We introduce a scheme through which the ground-state electronic density n and total energy Ev [n] of inhomogeneous interacting systems can be determined in a self-consistent (SC) way. It is based on a SC perturbation expansion for the single-particle Green function G. Our employed expansion is argued to be valid under all conditions provided that the ground-state charge density of the interacting system is pure-state non-interacting v representable. The formalism discussed in this paper has been applied to a quasi-one-dimensional periodic system. It has even been possible to carry out SC calculations with the dynamically screened exchange diagram (the so-called GW diagram, where W is the dynamically screened electron-electron interaction function) for the self-energy operator Σ. A coherent mode of interpretation of a number of our results entails that the overwhelming success of the GW framework in relatively accurately reproducing the experimental energy gaps of a number of semiconductors and insulators may be a direct consequence of non-self-consistency effects; whereas a non-self-consistent calculation in our tests increases the value of the local-density approximation (LDA) gap in the range of 40–65% of the ‘experimental’ gaps, self-consistency effects reduce this range to 14–17%. The same line of argument leads us to conclude that, in our model, 26–35% of the LDA error in the gap energies is due to the absence of a well known discontinuity in the LDA exchange-correlation potential, and the remaining part is due to the Kohn-Sham eigenvalues not being identifiable with one-particle excitation energies. A natural conclusion that may be drawn from these observations is that a correct description of excitation energies (e.g. of the gap energy) cannot be achieved without including, at least approximately, the vertex part of Σ, which is entirely absent in the GW scheme.
We introduce a scheme through which the ground-state electronic density n and total energy Ev [n] of inhomogeneous interacting systems can be determined in a self-consistent (SC) way. It is based on a SC perturbation expansion for the single-particle Green function G. Our employed expansion is argued to be valid under all conditions provided that the ground-state charge density of the interacting system is pure-state non-interacting v representable. The formalism discussed in this paper has been applied to a quasi-one-dimensional periodic system. It has even been possible to carry out SC calculations with the dynamically screened exchange diagram (the so-called GW diagram, where W is the dynamically screened electron-electron interaction function) for the self-energy operator Σ. A coherent mode of interpretation of a number of our results entails that the overwhelming success of the GW framework in relatively accurately reproducing the experimental energy gaps of a number of semiconductors and insulators may be a direct consequence of non-self-consistency effects; whereas a non-self-consistent calculation in our tests increases the value of the local-density approximation (LDA) gap in the range of 40–65% of the ‘experimental’ gaps, self-consistency effects reduce this range to 14–17%. The same line of argument leads us to conclude that, in our model, 26–35% of the LDA error in the gap energies is due to the absence of a well known discontinuity in the LDA exchange-correlation potential, and the remaining part is due to the Kohn-Sham eigenvalues not being identifiable with one-particle excitation energies. A natural conclusion that may be drawn from these observations is that a correct description of excitation energies (e.g. of the gap energy) cannot be achieved without including, at least approximately, the vertex part of Σ, which is entirely absent in the GW scheme.
We introduce a scheme through which the ground-state electronic density n and total energy Ey[n] of inhomogeneous interacting systems can be determined in a self-consistent (SC) way. It is based on a SC perturbation expansion for the single-particle Green function G. Our employed expansion is argued to be valid under all conditions provided that the ground-state charge density of the interacting system is pure-state non-interacting v representable. The formalism discussed in this paper has been applied to a quasi-one-dimensional periodic system. It has even been possible to carry out SC calculations with the dynamically screened exchange diagram (the so-called G W diagram, where W is the dynamically screened electron4ectron interaction function) for the selfenergy operator C. A coherent mode of interpretation of a number of our results entails that the overwhelming success of the G W framework in relatively accurately reproducing the experimental energy gaps of a number of semiconductors and insulators may be a direct consequence of non-selfconsistency effects; whereas a non-self-consistent calculation in our tests increases the value of the local-density approximation (LDA) gap in the range of 40-65% of the ‘experimental’ gaps, self-consistency effects reduce this range to 1 4 1 7 % . The same line of argument leads us to conclude that, in our model, 2635% of the LDA error in the gap energies is due to the absence of a well known discontinuity in the LDA exchange-correlation potential, and the remaining part is due to the Kohn-Sham eigenvalues not being identifiable with one-particle excitation energies. A natural conclusion that may be drawn from these observations is that a correct description of excitation energies (e.g. of the gap energy) cannot be achieved without including, at least approximately, the vertex part of C, which is entirely absent in the GW scheme. 5 1 . INTRODUCTION The ground-state (GS) density-functional theory (DFT) due to Hohenberg and Kohn (1964) provides a rigorous foundation for determining properties of systems in their ground states (for reviews see Lundqvist and March (1983), Jones and Gunnarsson (1989) and Dreizler and Gross (1990)). The central position in this theory is taken by the total-energy functional EY[n] which, owing to its unique absolute minimum a t the GS electronic density n, can be used to obtain not only the GS energy but also the GS charge density. tPresent address: Max-Planck-Institut fur Festkorperforschung, HeisenbergstraRe 1, 70569 Stuttgart, Federal Republic of Germany. 0141-8637/97 $12 .00
The most of the experimental measurable quantities in many-body electron system, such as the optical absorption, luminescent spectra and exciton effect, are related to the excitation states description. The density functional theory with local density approximation (DFT-LDA) as a ground state theory, i.e. the solutions based on the Kohn-Sham equation is a powerful tool for studying the ground state properties in many-particle systems. However, the first principles computation of excited states is more complexity than ground-state calculations. A key problem is that the exchange-correlation interaction in excited states is differ from the ground-states. In recent years, nevertheless, several electronic-excitation theories have been developed. The most important theoretical and computational method include the many-body perturbation theory which is based on the quasi-particle concept and the Green function equation, the time-dependent DFT and the Bethe-Salpeter equation for describing the electron-hole interaction. Among them the central ingredient is an electron's self-energy operator Σ that describes the exchange correlation effect beyond Hartree approximation. The implementations of the above theories are unavoidable to introduce some of approximations. A good approximation for Σ is the GW approach by Hedin. It is shown from computation simulations for many real condensed matter systems that the GW approach is a successful method for electronic-excitation problems. Another effective method for excited-states is so-called screen-exchange local-density approximation (sX-LDA). It is a combination of LDA and Hartree-Fock (HF)theory, in which the intrinsic local screen-exchange interaction has been replaced by a non-local one and a generalized Kohn-Sham equation (GKS) is given. Based on a PWscf package, the sX-LDA-PWscf method has been tested and indicates that an available approach for treating band-gap problem in semiconductors. In this paper, the many-body theories for the excitation process, its development and signification are reviewed. The relation and differences consist in different theoretical methods are discussed. On the based of above discussions, the applications of many-body theory to the band-band transition (band-gap underestimation problem) and exciton effect in semiconductors are presented.
Fully self-consistent GW (sc-GW) – based on the iterative solution of Dyson’s equation – incorporates many-body correlation effects in the electronic ground-state, providing a recipe to consistently describe ground- and excited-states on an equal footing. We have implemented sc-GW in the localized basis all-electron code FHIaims [1]. We show that, for finite systems, the ground-state properties as well as the excitation spectrum of the sc-GW Green’s function are unique and are not affected by different initialization of the sc-GW loop, curing the pathological dependence on the starting point of one-shot GW (G0W0). In addition, self-consistency systematically improves ionization energies and Galitskii-Migdal total energies of closed shell systems compared to G0W0 based on Hartree-Fock and density-functional theory in local and semilocal approximations. Since our implementation is based on interacting Green’s functions (unlike all G0W0 implementations) it is also a first step towards going beyond the GW approach.
Density functional theory (DFT) performs reasonably well for the determination of structural properties of many materials, but fails to predict electronic band gap values accurately. \nSuch a failure of DFT is not unexpected, since there exists no formal justification for interpreting the DFT eigenvalues as addition or removal energies of the many-body system (quasiparticle energies).\nAn alternative approach to the study of exchange-correlation effects in many-particle systems is provided by many-body perturbation theory (MBPT), which defines a rigorous approach to description of excited-state properties based on the Green's function formalism and the concept of quasiparticle electrons. Within MBPT, one can calculate the quasiparticle (QP) energies and the QP amplitudes by approximately solving, within the so-called GW approximation, the set of coupled integro-differential equations proposed by Hedin in 1965.\nThe GW method generally yields significantly better values for QP energies with respect to DFT as it accounts for the dynamic many-body effects in the electron-electron interaction going beyond the mean-field approximation of independent Kohn-Sham particles. The drawback of such an involved approach, however, is its large computational cost, which is mainly due to the evaluation of dielectric matrices, their inversion, and the solution of a non-Hermitian and nonlinear eigenvalue problem. \nThe first part of this PhD research is devoted to the implementation of an efficient and scalable coarse-grained parallel algorithms and the development of state-of-the-art methods to solve the GW equations.\nAll these techniques are then applied to the study of the quasiparticle band structures of SiO2 in the alpha quartz crystalline structure. The effects of the different approximations involved in the theory, the influence of self-consistency, and a systematic analysis of the reliability of the different plasmon-pole models are presented and discussed. \nThe second part of this PhD work is dedicated to the implementation of a methodological approach to the solution of the GW equations based on the so-called projector augmented wave method (PAW) proposed by Blochl in 1994. This new approach permits to remove many of the limitations intrinsic to the use of the pseudopotential technique widely used for ab-initio calculations, allowing one to reach GW results closer to an all-electron implementation while still maintaining computational efficiency, and flexibility. This allows us to achieve a coherent implementation of many different algorithms which enables a detailed comparison of GW calculations not yet performed until now. Results for the quasiparticle band structure and optical spectra of prototype s- and p-compounds are discussed and compared to recent studies reported in the scientific literature (when available).
In this lecture we present many-body perturbation theory as a method to determine quasiparticle excitations in solids, especially electronic band structures, accurately from first principles.The main ingredient is the electronic self-energy that, in principle, contains all many-body exchange and correlation effects beyond the Hartree potential.As its exact mathematical expression is unknown, approximations must be used in practical calculations.The approximation is obtained using a systematic algebraic approach on the basis of Green function techniques.It constitutes an expansion of the self-energy up to linear order in the screened Coulomb potential, which describes the interaction between the quasiparticles and includes dynamic screening through the creation of exchange-correlation holes around the bare particles.The implementation of the approximation relies on a perturbative treatment starting from density functional theory.Besides a detailed mathematical discussion we focus on the underlying physical concepts and show some illustrative applications.
We present self-consistent GW calculations for a quasi-one-dimensional model semiconductor, using multipole representations for W, the screened interaction, and G, the electron Green function. In the case of G, we distinguish between the quasiparticle contribution and the so-called incoherent contribution. We consider various strengths of the external potential. For two different starting points of the self-consistency cycle, local-density approximation and Hartree-Fock, the band gaps converge to the same values. The self-consistent GW band gaps differ very little from the Hartree-Fock band gaps. There is considerable disagreement with quantum Monte Carlo calculations for the same model, indicating the importance of vertex corrections.