Abstract:
In many-body perturbation theory (MBPT) the self-energy \Sigma=iGW\Gamma plays the key role since it contains all the many body effects of the system. The exact self-energy is not known; as first approximation one can set the vertex function \Gamma to unity which leads to the GW approximation. The latter properly describes the high-density regime, where screening is important; in the low-density regime, instead, other approximations are proposed, such as the T matrix, which describes multiple scattering between two particles. Here we combine the two approaches. Starting from the fundamental equations of MBPT we show how one can derive the T-matrix approximation to the self-energy in a common framework with GW. This allows us to elucidate several aspects of this formulation, including the origin of, and link between, the electron-hole and the particle-particle T matrix, the derivation of a screened T matrix, and the conversion of the T matrix into a vertex correction. The exactly solvable Hubbard molecule is used for illustration.

Abstract:
We present a new method for the computation of self-energy corrections in large supercells. It eliminates the explicit summation over unoccupied states, and uses an iterative scheme based on an expansion of the Green's function around a set of reference energies. This improves the scaling of the computational time from the fourth to the third power of the number of atoms for both the inverse dielectric matrix and the self-energy, yielding improved efficiency for 8 or more silicon atoms per unit cell.

Abstract:
We discuss the current status of a computational approach which allows to evaluate the dielectric matrix, and hence electronic excitations like optical properties, including local field and excitonic effects. We introduce a recent numerical development which greatly reduces the use of memory in such type of calculations, and hence eliminates one of the bottlenecks for the application to complex systems. We present recent applications of the method, focusing our interest on insulating oxides.

Abstract:
Several widely used methods for the calculation of band structures and photo emission spectra, such as the GW approximation, rely on Many-Body Perturbation Theory. They can be obtained by iterating a set of functional differential equations relating the one-particle Green's function to its functional derivative with respect to an external perturbing potential. In the present work we apply a linear response expansion in order to obtain insights in various approximations for Green's functions calculations. The expansion leads to an effective screening, while keeping the effects of the interaction to all orders. In order to study various aspects of the resulting equations we discretize them, and retain only one point in space, spin, and time for all variables. Within this one-point model we obtain an explicit solution for the Green's function, which allows us to explore the structure of the general family of solutions, and to determine the specific solution that corresponds to the physical one. Moreover we analyze the performances of established approaches like $GW$ over the whole range of interaction strength, and we explore alternative approximations. Finally we link certain approximations for the exact solution to the corresponding manipulations for the differential equation which produce them. This link is crucial in view of a generalization of our findings to the real (multidimensional functional) case where only the differential equation is known.

Abstract:
We present calculations of the absorption spectrum of semiconductors and insulators comparing various approaches: (i) the two-particle Bethe-Salpeter equation of Many-Body Perturbation Theory; (ii) time-dependent density-functional theory using a recently developed kernel that was derived from the Bethe-Salpeter equation; (iii) a scheme that we propose in the present work and that allows one to derive different parameter-free approximations to (ii). We show that all methods reproduce the series of bound excitons in the gap of solid argon, as well as continuum excitons in semiconductors. This is even true for the simplest static approximation, which allows us to reformulate the equations in a way such that the scaling of the calculations with number of atoms equals the one of the Random Phase Approximation.

Abstract:
Measurable spectra are theoretically very often derived from complicated many-body Green's functions. In this way, one calculates much more information than actually needed. Here we present an in principle exact approach to construct effective potentials and kernels for the direct calculation of electronic spectra. In particular, the potential that yields the spectral function needed to describe photoemission turns out to be dynamical but {\it local} and {\it real}. As example we illustrate this ``photoemission potential'' for sodium and aluminium, modelled as homogeneous electron gas, and discuss in particular its frequency dependence stemming from the nonlocality of the corresponding self-energy. We also show that our approach leads to a very short derivation of a kernel that is known to well describe absorption and energy-loss spectra of a wide range of materials.

Abstract:
An ab initio approach to the calculation of excitonic effects in the optical absorption spectra of semiconductors and insulators is formulated. It starts from a quasiparticle bandstructure calculation and is based on the relevant Bethe--Salpeter equation. An application to bulk silicon shows a substantial improvement with respect to previous calculations in the description of the experimental spectrum, for both peak positions and lineshape.

Abstract:
In the description of the interaction between electrons beyond the classical Hartree picture, bare exchange often yields a leading contribution. Here we discuss its effect on optical spectra of solids, comparing three different frameworks: time-dependent Hartree-Fock, a recently introduced combined density-functional and Green's functions approach applied to the bare exchange self-energy, and time-dependent exact-exchange within time-dependent density-functional theory (TD-EXX). We show that these three approximations give rise to identical excitonic effects in solids; these effects are drastically overestimated for semiconductors. They are partially compensated by the usual overestimation of the quasiparticle band gap within Hartree-Fock. The physics that lacks in these approaches can be formulated as screening. We show that the introduction of screening in TD-EXX indeed leads to a formulation that is equivalent to previously proposed functionals derived from Many-Body Perturbation Theory. It can be simulated by reducing the long-range part of the Coulomb interaction: this produces absorption spectra of semiconductors in good agreement with experiment.

Abstract:
In this work we explore the performance of approximations to electron correlation in reduced density-matrix functional theory (RDMFT) and of approximations to the observables calculated within this theory. Our analysis focuses on the calculation of total energies, occupation numbers, removal/addition energies, and spectral functions. We use the exactly solvable Hubbard molecule at 1/4 and 1/2 filling as test systems. This allows us to analyze the underlying physics and to elucidate the origin of the observed trends. For comparison we also report the results of the $GW$ approximation, where the self-energy functional is approximated, but no further hypothesis are made concerning the approximations of the observables. In particular we focus on the atomic limit, where the two sites of the molecule are pulled apart and electrons localize on either site with equal probability, unless a small perturbation is present: this is the regime of strong electron correlation. In this limit, using the Hubbard molecule at 1/2 filling with or without a spin-symmetry-broken ground state, allows us to explore how degeneracies and spin-symmetry breaking are treated in RDMFT. We find that, within the used approximations, neither in RDMFT nor in $GW$ the signature of strong correlation are present in the spin-singlet ground state, whereas both give the exact result for the spin-symmetry broken case. Moreover we show how the spectroscopic properties change from one spin structure to the other. Our findings can be generalized to other situations, which allows us to make connections to real materials and experiment.

Abstract:
Starting from the many-body Bethe-Salpeter equation we derive an exchange-correlation kernel $f_{xc}$ that reproduces excitonic effects in bulk materials within time-dependent density functional theory. The resulting $f_{xc}$ accounts for both self-energy corrections and the electron-hole interaction. It is {\em static}, {\em non-local} and has a long-range Coulomb tail. Taking the example of bulk silicon, we show that the $- \alpha / q^2$ divergency is crucial and can, in the case of continuum excitons, even be sufficient for reproducing the excitonic effects and yielding excellent agreement between the calculated and the experimental absorption spectrum.