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WIREs Comput Mol Sci
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The GW approximation: content, successes and limitations

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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
Electron removal spectrum of pyridazine, comparison of theory and experimental photo emission done in the gas phase, from Ref . Calculated results include Hartree–Fock and two density functional theory functionals, as well as G0W0 using the DFT calculations as starting points. A broadening of 0.3 eV was used. At the bottom the density of the frontier orbitals is shown (Adapted with permission from Ref. . Copyrighted by the American Physical Society).
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Dyson equation for the polarizability χ in the random‐phase approximation (RPA).
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Exchange–correlation self‐energy in the GWA. The test‐charge–test‐charge screened interaction is given by a wiggly line.
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Feynman diagrams for the Hartree–Fock self‐energy. The dashed line represents the bare Coulomb interaction, and the arrow a one‐body Green's function . The arrow closed as a circle is the density Eq. (3) (the diagonal of the Green's function Eq. (8)). The first contribution is the Hartree potential, and the second contribution is the exchange self‐energy.
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Typical electron removal spectral function matrix element in an extended system. The vertical line indicates an independent‐particle result such as HF: it consists of one sharp peak. The interacting spectral function, given by the fat dots, exhibits a broad quasi‐particle peak and a satellite due to excitations of the many‐body system. Shown are also the imaginary (continuous line) and shifted real (dotted line) parts of the self‐energy. The quasi‐particle peak appears where the shifted real part crosses zero. Relative shifts would be different if another independent‐particle system was chosen.
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Electron addition and removal spectral functions of the Hubbard dimer. There is one spin‐up electron in the ground state. The exact result is given by the continuous line, the GWA result by the dotted line. Left: The spin‐up spectral function for U/t = 5. The GWA suffers from self‐screening in the electron removal and creates spurious satellites. Right: The spin‐down spectral function for the atomic limit t → 0. The GWA cannot reproduce the gap in this limit.
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Band structure of bulk silicon. Left panel: Courtesy of Mark van Schilfgaarde. Blue dot‐dashed lines are local density approximation results, red dotted lines are results of a GW calculation using the quasi‐particle self‐consistent GW approximation (see The GW Approximation in Practice). Circles are a collection of experimental data. Right panel: Hartree–Fock band structure (Reproduced with permission from Ref Copyright 1985 IOP Publishing).
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Pictorial representation of the Hartree–Fock (left panel) and GW (right panel) approximations. Hartree–Fock eigenvalues reflects electron addition or removal with respect to a rigid system, in the GW approximation the system responds dynamically. (Credits to Andrea Cucca, 2017).
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Photo emission spectrum of bulk silicon, from Ref . Crosses are experimental results, the dashed line is from a G0W0 calculation. The thick continuous line is the intrinsic spectral function from a cumulant expansion calculation. The dot‐dashed line includes moreover extrinsic losses of the outgoing photoelectron and interference effects. Cross sections and a secondary electrons background have been added to all calculated spectral functions (Reproduced with permission from Ref . Copyright 2011 American Physical Society).
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Effect of the update of quasi‐particle wavefunctions in a GWA calculation for hydrogen fluoride: shown is the difference between the densities calculated in density functional theory (DFT)‐PBE and quasi‐particle self‐consistent GW approximation, respectively. The result is normalized with respect to the DFT density. The fluoride atom is indicated by a green cross and the hydrogen atom by a gray cross (Reproduced with permission Ref . Copyright 2016 American Chemical Society).
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Structure and Mechanism > Computational Materials Science
Theoretical and Physical Chemistry > Spectroscopy
Electronic Structure Theory > Ab Initio Electronic Structure Methods
Electronic Structure Theory > Density Functional Theory

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