Electronic excitations lie at the origin of most of the commonly measured spectra. However, the first-principles computation of excited states requires a larger effort than ground-state calculations, which can be very efficiently carried out within density-functional theory. On the other hand, two theoretical and computational tools have come to prominence for the description of electronic excitations. One of them, many-body perturbation theory, is based on a set of Green’s-function equations, starting with a one-electron propagator and considering the electron-hole Green’s function for the response. Key ingredients are the electron’s self-energy S and the electron-hole interaction. A good approximation for S is obtained with Hedin’s GW approach, using density-functional theory as a zero-order solution. First-principles GW calculations for real systems have been successfully carried out since the 1980s. Similarly, the electron-hole interaction is well described by the Bethe-Salpeter equation, via a functional derivative of S. An alternative approach to calculating electronic excitations is the time-dependent density-functional theory (TDDFT), which offers the important practical advantage of a dependence on density rather than on multivariable Green’s functions. This approach leads to a screening equation similar to the Bethe-Salpeter one, but with a two-point, rather than a four-point, interaction kernel. At present, the simple adiabatic local-density approximation has given promising results for finite systems, but has significant deficiencies in the description of absorption spectra in solids, leading to wrong excitation energies, the absence of bound excitonic states, and appreciable distortions of the spectral line shapes. The search for improved TDDFT potentials and kernels is hence a subject of increasing interest. It can be addressed within the framework of many-body perturbation theory: in fact, both the Green’s functions and the TDDFT approaches profit from mutual insight. This review compares the theoretical and practical aspects of the two approaches and their specific numerical implementations, and presents an overview of accomplishments and work in progress.

/pdf/electronic-excitations-density-functional-versus-many-body-2tgqd9dfpy.pdf

Electronic excitations: density-functional versus many-body Green's-function approaches

http://nano-bio.ehu.es/files/articles/Gonze_CPC_2009_590.pdf

ABINIT: First-principles approach to material and nanosystem properties

Calculations of ground-state and excited-state properties of materials have been one of the major goals of condensed matter physics. Ground-state properties of solids have been extensively investigated for several decades within the standard density functional theory. Excited-state properties, on the other hand, were relatively unexplored in ab initio calculations until a decade ago. The most suitable approach up to now for studying excited-state properties of extended systems is the Green function method. To calculate the Green function one requires the self-energy operator which is non-local and energy dependent. In this article we describe the GW approximation which has turned out to be a fruitful approximation to the self-energy. The Green function theory, numerical methods for carrying out the self-energy calculations, simplified schemes, and applications to various systems are described. Self-consistency issue and new developments beyond the GW approximation are also discussed as well as the success and shortcomings of the GW approximation.

https://iopscience.iop.org/article/10.1088/0034-4885/61/3/002/pdf

The GW method

The theoretical description of charge distribution, and related properties, such as chemical reactivity descriptors of chemical compounds, has greatly benefited from the development of density functional theory (DFT) methods. Indeed, most concepts stemmed from DFT but, up to now, they have been used mostly within semiempirical MO methods, Hartree–Fock, or post-Hartree–Fock methods. During the last decade, however, DFT has enabled theoretical chemistry to predict accurately structures and energetics of clusters and molecules. Therefore, more attention should also now be paid to these reactivity descriptors determined directly from DFT calculations. In this work, chemical reactivity is explored in DFT through a functional Taylor expansion of energy that introduces various energy derivatives of chemical significance. This review summarizes their main features and examines the limitations of some indexes presently used for the characterization of reactivity. Also, several perspectives are given. © 1999 John Wiley & Sons, Inc. J Comput Chem 20: 129–154, 1999

Chemical reactivity indexes in density functional theory

Algorithmic details and results of fully frequency-dependent ${G}_{0}{W}_{0}$ calculations are presented. The implementation relies on the spectral representation of the involved matrices and their Hilbert or Kramers-Kronig transforms to obtain the polarizability and self-energy matrices at each frequency. Using this approach, the computational time for the calculation of polarizability matrices and quasiparticle energies is twice as that for a single frequency, plus Hilbert transforms. In addition, the implementation relies on the PAW method, which allows to treat $d$-states with relatively modest effort and permits the reevaluation of the core-valence interaction on the level of the Hartree-Fock approximation. Tests performed on an $sp$ material (Si) and materials with $d$ electrons (GaAs and CdS) yield quasiparticle energies that are very close to previous all-electron pseudopotential and all-electron full-potential linear muffin-tin-orbital calculations.

Implementation and performance of the frequency-dependent GW method within the PAW framework

A generalized plasmon-pole model is presented to describe the energy dependence of the polarizability of semiconductors and insulators. This model leads in a natural way to a simple prescription for determining plasmon band structures. As an application, the plasmon band structure of silicon is calculated within the random-phase approximation. Our plasmon-pole model offers considerable computational advantages over earlier such models in the context of self-energy calculations within the Hedin GW framework. Furthermore, we comment on the purely classical origin of the f-sum rule for the polarizability.

Generalized plasmon-pole model and plasmon band structures of crystals.

Plasmon excitations in crystals.

Plasmon energies are closely related to the complex zeros of the determinant of the dielectric matrix or, equivalently, to the complex poles of the dynamically screened interaction matrix of electronic systems. In this paper we first clarify the true positions of these zeros, or poles, in the complex energy plane. Concluding that these points are on the non-physical Riemann sheets associated with the above-mentioned matrices, we present two methods which enable us to determine not only the plasmon energy bands but also their lifetimes throughout the whole Brillouin zone. We present, among other things, the plasmon energy bands of silicon in the diamond structure (within the norm-conserving pseudopotential framework) along a symmetry axis. Finally, we demonstrate how the electronic self-energy operator within the Hedin GW framework can be accurately and relatively easily calculated.

Plasmons and dynamic screening effects in crystals

We report on numerical calculations of the dynamical dielectric function in silicon, using a continued-fraction expansion of the polarizability and a recently proposed representation of the inverse dielectric function in terms of plasmonlike excitations A number of important technical refinements to further improve the computational efficiency of the method are introduced, making the ab initio calculation of the full energy dependence of the dielectric function comparable in cost to calculation of its static value Physical results include the observation of previously unresolved features in the random-phase approximated dielectric function and its inverse within the framework of density-functional theory in the local-density approximation, which may be accessible to experiment We discuss the dispersion of plasmon energies in silicon along the \ensuremath{\Lambda} and \ensuremath{\Delta} directions and find improved agreement with experiment compared to earlier calculations We also present quantitative evidence indicating the degree of violation of the Johnson f-sum rule for the dielectric function due to the nonlocality of the one-electron potential used in the underlying band-structure calculations

Calculation of the dielectric properties of semiconductors.

In this paper we demonstrate that the extremal properties of the Harris-Foulkes energy functional in combination with the exact form for the exchange and correlation energy functional are qualitatively the same as those using the local-density approximation for the latter functional. Foremost among these properties is that the Harris-Foulkes functional does not have a local maximum at the exact ground-state charge density. The extremal properties are shown to depend on whether the static local-field-correction function G(q;\ensuremath{\omega}=0) is greater or less than unity. Our results are based on an improved and very accurate expression for G(q;\ensuremath{\omega}=0), which we present in this paper. This expression satisfies the exact asymptotic results for the short- and long-wavelength limits as determined in terms of some exact frequency moments of the density-density correlation function. One of the major ingredients of this G(q;0) is the quantum-Monte-Carlo results of Ceperley and Alder for the correlation energy in the paramagnetic state of the uniform electron gas. Although the present G(q;0) has some similarities with the Utsumi-Ichimaru G(q), it differs from the latter in a fundamental way. For comparison, the consequences of using the latter G(q) in the context of the present work are indicated. Further, we discuss a number of issues of a general nature pertinent to the practical application of the Harris-Foulkes functional. As G(q;0) involves momentum moments of the momentum distribution function of the interacting electron gas, we also present a model for this function, along with some very accurate interpolation expressions for a number of the coefficients both in this and in G(q;0). Consequences of the use of various local-field-correction functions on the static screening properties of interacting electron gas in the paramagnetic phase are also discussed.

Extremal properties of the Harris-Foulkes functional and an improved screening calculation for the electron gas.

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