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arXiv:0704.1542 · Physical Review A

Quasiparticles in neon using the Faddeev random-phase approximation

C. Barbieri, Dimitri Van Neck, Willem H Dickhoff

Published Nov 6, 2007• 47 citations
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Abstract

The spectral function of the closed-shell neon atom is computed by expanding the electron self-energy through a set of Faddeev equations. This method describes the coupling of single-particle degrees of freedom with correlated two-electron, two-hole, and electron-hole pairs. The excitation spectra are obtained using the random-phase approximation (RPA), rather than the Tamm-Dancoff framework employed in the third-order algebraic diagrammatic construction method. The difference between these two approaches is studied, as well as the interplay between ladder and ring diagrams in the self-energy. Satisfactory results are obtained for the ionization energies as well as the energy of the ground state with the Faddeev RPA scheme, which is also appropriate for the high-density electron gas.

DOI
10.1103/physreva.76.052503

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The spectral function of the closed-shell neon atom is computed by expanding the electron self-energy through a set of Faddeev equations. This method describes the coupling of single-particle degrees of freedom with correlated two-electron, two-hole, and electron-hole pairs. The excitation spectra are obtained using the random-phase approximation (RPA), rather than the Tamm-Dancoff framework employed in the third-order algebraic diagrammatic construction method. The difference between these two approaches is studied, as well as the interplay between ladder and ring diagrams in the self-energy. Satisfactory results are obtained for the ionization energies as well as the energy of the ground state with the Faddeev RPA scheme, which is also appropriate for the high-density electron gas.
The accuracy of the Faddeev random phase approximation (FRPA) method is tested by evaluating total and ionization energies in the basis-set limit. A set of light atoms up to Ar is considered. Comparisons are made with the results of coupled-cluster singles and doubles (CCSD), with third-order algebraic diagrammatic construction [ADC(3)], and with the experiment. It is seen that even for two-electron systems, He and Be2+{}^{2+}2+, the inclusion of RPA effects leads to satisfactory results, and therefore it does not overcorrelate the ground state. The FRPA becomes progressively better for larger atomic numbers, where it gives $\ensuremath{\approx}$5 mH more correlation energy, and it shifts ionization potentials by 2--10 mH with respect to the similar ADC(3) method. The ionization potentials from FRPA tend to reduce the discrepancies with the experiment.
The coupled-cluster or exp(S)\mathrm{exp}(S)exp(S) formalism is applied to the problem of correlations in a many-electron system. In this first work in a series of papers we concentrate on exact results and restrict ourselves to various ring approximations applicable essentially to the high-density regime. It is shown that the well-known random-phase approximation (RPA) can be formulated as a nonlinear three-dimensional integral equation for the four-point function S2{S}_{2}S2​ which provides a measure of the two-particle---hole pair component in the true ground-state wave function. Exact analytic solutions are presented for both S2{S}_{2}S2​ and the three-point particle-hole vertex function in the RPA, and their properties are discussed. The Tamm-Dancoff approximation to the ring summation is also formulated, and the analagous exact solutions are presented for the first time. It is demonstrated that both solutions are unique, and a comparison of the two approximations is presented in terms of the correlation energy.
We have evaluated the density-density response of the two-dimensional electron gas at zero temperature by solving the Dyson equation for the particle-hole Green's function, including exchange Coulomb matrix elements and short-range contributions in the ladder approximation. We study the effect of these correlations on the total energy, compressibility per particle, local field factor G ( q ), static structure factor and pair-correlation function. Results are compared with the normal random-phase approximation, local field theories and quantum Monte Carlo calculations.
The ground-state correlation energy calculated in the random-phase approximation (RPA) is known to be identical to that calculated using a subset of terms appearing in coupled-cluster theory with double excitations (CCD). In particular, for particle-hole (ph) RPA this equivalence requires keeping only those terms that generate time-independent ring diagrams, and for particle-particle (pp) RPA it requires keeping only those terms that generate ladder diagrams. Here I show that these identities extend to excitation energies, for which those calculated in each RPA are identical to those calculated using approximations to equation-of-motion coupled-cluster theory with double excitations (EOM-CCD). The equivalence requires three approximations to EOM-CCD: first, the ground-state CCD amplitudes are obtained from the ring-CCD or ladder-CCD equations (the same as for the correlation energy); second, the EOM eigenvalue problem is truncated to the minimal subspace, which is one particle + one hole for ph-RPA and two particles or two holes for pp-RPA; third, the similarity transformation of the Fock operator must be neglected, as it corresponds to a Brueckner-like dressing of the single-particle propagator, which is not present in the conventional RPA.
The well-known and well-used Density Functional Theory (DFT) is a variational theory for the ground-state. There are no guarantees that it delivers good results for excited state properties such as quasiparticle energies or wave function. All that is guaranteed is that the total energy is exact, the chemical potential is correct, and that the eective electronic states (Kohn-Sham states) generate the right electron density at the variational minimum. Luttinger-Ward approaches to electronic structure are attractive as beyond-DFT schemes since, in principle, they oer a variational functional of the one-electron Green’s function that yields both the correct total energy and all one-particle properties at the variational extremum. However, compared to DFT, these methods face a number of additional challenges beyond the familiar problem of choosing an exchange-correlation functional: (a) they have a much greater computational cost compared to DFT, (b) one must choose some set of trial Green’s functions, and (c) one must locate an extremum (not necessarily a minimum). This talk aims both to be pedagogical and to provide a summary of our most recent work in this area. We focus mainly on the random-phase approximation (RPA) for the correlation functional which yields the GW self-energy for the Green’s function. We will describe ecient approaches for computing the RPA correlation energy; how various types of approximations to the correlation energy create a ladder of approximate self-energy operators and what type of physics is included in these approximations (exchange, screening, Hubbard U, etc.); and how the extremization is in fact highly problematic. We will end with some ideas for making practical progress with the extremization and their relation to the successful Quasiparticle Self-consistent GW (QSGW) scheme [1].
We present a new extension of the random-phase approximation method: the quasiboson approximation is avoided and correlations are included in the ground state without resorting to renormalized operators or renormalized matrix elements; the configuration space is enlarged by considering also elementary excitations corresponding to the annihilation of a particle (hole) and the creation of another particle (hole) on the correlated ground state, together with the particle-hole ones. Two new and relevant advantages of this method with respect to the existing extensions of random-phase approximation are highlighted: (i) the energy weighted sum rules are completely satisfied; (ii) the problem of the existence of nonphysical states in the spectrum, related to the inclusion of particle-particle and hole-hole configurations, is solved: a way to unambiguously disentangle physical from nonphysical states in the excitation spectrum is presented. The method is applied here to a three-level Lipkin model where its quality can be judged by comparing with the exact results. Both advantages (i) and (ii) shall lead to feasible future applications of this extended RPA to several realistic cases.