The GW Approximation for the Electronic Self-Energy

Lucia Reining

Many observables such as the density, total energy, or electric current, can be expressed explicitly in terms of the one‐body Green's function, which describes electron addition or removal to or from a system. An efficient way to determine such a Green's function is to introduce a self‐energy, which is a nonlocal and dynamic effective potential that influences the propagation of particles in an interacting system. The state‐of‐the art approximation for the self‐energy is the GW approximation, where the system to (or from) which the electron is added (or removed) is described as a polarizable, screening, medium. This is expressed by the name of the approximation: ‘GW’ stands for the one‐body Green's function G and for W , the dynamically screened Coulomb interaction. The GW approximation is very popular for the calculation of band structures in solids, and increasingly used also to describe nanostructures, clusters, and molecules. As compared to static mean‐field approximations for the effective potential, the dynamical screening of the Coulomb interaction in GW leads to a renormalization of energies, to broadening and/or to the observation of additional excitations. An analysis of the approximations that lead to the GW self‐energy, and of the underlying picture, explains the successes and the limitations of the approach. This article is categorized under: Electronic Structure Theory > Density Functional Theory Electronic Structure Theory > Ab Initio Electronic Structure Methods Theoretical and Physical Chemistry > Spectroscopy Structure and Mechanism > Computational Materials Science

Antoine Marie, Abdallah Ammar, Pierre‐François Loos

Günther Engel, Behnam Farid, C.M.M. Nex +1 more

The concepts of quasiparticle and collective excitation are considered in the context of continued-fraction expansions. This leads to a practical scheme allowing a highly accurate description of the screened interaction function W and the electron self-energy \ensuremath{\Sigma}, which can be achieved at a reasonable computational cost. The usefulness of this scheme is demonstrated on a simple two-band model of a semiconductor, where it leads to an unambiguous calculation of the electron self-energy in the GW approximation. Conceptually, the theory clarifies some of the limitations of the standard quasiparticle concept in many-body theory. Furthermore, it allows a proper assessment of the quality of the standard approximations employed in GW calculations.

Martin M. Rieger, Lutz Steinbeck, I.D. White +2 more

S. Di Sabatino, Pierre‐François Loos, Pina Romaniello

Using the simple (symmetric) Hubbard dimer, we analyze some important features of the GW approximation. We show that the problem of the existence of multiple quasiparticle solutions in the (perturbative) one-shot GW method and its partially self-consistent version is solved by full self-consistency. We also analyze the neutral excitation spectrum using the Bethe-Salpeter equation (BSE) formalism within the standard GW approximation and find, in particular, that 1) some neutral excitation energies become complex when the electron-electron interaction U increases, which can be traced back to the approximate nature of the GW quasiparticle energies; 2) the BSE formalism yields accurate correlation energies over a wide range of U when the trace (or plasmon) formula is employed; 3) the trace formula is sensitive to the occurrence of complex excitation energies (especially singlet), while the expression obtained from the adiabatic-connection fluctuation-dissipation theorem (ACFDT) is more stable (yet less accurate); 4) the trace formula has the correct behavior for weak (i.e., small U) interaction, unlike the ACFDT expression.