strong nonlinear reduction of the electronic stopping power at ion velocities below 0.2 a.u. 

In the random phase approximation, the matrix elements Bn, which were neglected in the TDA, are retained. 

The FOSEP approximation has to be compared with two other well-known first order approximation schemes, the Tamm-Dancoff approximation (TDA) [11,30] and the random phase approximation (RPA) [31,32,11,30]. The TDA leads to a hermitian eigenvalue problem of half the dimension of FOSEP. In fact the upper left (ph-ph) block of the FOSEP matrix coincides with... 

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)... 

The photoabsorption spectrum a(co) of a cluster measures the cross-section for electronic excitations induced by an external electromagnetic field oscillating at frequency co. Experimental measurements of a(co) of free clusters in a beam have been reported, most notably for size-selected alkali-metal clusters [4]. Data for size-selected silver aggregates are also available, both for free clusters and for clusters in a frozen argon matrix [94]. The experimental results for the very small species (dimers and trimers) display the variety of excitations that are characteristic of molecular spectra. Beyond these sizes, the spectra are dominated by collective modes, precursors of plasma excitations in the metal. This distinction provides a clear indication of which theoretical method is best suited to analyze the experimental data for the very small systems, standard chemical approaches are required (Cl, coupled clusters), whereas for larger aggregates the many-body perturbation methods developed by the solid-state community provide a computationally more appealing alternative. We briefly sketch two of these approaches, which can be adapted to a DFT framework (1) the random phase approximation (RPA) of Bohm and Pines [95] and the closely related time-dependent density functional theory (TD-DFT) [96], and (2) the GW method of Hedin and Lundqvist [97]. 

This approximation is better known as the time-dependent Hartree—Fock approximation (TDHF) (McLachlan and Ball, 1964) (see Section 11.1) or random phase approximation (RPA) (Rowe, 1968) and can also be derived as the linear response of an SCF wavefunction, as described in Section 11.2. Furthermore, the structure of the equations is the same as in time-dependent density functional theory (TD-DFT), although they differ in the expressions for the elements of the Hessian matrix E22. The polarization propagator in the RPA is then given as... 

During this same period theoretical techniques which can account for electron correlations have been developed and refined. Many calculations have been carried out using many-body perturbation theory (MBPT), R-matrix theory,6 the random phase approximation with exchange (RPAE), and other related techniques. This article will focus on nonrelativistic calculations since relativistic calculations such as the RRPA will be covered in the article by W, Johnson in this volume. 

Motivated by a puzzling shape of the coexistence line, Kierlik et al. [27] have investigated the model with Lennard-Jones attractive forces between fluid particles as well as matrix particles and have shown that the mean spherical approximation (MSA) for the ROZ equations provides a qualitatively similar behavior to the MFA for adsorption isotherms. It has been shown, however, that the optimized random phase (ORPA) approximation (the MSA represents a particular case of this theory), if supplemented by the contribution of the second and third virial coefficients, yields a peculiar coexistence curve. It exhibits much more similarity to trends observed in... 

Using the long time-weak coupling approximation and the hypothesis of random phases for the thermostat, Bloch and Wangsness find, after taking the trace over the heat bath, the following equations for the reduced density matrix a ... 

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