The GW compendium: A practical guide to theoretical photoemission spectroscopy

Abstract

The GW approximation in electronic structure theory has become a widespread tool for predicting electronic excitations in chemical compounds and materials. In the realm of theoretical spectroscopy, the GW method provides access to charged excitations as measured in direct or inverse photoemission spectroscopy. The number of GW calculations in the past two decades has exploded with increased computing power and modern codes. The success of GW can be attributed to many factors: favorable scaling with respect to system size, a formal interpretation for charged excitation energies, the importance of dynamical screening in real systems, and its practical combination with other theories. In this review, we provide an overview of these formal and practical considerations. We expand, in detail, on the choices presented to the scientist performing GW calculations for the first time. We also give an introduction to the many-body theory behind GW , a review of modern applications like molecules and surfaces, and a perspective on methods which go beyond conventional GW calculations. This review addresses chemists, physicists and material scientists with an interest in theoretical spectroscopy. It is intended for newcomers to GW calculations but can also serve as an alternative perspective for experts and an up-to-date source of computational techniques. 26 4. IP-theorem schemes 27 H. Computational complexity and cost 28 I. Practical guidelines 29 V. Beyond G 0 W 0 : self-consistency schemes 30 A. Fully self-consistent GW 30 B. Eigenvalue self-consistency and level alignment 32 C. Self-consistency via a new ground state 33 VI. Solids 34 A. Band gaps 35 B. Band structures and band parameters 36 C. Lifetimes 37 D. More challenging solids 38 E. Defects in solids 39 F. Outlook on solids 40 VII. Surfaces 40 VIII. Two-dimensional materials 42 IX. Molecules 44 A. First ionization potentials and electron affinities 44 B. Ionization spectra 45 C. The GW 100 benchmark set 46 D. Molecular crystals 47 X. Total energy and the electronic ground state 48 A. Electron density 48 B. Total energy 49 XI. Current challenges and beyond GW 50 A. Challenges 50 B. Quantum chemistry 50 C. Non-equilibrium Green's functions 51 D. Vertex corrections 51 E. Optical properties 53 F. T -matrix 53 G. Cumulant expansion 54 H. Local vertex 55 XII. Conclusion 55 XIII. Acknowledgements 55 A. Integral notation 56

Far-from-Equilibrium Quasiparticle Dynamics in Graphene

“…Many-body interactions are described in the the spectral function A(k, ω) in Refs. 9,10,38 . In ARPES, we can write the photoemitted intensity for two dimensional single band systems as 9 I(k, ω) = I 0 (k, ν, A)A(k, ω)f (ω).…”

“…An established approach to model many-body interactions is the use of the Green's function formalism 9,38 . The time-ordered one-electron Green's-function G(t − t ′ ) describes the probability that an electron (hole) added (removed) in a many-body system in a Bloch state is still there after a time t − t ′ .…”

“…for a linear bare band 40 with the Fermi velocity v F . The self-energy satisfies the Kramers-Kronig relation, and thus real and imaginary parts can be transformed into each other by the Hilbert transformation 9,38 . These relations will be necessary for the self-consistent extraction of the self-energy (see appendix A.3).…”

“…The spectral function depends on the self-energy and the bare bandstructure 9,38 (see chapter 2.1.2). We can extract the FWHM and dispersion (i.e., peak positions) with the usual MDC Lorentz fit evaluation.…”

“…The bare band structure could also be calculated from the single-particle band structure calculations such as density-functional-theory 38 . However, this introduces uncertainties to the self-energies extracted with such assumptions.…”